# Cohomology of toric diagrams

Grigory Solomadin  IRMA, University of Strasbourg, France
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this note, integral cohomology of homotopy colimits for toric diagrams and
their classifying spaces over regular CW posets are described in terms of
sheaf cohomology. Split $T$-CW-complexes with CW orbit poset $C$ have such
decomposition (up to a homeomorphism) in terms of a $T$-diagram $D$ over $C$.
Equivariant formality for $\operatorname{hocolim}D$ is equivalent to
$H^{odd}(\operatorname{hocolim}D)=0$ (over $\mathbb{Q}$, or over $\mathbb{Z}$
for connected stabilizers) provided that $C$ is an oriented homology manifold.
The integral singular cohomology groups and bigraded Betti numbers are
computed in this setting. Similar descriptions are provided for skeleta of
toric manifolds and compact nonsingular toric varieties. The cohomological
orbit spectral sequence collapse over $\mathbb{Z}$ at the second page is
proved for any compact toric variety.

###### Key words and phrases:

Homotopy colimit, toric diagram, toric variety, simplicial set

###### 2020 Mathematics Subject Classification:

57S12, 55U10, 55N91

##  1\. Introduction

Let $X^{2n}$ be a compact oriented manifold with an effective action of the
compact torus $T=(S^{1})^{k}$, with a non-empty and finite $T$-fixed point
set. Properties of the orbit space $Q=X/T$ (a Hausdorff space) and of the
Borel equivariant cohomology $H^{*}_{T}(X)$ are closely related to the
properties of the cohomology ring $H^{*}(X;\ \mathbb{Z})$ under certain
assumptions. For instance, consider the case of a torus manifold introduced by
Hattori and Masuda in [ha-ma-03], i.e. when the complexity $n-k$ of the action
is zero. The condition on the isotropy subgroup of $x\in X$ to have a fixed
dimension cuts a subset in $X$. By definition, the connected components of
this subset project onto open faces by the map $X\to Q$. The closure of an
open face in $Q$ is called a face of the $T$-space $X$. The orbit poset of $X$
is given by the partial order opposite to inclusion on the set of faces of
$X$. The $T$-action on $X$ is called locally standard if there exists an atlas
of $T$-invariant charts covering $X$ such that each chart of the atlas is
$T$-equivariantly homeomorphic to the standard rotational $T^{n}$-action on
$\mathbb{C}^{n}$ at the origin. The vanishing $H^{odd}(X;\ \mathbb{Z})=0$ for
a torus manifold is equivalent to acyclicity of proper (i.e $\neq Q$) faces of
$X$, together with local standardness condition by [ma-pa-06]. The orbit space
of an equivariantly formal torus manifold is acyclic by [ma-pa-06]. Local
standardness implies that $Q$ is a manifold with corners and therefore a
homology manifold. The class of torus manifolds with acyclic proper faces was
studied in a series of works [ay-17, ay-16]. Any $T$-space from this class is
equivariantly homeomorphic to the quotient construction $Q\times T/\sim$, see
[da-ja-91, yo-11]. In [ay-17, ay-16], the orbit spectral sequences for the
quotient construction and $Q\times T$ were compared, and cellular sheaves over
Alexandrov spaces (see [cu-14]) were applied. Betti numbers for those of such
spaces that admit a characteristic function were computed in [ay-17, ay-16].
The orbit spectral sequence for a torus manifold has nontrivial differentials
of order $\geqslant 2$, in general. In the particular case of a complete
(possibly, singular) toric variety the orbit spectral sequence (over the field
of complex numbers) arose in the work of Danilov [da-78]. He proved the
collapse of this spectral sequence at the second page (i.e. all differentials
of order $\geqslant 2$ are zero). The similar result in Borel-Moore homology
with rational coefficients was proved in [to-14, §5].

The cohomological Eilenberg-Moore spectral sequence, e.g. see [mcc-01], (EMSS,
for short) of the Borel fibration for a $T$-space $X$ starts from the bigraded
components of the $\operatorname{Tor}$-module
$\operatorname{Tor}_{H^{*}(BT)}(H^{*}_{T}(X),\mathbb{Z})$ at the second page
and converges to $H^{*}(X)$. The multiplication of this spectral sequence is
known to have a nontrivial extension, in general (e.g. see [fr-21]). In the
case of a partial quotient of a moment-angle complex the rigorous formula
identifying the cup-product with the twisted product on
$\operatorname{Tor}$-groups was recently obtained in [fr-21]. The homotopy
type of the skeleton for a complex projective space $\mathbb{C}P^{m-1}$ (a
partial quotient with a $T^{m-1}$-action) was obtained in [fu-19]. Together
with [gr-th-07], this leads to the computation of Betti numbers of such space.

By definition [go-ko-mc-98], a $T$-space is equivariantly formal if the Serre
spectral sequence of the corresponding Borel fibration collapses at the second
page. Equivariant formality is equivalent to vanishing of all higher
$\operatorname{Tor}$-groups from the second page of the corresponding
Eilenberg-Moore spectral sequence. The equivariant cohomology ring of any
compact equivariantly formal toric orbifold with vanishing odd cohomology was
described in [fr-10] in terms of the ring of piecewise polynomial functions on
the corresponding fan. Furthermore, it was shown in [fr-10] that
$H^{*}(X_{\Sigma};\ \mathbb{Z})$ is torsion-free under the same conditions on
$X_{\Sigma}$. Recently, a criterion of equivariant formality (over rational
numbers) for partial quotients over toric manifolds (in the sense of [da-
ja-91]) was provided in [ku-23]. This result together with [ma-pa-06] implies
that any partial quotient $X\neq M$ over any toric manifold $M$ is not
equivariantly formal.

A toric diagram $D\in[C,\operatorname{Top}]$ over a small category $C$ is a
functor such that $D(c)$ is a compact torus and $D(c\to c^{\prime})$ is a
group homomorphism, $c\in C_{0}$, $(c\to c^{\prime})\in C_{1}$, see [we-zi-
zi-99]. A toric diagram is called a $T$-diagram if there is an object-wise
epimorphism $\kappa T\to D$ of diagrams, where $\kappa T$ denotes the constant
diagram. Consider any $T$-CW-complex $X$ such that $X$ is split, i.e. the
projection $X\to X/T$ admits an equivariant section, and the orbit poset of
$X$ is a CW poset. Then the $T$-space $X$ is equivariantly homeomorphic to the
homotopy colimit $\operatorname{hocolim}D$ (in the standard realization) of a
$T$-diagram $D$ over the orbit poset $C$ of $X$. (Every face of the space
$\operatorname{hocolim}D$ is contractible.) The converse also holds provided
that $D$ is reduced, i.e. any two $T$-cells $\sigma\subset\sigma^{\prime}$ in
$X$ have different stabilizers (see Theorem 6.12). The Borel fibration of
$\operatorname{hocolim}D$ takes form $\operatorname{hocolim}BS\to BT$, where
the diagram $S$ is given by the stabilizers for the $T$-action on
$\operatorname{hocolim}D$ (see Proposition 7.2).

For any diagram $D\in[C,\operatorname{Top}]$ over a CW poset $C$, the skeletal
filtration of the nerve of $C$ induces the filtration on
$\operatorname{hocolim}D$. The cohomological Bousfield-Kan spectral sequence
of $D$ (BKSS, for short), i.e. the spectral sequence of this filtration,
converges to $H^{*}(\operatorname{hocolim}D;\ \Bbbk)$, where
$\Bbbk=\mathbb{Q},\mathbb{Z}$ [bo-ka-72, bo-87]. The second page of BKSS is
given by sheaf cohomology of the refinement sheaf $sH^{*}(D)$ for the cellular
cosheaf $H^{*}(D)$, or equivalently by higher derived functors of $\lim
sH^{*}(D)$. Based on the ideas from [la-tu-vo-10], below we give the
cohomological version of BKSS collapse at the second page for any formal
diagram $D$ over a graded finite poset. (The extension of BKSS is trivial in
the category of $\Bbbk$-modules.) The functorial models of a simplicial torus
(in particular, a simplicial abelian group) and its classifying space [fr-21]
are used to show formality of any toric diagram $D$ over
$\mathbb{Z},\mathbb{Q}$, and formality of any quasitoric diagram $BD$ over
$\Bbbk$ (i.e. objects of $D$ are quasitori). Here,
$\Bbbk=\mathbb{Z},\mathbb{Q}$, if any object of $D$ is connected, and
$\Bbbk=\mathbb{Q}$ otherwise.

Below we derive three main results of this note from the BKSS collapse. Let
$X_{\Sigma}$ be any compact toric variety with the corresponding complete
rational polyhedral fan $\Sigma$ of dimension $d$ consisting of salient cones.
There is the $T$-diagram $D_{\Sigma}$ such that $X_{\Sigma}$ is equivariantly
homotopy equivalent to $\operatorname{hocolim}D_{\Sigma}$, as described in
[we-zi-zi-99, li-so-22].

######  Theorem 1.

The orbit spectral sequence of $X_{\Sigma}$ collapses at the second page with
trivial extension. There is the abelian group isomorphism

$H^{i}(X_{\Sigma};\ \mathbb{Z})\cong\bigoplus_{s+t=i}H^{cw}_{d-s}(\Sigma;\
H^{t}(D_{\Sigma})),$

with the direct sum of cellular homology groups for the poset $\Sigma$ with
coefficients in the cosheaf.

Theorem 1 (see Theorem 6.15) provides a possible explanation of the collapse
for the Danilov spectral sequence [da-78]. This theorem positively resolves
the conjecture (for cohomology with integral coefficients) of M. Franz [fr-06,
Conjecture 6.4] in the case of a compact toric variety. (In [fr-06] the
collapse result was obtained over any commutative unital ring $R$ for any
nonsingular toric variety). A more general result holds for the spectral
sequence induced by the coskeletal filtration on $C$ for any toric diagram
over any oriented homology manifold poset (see Theorem 5.12). This filtration
coincides with the orbit filtration for any strongly reduced $T$-diagram (see
Section 6). The proof of Theorem 1 relies on the Poincaré-Lefschetz type
duality for sheaf cohomology using the Zeeman-McCrory spectral sequence (ZMSS,
for short) collapse (e.g. see [ay-17, ch-23]). It is expected that Theorem 1
will be helpful for explicit computations of cohomology for singular toric
varieties (compare with [fr-10]). For instance, the orbit spectral sequence
collapse rules out one of the two cases considered in [Ho-08].

######  Theorem 2.

Let $D$ be any reduced $T$-diagram over an oriented homology manifold $C$ of
dimension $d$. Let $\Bbbk=\mathbb{Q}$, or let $\Bbbk=\mathbb{Z}$ if all
stabilizers of the $T$-action on $\operatorname{hocolim}D$ are connected. Then
$\operatorname{hocolim}D$ is equivariantly formal over $\Bbbk$ iff
$H^{odd}(\operatorname{hocolim}D;\ \Bbbk)=0$. In such case, the sheaf
$sH^{*}(BS;\ \Bbbk)$ is acyclic, and the orbit space $\operatorname{ord}C$ is
$\Bbbk$-acyclic. Then one has the isomorphisms

(1) $H^{*}_{T}(\operatorname{hocolim}D;\ \Bbbk)\cong\lim sH^{*}(BS),\
H^{2i}(\operatorname{hocolim}D;\ \Bbbk)\cong\operatorname{lim}^{i}sH^{i}(D),$

of modules over $H^{*}(BT)$ and $\Bbbk$, respectively. Furthermore,
$H^{*}(\operatorname{hocolim}D;\ \Bbbk)$ is torsion-free.

Theorem 2 implies that the closure of any orbit in $\operatorname{hocolim}D$
is equivariantly formal, has vanishing odd cohomology and is torsion-free (see
Corollary 9.10). Assuming that $X$ is a torus manifold such that its orbit
poset is a regular CW poset, the criterion of $H^{odd}(X)=0$ from [ma-pa-06]
follows in one direction from Theorem 2 in such a particular case (see Theorem
9.7). The proof of Theorem 2 relies in particular on ABFP exact sequence [fr-
pu-06], BKSS and ZMSS collapse, mentioned above. Another key tool of the proof
is the $t$-graded comparison spectral sequence (CSS, for short) of a
$T$-diagram $D$ introduced in Section 8. This spectral sequence compares EMSS
and BKSS in the following sense. The target of the CSS is given by the
respective line from the second page of BKSS. The second page of CSS is
concentrated in a line coinciding with the line from the second page of the
respective EMSS (up to a certain change of indices), provided that the sheaf
$sH^{*}(BS)$ is acyclic. As a byproduct, we obtain the description of bigraged
Betti numbers of $\operatorname{hocolim}D$ (i.e. the dimensions of the second
page for EMSS) when $sH^{*}(BS)$ is acyclic.

We derive the explicit formulas for the Betti numbers for
$H^{*}(\operatorname{hocolim}D)$ below under slightly more general conditions
compared to those of Theorem 2. This computation follows from the above
results by relating the Euler characteristic of the first and second pages
from BKSS. Betti numbers of tropical toric varieties are expressed by a
similar formula, see [ch-23]. Furthermore, we describe the cohomology groups
and bigraded Betti numbers of skeleta for toric manifolds. This is the third
main result of the present note. (In fact, the computation holds in a more
general setting of skeleta for a $T$-characteristic diagram over an oriented
homology manifold, see Section 10). Notice that skeleta for toric varieties
are different from broken toric varieties over skeleta considered in [su-23].
In the case of a skeleton for $\mathbb{C}P^{m-1}$ we compare these formulas
with those implied by [fu-19], [gr-th-07]. Our computation is based on the
dual to the Taylor-type complex introduced in [ay-17], combined with the
aforementioned methods of BKSS collapse and comparison SS.

This note is organized as follows. In Section 2, the BKSS collapse for formal
diagrams is proved after introducing the necessary definitions. Section 3 is
devoted to the proof of formality for simplicial (quasi)tori and their
classifying spaces. The necessary notions of (co)sheaf theory on posets are
given in Section 4. Section 5 contains a short recollection on Zeeman-McCrory
spectral sequence for sheaves on posets, coskeletal spectral sequence and
related collapse results. In Section 6 we characterize homotopy colimits of
$T$-diagrams over CW posets in terms of a certain class of $T$-spaces (up to a
homeomorphism). In the same section we compare the orbit and coskeletal
spectral sequences for strongly reduced $T$-diagrams and prove Theorem 1. In
Section 7 we give a diagrammatic description of the Borel space for
$\operatorname{hocolim}D$ of a $T$-diagram $D$ and the respective EMSS. The
comparison spectral sequence is introduced in Section 8. Section 9 contains a
recollection on equivariant formality, as well as the proof of Theorem 2. The
concluding Section 10 is devoted to computations of Betti numbers for
equivariantly formal spaces and skeleta of $T$-characteristic diagrams over
(oriented) homology manifolds.

_Acknowledgements._ The author thanks Okayama University of Science for a
comfortable work atmosphere and assistance during his research visit. Support
and communication with Katsuhiko Kuribayashi, Shintarô Kuroki, Mikiya Masuda
and Naoto Yotsutani were very helpful during the writing process for the
present note, many thanks to these people. It is a pleasure to thank Anton
Ayzenberg and Matthias Franz for several inspiring discussions on the subject
of this note. Finally, the author is grateful to Vladimir Dotsenko for the
interest to this project. This work was supported by the French national
research agency [grant ANR-20-CE40-0016]. The author is a Young Russian
Mathematics award winner and would like to thank its sponsors and jury.

##  2\. BKSS collapse

In this section we recall the definition of the cohomological Bousfield-Kan
spectral sequence (BKSS, for short). Then we prove the collapse result for
BKSS of any formal diagram over any finite graded poset $C$ (Theorem 2.12). We
use the ideas (formality of diagrams) from [la-tu-vo-10] adjusted to
computation for cohomology of simplicial spaces. In what follows, we refer to
the standard notions of simplicial homotopy theory from [go-ja-09].

######  Definition 2.1.

Let $\Delta$, $\Delta_{n}$ be the categories of all finite ordered sets, and
of all ordered sets of cardinality $\leqslant n$, respectively. Let
$\operatorname{Top}$ denote the category of compactly generated weakly
Hausdorff topological spaces and continuous maps between them. For a
simplicial space $X_{\bullet}\in[\Delta^{op},\operatorname{Top}]$ let

$\operatorname{tr}_{i}X_{\bullet}:=\iota^{n}_{*}X_{\bullet}\in[\Delta_{n}^{op},\operatorname{Top}],$

where $\iota^{n}\colon\Delta_{n}\to\Delta$ is the natural functor. Let

$\mathrm{sk}_{i}\colon[\Delta_{n}^{op},\operatorname{Top}]\to[\Delta^{op},\operatorname{Top}],$

be the left adjoint of $\operatorname{tr}_{i}$ given by the respective left
Kan extension. The functor

$\operatorname{Sk}_{i}:=\mathrm{sk}_{i}\circ\operatorname{tr}_{i}\colon[\Delta^{op},\operatorname{Top}]\to[\Delta^{op},\operatorname{Top}],$

is called the $i$-skeleton of $X_{\bullet}$.

Let $\tilde{X}_{\bullet}\to X_{\bullet}$ be a cofibrant replacement of a
simplicial space. The counit of the adjunction
$sk_{i}\dashv\operatorname{tr}_{i}$ is the map
$\operatorname{Sk}_{i}\tilde{X}_{\bullet}\to\tilde{X}_{\bullet}$ which is a
cofibration. The simplicial space $X_{\bullet}$ has dimension $n$ if
$X_{\bullet}=\operatorname{Sk}^{n}X_{\bullet}$. In what follows, we consider
only simplicial spaces of finite dimension, whose every object is cofibrant.
Using the Quillen model structure, we often replace a weak equivalence in
$\operatorname{Top}$ with a homotopy equivalence by Whitehead theorem for
diagrams consisting of CW-complexes. By taking the geometric realization of an
$n$-dimensional simplicial space, one obtains the filtration

(2)
$\lvert\operatorname{Sk}_{0}\tilde{X}_{\bullet}\rvert\subset\cdots\subset\lvert\operatorname{Sk}_{n}\tilde{X}_{\bullet}\rvert=\lvert\tilde{X}_{\bullet}\rvert,$

where $\lvert X_{\bullet}\rvert$ denotes the realization of a simplicial
space. (Notice that $\lvert\tilde{X}_{\bullet}\rvert\to\lvert
X_{\bullet}\rvert$ is a weak equivalence.)

######  Definition 2.2.

For a diagram $D\in[C,\operatorname{Top}]$ over a small category $C$ the
simplicial replacement
$\operatorname{srep}_{\bullet}D\in[\Delta^{op},\operatorname{Top}]$ of $D$ is
given by the set of $n$-simplices

$\biggl{\\{}\bigsqcup_{\sigma\in N_{n}(C)}D(c_{n})|\
\sigma=(c_{0}\leftarrow\cdots\leftarrow c_{n})\in N_{n}(C)\biggr{\\}},$

where $N_{\bullet}(C)$ is the nerve category of $C$. The face and degeneracy
maps are given by the usual formulas [go-ja-09].

Let $\Bbbk=\mathbb{Z},\mathbb{Q}$. From now on we assume that $C$ is a finite
graded poset (or, more generally, a finite Reedy category).

######  Definition 2.3 ([bo-ka-72, bo-87]).

The cohomological Bousfield-Kan spectral sequence (BKSS, for short) of a
simplicial set $X_{\bullet}$ is the first quadrant spectral sequence
$((E_{X_{\bullet}})_{r}^{p,q},d_{r})$, $r\geqslant 1$, in cohomology with
coefficients in a ring $\Bbbk$, corresponding to the filtration (2). The
cohomological Bousfield-Kan spectral sequence $((E_{D})_{r}^{p,q},d_{r})$ of a
diagram $D\in[C,\operatorname{Top}]$ over $C$ with values in
$\operatorname{Top}$ is the BKSS of the simplicial replacement
$\operatorname{srep}_{\bullet}D$ [go-ja-09].

######  Definition 2.4.

The realization $\lvert\operatorname{srep}_{\bullet}D\rvert$ of the simplicial
replacement for a $C$-diagram $D$ of spaces is called the homotopy colimit
$\operatorname{hocolim}D$ of $D$.

######  Remark 2.5.

The above definition agrees up to a weak equivalence with the colimit
$\operatorname{colim}\tilde{D}$ of a cofibrant replacement $\tilde{D}\to D$,
provided that $D$ is objectwise cofibrant [du]. In particular,
$\operatorname{hocolim}$ is the left derived functor of
$\operatorname{colim}$. More explicitly,

$\tilde{D}(c)=\lvert\operatorname{srep}_{\bullet}u^{*}_{c}D\rvert,$

where $u_{c}\colon C\downarrow c\to C$ is the forgetting functor from the
respective arrow category. Furthermore, for every $f\colon c\to c^{\prime}$
the corresponding map $\tilde{D}(f)=(u_{f})_{*}$ is induced by the functor
$u_{f}\colon C\downarrow c\to C\downarrow c^{\prime}$ between the comma-
categories. By considering only nondegenerate simplices in the nerve of $C$,
one can describe $\operatorname{hocolim}D$ as follows

(3)
$\operatorname{hocolim}D\simeq\biggl{(}\bigsqcup_{\sigma=c_{0}\leftarrow\cdots\leftarrow
c_{i}}\Delta_{\sigma}\times D(c_{i})\biggr{)}/\sim,$ $\bigl{(}x,D(c_{i}\to
c^{\prime}_{j})y\bigr{)}\sim(\operatorname{Inc}_{\sigma^{\prime}\to\sigma}x,y),\
x\in\Delta_{\sigma^{\prime}},\ y\in D(c_{i}),$

where the coproduct is taken by all strictly increasing chains in $C$ and

$\operatorname{Inc}_{\sigma^{\prime}\to\sigma}\colon\Delta_{\sigma^{\prime}}\to\Delta_{\sigma},\
\sigma^{\prime}=(c^{\prime}_{0}\leftarrow\cdots\leftarrow
c^{\prime}_{j})\to\sigma,$

is the natural inclusion of faces.

######  Remark 2.6.

The simplicial space $\operatorname{srep}D_{\bullet}$ is cofibrant [go-ja-09].
For a simplicial space $X_{\bullet}$ the spectral sequence
$((E_{X_{\bullet}})_{r}^{p,q},d)$ is isomorphic (from the second page) to the
spectral sequence of the simplicial dga $C^{*}(X_{\bullet})$ of normalized
cochains with the filtration by simplicial degree (the proof is similar to
[bo-87, Lemma 2.2]). The BKSS second page is well-known to be expressed in
terms of higher limits (e.g. see [no-ra-05]). Therefore, the BKSS for a
diagram $D\in[C,\operatorname{Top}]$ takes the following form (see Section 4):

${\lim}^{i}sH^{j}(D)\Rightarrow H^{i+j}(\operatorname{hocolim}\ D).$

The proof of the following proposition is obtained trivially from naturality
of BKSS with respect to morphism of diagrams.

######  Proposition 2.7.

For any morphism of diagrams $\alpha\colon D\to X$, where
$D\in[C,\operatorname{Top}]$, $X\in[\Delta_{1},\operatorname{Top}]$, the
spectral sequence $((E_{D})_{r}^{i,j},d_{r})$ carries the structure of graded
$H^{*}(X;\ \Bbbk)$-module. This structure agrees with the $H^{*}(X;\
\Bbbk)$-module structure on the BKSS target $H^{*}(\operatorname{hocolim}D)$
induced by $\alpha$.

In the sequel we use the line of argument similar to that of [la-tu-vo-10].
The difference in our situation is that we take a simplicial replacement,
compared to the cosimplicial replacement from [la-tu-vo-10]. Another feature
of the proof below is the cohomological setting which is similar to that of
[go-ja-09].

######  Definition 2.8 ([la-tu-vo-10]).

A diagram $D\in[C,\operatorname{Top}]$ is called formal (over $\Bbbk$) if
there is a chain of quasiisomorphisms between the diagrams $C^{*}(D;\Bbbk)$
and $H^{*}(D;\Bbbk)$ (with zero differentials) in
$[C^{op},\operatorname{dgm}_{\Bbbk}]$.

######  Lemma 2.9 ([la-tu-vo-10]).

For any formal diagram $D\in[C,\operatorname{Top}]$ the diagram
$\operatorname{srep}_{\bullet}D\in s\operatorname{Top}$ is formal.

###### Proof.

For any diagram $A\in[C^{op},\operatorname{dgm}_{\Bbbk}]$ denote by
$\oplus^{\bullet}A\in[\Delta,\operatorname{dgm}_{\Bbbk}]$ its cosimplicial
replacement. More precisely, let

${\oplus}^{n}A:=\bigoplus_{\sigma\in N(C)}A(c_{n}),\
\sigma=(c_{0}\leftarrow\cdots\leftarrow c_{n})\in N_{n}(C).$

To any monotone map of finite ordinals $\theta\colon[m]\to[n]$ we associate
(component-wisely) the respective structure map $\theta^{\dagger}$ in
$\oplus^{\bullet}A$ by the following commutative diagram

${A(c^{\prime}_{\theta(m)})}$${A(c^{\prime}_{m})}$${\underset{\sigma\in
N_{n}(C)}{\bigoplus}A(c_{n})}$${\underset{\tau\in
N_{m}(C)}{\bigoplus}A(c_{m}),}$$\scriptstyle{A(\alpha)}$$\scriptstyle{in_{\theta_{*}(\tau)}}$$\scriptstyle{in_{\tau}}$$\scriptstyle{\theta^{\dagger}}$

where $\sigma$ is as above, $\tau=(c^{\prime}_{0}\leftarrow\cdots\leftarrow
c^{\prime}_{m})$, $\alpha\colon c^{\prime}_{m}\to c^{\prime}_{\theta(m)}$ is
defined by the arrow in $C$ corresponding to $\theta$.

By the assumption of formality, there is the zig-zag of quasiisomorphisms

${H^{*}(D)}$${\cdots}$${C^{*}(D),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

for diagrams in $[C^{op},\operatorname{dgm}_{\Bbbk}]$. It induces the zig-zag
of quasiisomorphisms

(4)
${{\oplus}^{\bullet}H^{*}(D)}$${\cdots}$${{\oplus}^{\bullet}C^{*}(D),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

for $[\Delta,\operatorname{dgm}_{\Bbbk}]$. The isomorphisms

$\underset{\sigma\in N_{n}(C)}{\bigoplus}H^{*}(D(c_{n}))\to
H^{*}\biggl{(}\bigsqcup_{\sigma\in N_{n}(C)}D(c_{n})\biggr{)},\
\underset{\sigma\in N_{n}(C)}{\bigoplus}C^{*}(D(c_{n}))\to
C^{*}\biggl{(}\bigsqcup_{\sigma\in N_{n}(C)}D(c_{n})\biggl{)},$

induce the isomorphisms

(5) ${\oplus}^{\bullet}H^{*}(D)\to H^{*}(\operatorname{srep}_{\bullet}D),\
{\oplus}^{\bullet}C^{*}(D)\to C^{*}(\operatorname{srep}_{\bullet}D),$

in $[\Delta,\operatorname{dgm}_{\Bbbk}]$. Now the claim follows by combining
the isomorphisms (5) with zigzag (4) of quasiisomorphisms. ∎

######  Remark 2.10.

The original statement [la-tu-vo-10] for the homology of cosimplicial spaces
was given for rational coefficients. For cosimplicial spaces it can be
generalized (by using Künneth formula) to integral coefficients under the
additional assumption that any object of the diagram $D$ has torsion-free
homology.

Consider any cosimplicial dgm $V^{*}=(V^{*})^{\bullet}$ of finite dimension
(i.e. $(V)^{i}$ is degenerate for $i\geqslant n$ for some $n$) with
cohomological differential of degree $1$. The simplicial filtration on $V^{*}$
induces the spectral sequence $(E^{*,*}_{*}(V^{*}),d_{*})$. Due to assumption
of finite dimension on $V^{*}$, this filtration is assumed to be finite
without loss of generality.

######  Lemma 2.11.

Let $V^{*}\to W^{*}$ be any quasiisomorphism of cosimplicial
$\operatorname{dgm}_{\Bbbk}$ of finite dimension. Then the extension is
trivial for $(E^{*,*}_{*}(V^{*}),d_{*})$ iff it is trivial for
$(E^{*,*}_{*}(W^{*}),d_{*})$.

###### Proof.

Follows directly by the induction on the length of the filtration by
$5$-lemma. ∎

Let $\Bbbk=\mathbb{Q},\mathbb{Z}$.

######  Theorem 2.12 ([la-tu-vo-10]).

For any formal (over $\Bbbk$) diagram $X_{\bullet}\in\operatorname{sTop}$, the
spectral sequence $((E_{X^{\bullet}})_{r}^{i,j},d_{r})$ of $\Bbbk$-modules
collapses at the second page with trivial extension.

###### Proof.

By Remark 2.6, $(E^{*,*}_{*}(C^{*}(X)),d_{*})$ coincides from the second page
with the Bousfield-Kan spectral sequence of
$C^{*}(X_{\bullet})\in[\Delta,\operatorname{dga}_{\Bbbk}]$. The spectral
sequence $(E^{*,*}_{*}(C^{*}(X_{\bullet})),d_{*})$ of $\Bbbk$-modules
coincides from the first page with the spectral sequence
$(E^{*,*}_{*}(H^{*}(X_{\bullet})),d_{*})$ by the assumption of formality for
$X_{\bullet}$. However, $(E^{*,*}_{*}H^{*}(X_{\bullet}),d_{*})$ collapses at
the second page, because the vertical differential in the associated bicomplex
is trivial.

Notice that the extension of $\Bbbk$-modules is trivial for
$(E^{*,*}_{*}H^{*}(X_{\bullet}),d_{*})$. Indeed, $H^{*}(X_{\bullet})$
decomposes into a direct sum of sub-dg-modules by the cohomological degree.
Therefore, $(E^{*,*}_{*}(H^{*}(X_{\bullet})),d_{*})$ is a direct sum of
spectral sequences, each concentrated at one line. We conclude that the
extension for $(E^{*,*}_{*}(H^{*}(X_{\bullet})),d_{*})$ is trivial. Now the
triviality of $\Bbbk$-module extension for
$(E^{*,*}_{*}(C^{*}(X_{\bullet})),d_{*})$ follows by Lemma 2.11. ∎

##  3\. Formality of toric diagrams

The purpose of the current section is to prove the BKSS collapse result for
any toric diagram (Theorem 3.14) and its classifying space (Theorem 3.15). By
the results of the previous section this is done by establishing formality for
any toric (over $\Bbbk=\mathbb{Z},\mathbb{Q}$) and quasitoric diagram (over
$\Bbbk=\mathbb{Z}$ or $\mathbb{Q}$ depending on the connectedness of its
objects), see Theorems 3.12, 3.13. The line of argument is similar to that
from [li-so-22] and is based on the notion of a simplicial torus (a particular
case of a simplicial abelian group) and its classifying space [ma-92, fr-10].

######  Definition 3.1 ([we-zi-zi-99]).

We call a diagram $D\in[C,\operatorname{Top}]$ a _quasitoric diagram_ , if
$D(c)$ is a quasitorus (i.e. the product of a compact torus with a finite
abelian group) for any $c\in C$ and any arrow of $D$ is a group homomorphism.
A quasitoric diagram $D\in[C,\operatorname{Top}]$ consisting of compact tori
(as objects) is called a _toric diagram_.

Let $N$ be any finitely generated abelian group. Consider
$N_{\bullet}=\kappa_{\bullet}N$ as a constant simplicial group.

######  Definition 3.2 ([ma-92, fr-21]).

Let $BN_{\bullet}$ be the $0$-reduced simplicial abelian group given by

$BN_{n}:=\\{[a_{0},\dots,a_{n-1}]|\ a_{0},\dots,a_{n-1}\in N\\},\ n>0,\
BN_{0}:=\\{[\ ]\\},$

and the explicit formulas for faces and degenerations of $BN_{\bullet}$ are
given in [ma-92, p.87]. The simplicial group structure is defined by
coordinatewise summation in $N$. If $N$ is torsion-free then $BN_{\bullet}$ is
called a simplicial torus.

Given any finitely generated abelian group $N$, let $N=N_{f}\times N_{t}$ be
its decomposition into the direct product of the free $N_{f}$ and torsion
$N_{t}$ subgroups. Then one has a homotopy equivalence

$|BN_{\bullet}|\simeq BN\simeq T\times BN_{t},$

with the product of Eilenberg-Maclane spaces: the torus $T=BN_{f}$ and the
product $BN_{f}$ of infinite-dimensional lens spaces $L_{p^{n}}^{\infty}$.

######  Remark 3.3.

In this note we need to consider the classifying spaces of a torus and a
torsion group. Therefore, we avoid considering the classifying space of the
simplicial group $(BN_{t})_{\bullet}$.

######  Definition 3.4.

The bar-construction $B(BN_{f}\times N_{t})_{\bullet}$ of the simplicial
abelian group $(BN_{f}\times N_{t})_{\bullet}$ is the simplicial set defined
by

$B(BN_{f}\times N_{t})_{n}:=\\{[(b^{n-1},[\ ]),\dots,([\ ],c_{n-1})]|\
b^{i}\in(BN_{f})_{i},\ c_{i}\in N_{t}\\},$

and nothing in degree 0. We refer to [ma-92, p.87] for the definition of face
and degeneration maps of $B(BN_{f}\times N_{t})_{\bullet}$.

One has an isomorphism of simplicial sets

(6) $B(BN_{f}\times N_{t})_{\bullet}\cong
B(BN_{f})_{\bullet}\times(BN_{t})_{\bullet}.$

######  Definition 3.5.

For any quasitorus $G$ let

$N(G):=N_{f}(G)\times N_{t}(G),\ N_{f}(G):=\pi_{1}(G,e),\
N_{t}(G):=\pi_{0}(G).$

Let

$\overline{W}(G):=BN_{t}(G)_{\bullet}\times N_{f}(G)_{\bullet},\
\overline{W}(BG):=B(BN_{f}(G))_{\bullet}\times BN_{t}(G)_{\bullet}.$

be the simplicial abelian group and simplicial set, respectively. Denote the
normalized chain and cochain complexes of $\overline{W}(G)$ and
$\overline{W}(BG)$ by $\overline{W}_{*}(G)$, $\overline{W}^{*}(G)$,
$\overline{W}_{*}(BG)$, $\overline{W}^{*}(BG)$, respectively.

Notice that for a torsion group $G$ one has $\overline{W}(G)=\overline{W}(BG)$
by the definition. The cochain complexes $\overline{W}^{*}(G)$,
$\overline{W}^{*}(BG)$ are dga with respect to the $\cup$-product. Any
homomorphism of tori $f\colon T\to T^{\prime}$ of ranks $r$, $r^{\prime}$,
respectively, is given by the formula

(7) $(e^{2\pi i\varphi_{1}},\dots,e^{2\pi i\varphi_{r}})\mapsto(e^{2\pi
i\sum_{q}a_{1,q}\varphi_{q}},\dots,e^{2\pi
i\sum_{q}a_{r^{\prime},q}\varphi_{q}}),\
\varphi_{1},\dots,\varphi_{r}\in[0,1),$

for some integer matrix
$A=(a_{i,j})\in\operatorname{Mat}_{r^{\prime},r}(\mathbb{Z})$ (which we denote
by $A=A(f)$), and vice versa. For any cyclic finite group $N$ of order $p^{n}$
consider the embedding $N\to S^{1}$ given by mapping the generator to
$\exp{(2\pi i/p^{n})}$. The product of such maps induces the embedding
$G\to\tilde{T}$ of the quasitorus $G$ to the torus $\tilde{T}$. Given any
morphism of quasitori $f\colon G\to G^{\prime}$, there is a unique extension
of $f$ to the homomorphism of tori
$\tilde{f}\colon\tilde{T}\to\tilde{T^{\prime}}$. Therefore, $f$ is given by
the integral matrix denoted by $A$ by abuse of the notation. (Notice that $A$
is not uniquely defined due in presence of torsion in $G^{\prime}$.)

######  Definition 3.6.

Define the morphism $f_{*}\colon\overline{W}(G)\to\overline{W}(G^{\prime})$ by

(8) $f_{*}(a^{n})=Aa^{n}=[A(f)(a_{0}),\dots,A(f)(a_{n-1})],\
a^{n}=[a_{0},\dots,a_{n-1}].$

Define the $Bf_{*}\colon\overline{W}(BG)\to\overline{W}(BG^{\prime})$ by

(9) $Bf_{*}([(b^{n-1},[\ ]),(b^{n-2},c_{1}),\dots,([\
],c_{n-1})])=[(A_{f,1}b^{n-1},A_{t}c_{1}+A_{f,2}b^{n-2}_{n-2}),\dots,([\
],A_{t}c_{n-1})],$

where

$b^{i}=[b^{i}_{0},\dots,b^{i}_{i-1}]\in BN_{f}(G)_{i},\ c_{j}\in N_{t},\ i>0,\
j\geqslant 0,$

and $A$ has the block form

$\begin{pmatrix}A_{f,1}&&0\\\ A_{f,2}&&A_{t}\end{pmatrix},$

with respect to the decompositon $N(G)=N_{f}(G)\times N_{t}(G)$ and
$N(G^{\prime})=N_{f}(G^{\prime})\times N_{t}(G^{\prime})$. The formula (9)
gives a well defined morphism of simplicial sets.

######  Remark 3.7.

For any compact torus $T=(S^{1})^{r}$ let
$T_{\mathbb{C}}=(\mathbb{C}^{\times})^{r}$, and for a homomorphism $f\colon
T\to T^{\prime}$ given by matrix $A\in\operatorname{Mat}_{r^{\prime}\times
r}(\mathbb{Z})$ (see (7)), let $f_{\mathbb{C}}\colon T_{\mathbb{C}}\to
T^{\prime}_{\mathbb{C}}$ be the homomorphism given by

(10)
$(z_{1},\dots,z_{r})\mapsto(\prod_{q}z_{q}^{a_{1,q}},\dots,\prod_{q}z_{q}^{a_{r^{\prime},q}}).$

Notice that the restriction of $f_{\mathbb{C}}$ to compact tori is equal to
$f$. This defines the complexification functor from the category of compact
tori to non-compact (algebraic) tori. For any toric diagram $D$, the
complexification $D_{\mathbb{C}}$ of $D$ is by definition given by applying
the complexification functor. The natural embedding of compact tori to
algebraic tori induces a homotopy equivalence of diagrams $D\to
D_{\mathbb{C}}$.

Consider the Quillen equivalence [go-ja-09]

$\operatorname{Sing}\colon\operatorname{Top}_{Quillen}\rightleftarrows\operatorname{sSet}_{Quillen}\colon|-|,$

given by the functors of singular simplicial complex and geometric
realization. Let $S(X):=\operatorname{Sing}(X)$ be the simplicial set of
singular simplices for any topological space $X$.

######  Lemma 3.8 ([fr-21, li-so-22]).

There are the following commutative diagrams of simplicial sets

${\overline{W}(G)}$${S(G)}$${\overline{W}(G^{\prime})}$${S(G^{\prime}),}$$\scriptstyle{\sim}$$\scriptstyle{f_{*}}$$\scriptstyle{f_{*}}$$\scriptstyle{\sim}$
${\overline{W}(BG)}$${S(BG)}$${\overline{W}(BG^{\prime})}$${S(BG^{\prime}),}$$\scriptstyle{\sim}$$\scriptstyle{Bf_{*}}$$\scriptstyle{Bf_{*}}$$\scriptstyle{\sim}$

where any horizontal arrow is a homotopy equivalence.

###### Proof.

Let $G=T\times G_{t}$, $G^{\prime}=T^{\prime}\times G^{\prime}_{t}$ be the
decomposition of $G$, $G^{\prime}$ into the direct products of the tori $T$,
$T^{\prime}$ and torsion subgroups $G_{t}$, $G^{\prime}_{t}$. First, we prove
commutation of the left-hand diagram above. Any nondegenerate simplex in
$\overline{W}(G_{t})$ has dimension $1$, and $\overline{W}(G_{t})_{1}=G_{t}$.
Furthermore, any simplex in $S(G_{t})$ is a constant map. Therefore, there is
a well-defined tautological map $\overline{W}(G_{t})\to S(G_{t})$ defined on
$\overline{W}(G_{t})_{n}$ by sending any element $g\in G_{t}$ to the constant
map $\Delta^{n}\mapsto g$, $n>0$. Clearly, it is a homotopy equivalence.
Consider the commutative diagram

(11)
${\overline{W}(G)}$${\overline{W}(T)\times\overline{W}(G_{t})}$${S(T)\times
S(G_{t})}$${S(G)}$${\overline{W}(G^{\prime})}$${\overline{W}(T^{\prime})\times\overline{W}(G^{\prime}_{t})}$${S(T^{\prime})\times
S(G^{\prime}_{t})}$${S(G^{\prime}).}$$\scriptstyle{\sim}$$\scriptstyle{f_{*}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{f_{*}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Here, the upper left horizontal arrow is given by (6). It induces an
isomorphism in homology by Künneth formula, therefore it is a homotopy
equivalence by Whitehead theorem. The upper middle horizontal arrow in (11) is
the product of the homotopy equivalence $\overline{W}(T)\to S(T)$ defined in
[fr-21] and of the tautological map $\overline{W}(G_{t})\to S(G_{t})$. The
right square horizontal arrows from (11) are well defined because the image of
any continuous map $\Delta^{n}\to G$ is connected. One can check that the
diagram (11) is commutative. Each of the middle arrows in (11) is a product of
homotopy equivalences. For the connected component of the identity this holds
by [fr-21], and for the torsion component this follows by the construction.
This proves the claim for the left diagram. The claim for the right diagram
follows from the first by the same argument from [li-so-22] applied to
$B(BN_{f}\times N_{t})_{\bullet}$. ∎

Recall that $C$ is any finite graded poset (or more generally, any Reedy
category). There is the Quillen equivalence

$\operatorname{Sing}^{C}\colon\operatorname{Top}^{C}\rightleftarrows\operatorname{sSet}^{C}\colon|-|^{C},$

of Reedy closed model structures [hi-03].

######  Proposition 3.9.

For any toric diagram $D$ there are weak equivalences

$\overline{W}(D)\overset{\sim}{\to}S(D),\
\overline{W}(BD)\overset{\sim}{\to}S(BD),$

for diagrams in $[C,s\operatorname{Top}]$.

###### Proof.

Since the weak equivalences are defined objectwise in the Reedy closed model
structure, Lemma 3.8 gives the necessary result. ∎

So far we have provided a model for the normalized singular (co)chain functor
for any (quasi)toric diagram and its composition with the classifying space
functor. Our next task is to construct a chain of quasiisomorphisms
establishing formality for toric diagrams over $\Bbbk=\mathbb{Z},\mathbb{Q}$
and formality for classifying space functor composed with quasitoric diagrams
with coefficients $\Bbbk=\mathbb{Z}$ or $\mathbb{Q}$, depending on the number
of prime-power factors. Here we follow a variant of the argument from [no-
ra-05] for simplicial sets in a similar way to [li-so-22].

Let $H^{*}(S^{1})\cong\Lambda[u]$. Choose a cocycle
$\varphi_{u}\in\overline{W}^{1}(S^{1})$ representing $u$ (see Lemma 3.8). This
choice defines the homomorphism $\varphi\colon
H^{*}(S^{1})\to\overline{W}^{*}(S^{1})$, where the product on the right is
trivial. Let

${\kappa\colon H^{*}(T)}$${H^{*}(S^{1})^{\otimes
r}}$${\overline{W}^{*}(S^{1})^{\otimes
r},}$$\scriptstyle{\cong}$$\scriptstyle{\otimes\varphi}$

be given by the composition of Künneth isomorphism and $\otimes\varphi$.

Similarly, given a cyclic group $G=\mathbb{Z}$, choose a generator $v$ (of
degree $2$) of the ring
$H^{*}(BG;\Bbbk)\cong G\otimes\Bbbk[v]$ for the Eilenberg-MacLane space $BG$.
For $G=\mathbb{Z}_{p^{n}}$ the ring $H^{*}(BG;\mathbb{Q})=0$ is trivial.
Choose a cocycle $\psi_{v}\in\overline{W}^{2}(BG)$ representing $v$ in sense
of homotopy equivalence from Lemma 3.8. Let $\psi\colon
H^{*}(BG)\to\overline{W}^{*}(BG)$ be given by $v^{q}\mapsto(\psi_{v})^{q}$,
where the product on the right is with respect to $\cup$-product.

Now let $G=\prod_{i=1}^{r}G_{i}$ be any direct product of $S^{1}$’s and prime-
power finite groups. Let $\Bbbk=\mathbb{Z}$ or $\mathbb{Q}$ if $G$ is
connected, and let $\Bbbk=\mathbb{Q}$, otherwise. Then there is a composition
of Künneth isomorphism and $\otimes\psi$ given as follows

${\kappa\colon
H^{*}(BG)}$${\bigotimes_{i}H^{*}(BG_{i})}$${\bigotimes_{i}\overline{W}^{*}(BG_{i}).}$$\scriptstyle{\cong}$$\scriptstyle{\otimes\psi}$

There is the following zig-zag of quasiisomorphisms

${\bigotimes_{i}\operatorname{Hom}(\overline{W}_{*}(G_{i});\
\Bbbk)}$${\operatorname{Hom}(\bigotimes_{i}\overline{W}_{*}(G_{i});\
\Bbbk)}$${\overline{W}^{*}(G),}$$\scriptstyle{\nabla^{*}}$

where $\nabla^{*}$ is the dual to the shuffle map of Eilenberg and Zilber. Let
$f\colon G\to G^{\prime}=\prod_{j=1}^{r^{\prime}}G^{\prime}_{j}$ be any group
homomorphism with the corresponding (not uniquely defined) matrix
$A=(a_{i,j})\in\operatorname{Mat}_{r^{\prime},r}(\mathbb{Z})$. Let

$A_{*}\colon\overline{W}_{*}(G_{i})^{\otimes
r}\to\overline{W}_{*}(G^{\prime}_{i})^{\otimes r^{\prime}},\
A^{*}\colon\overline{W}^{*}(G^{\prime}_{i})^{\otimes
r^{\prime}}\to\overline{W}^{*}(G_{i})^{\otimes r},$
$BA_{*}\colon\overline{W}_{*}(BG_{i})^{\otimes
r}\to\overline{W}_{*}(BG^{\prime}_{i})^{\otimes r^{\prime}},\
BA^{*}\colon\overline{W}^{*}(BG^{\prime}_{i})^{\otimes
r^{\prime}}\to\overline{W}^{*}(BG_{i})^{\otimes r},$

be chain complex morphisms induced on the degree $1$ and $2$, respectively
components, respectively, by the lattice morphisms with matrices $A$ and
$A^{t}$, respectively.

Let $f\colon T\to T^{\prime}$ be any homomorphism of compact tori of dimension
$r$, $r^{\prime}$, respectively. Let $f\colon G\to G^{\prime}$ be any
homomorphism of quasitori of ranks $r$, $r^{\prime}$, respectively.

######  Lemma 3.10.

[li-so-22, Lemma 4.19] There is the following commutative diagram of
$\operatorname{dgm}_{\Bbbk}$, $\Bbbk=\mathbb{Z},\mathbb{Q}$,

(12) ${H^{*}(T^{\prime})}$${\overline{W}^{*}(S^{1})^{\otimes
r^{\prime}}}$${\operatorname{Hom}(\overline{W}_{*}(S^{1})^{\otimes
r^{\prime}};\
\Bbbk)}$${\overline{W}^{*}(T^{\prime})}$${H^{*}(T)}$${\overline{W}^{*}(S^{1})^{\otimes
r}}$${\operatorname{Hom}(\overline{W}_{*}(S^{1})^{\otimes r};\
\Bbbk)}$${\overline{W}^{*}(T),}$$\scriptstyle{\kappa}$$\scriptstyle{f^{*}}$$\scriptstyle{A^{*}}$$\scriptstyle{\operatorname{Hom}(A_{*},\operatorname{Id})}$$\scriptstyle{\nabla^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{\kappa}$$\scriptstyle{\nabla^{*}}$

where any horizontal arrow is a quasiisomophism. There is the following
commutative diagram of $\operatorname{dgm}_{\Bbbk}$

(13)
${H^{*}(BG^{\prime})}$${\bigotimes_{j=1}^{r^{\prime}}\overline{W}^{*}(BG^{\prime}_{j})}$${\operatorname{Hom}(\bigotimes_{j=1}^{r^{\prime}}\overline{W}_{*}(BG^{\prime}_{j});\
\Bbbk)}$${\overline{W}^{*}(BG^{\prime})}$${H^{*}(BG)}$${\bigotimes_{i=1}^{r}\overline{W}^{*}(BG_{i})}$${\operatorname{Hom}(\bigotimes_{i=1}^{r}\overline{W}_{*}(BG_{i});\
\Bbbk)}$${\overline{W}^{*}(BG),}$$\scriptstyle{\kappa}$$\scriptstyle{Bf^{*}}$$\scriptstyle{BA^{*}}$$\scriptstyle{\operatorname{Hom}(BA_{*},\operatorname{Id})}$$\scriptstyle{\nabla^{*}}$$\scriptstyle{Bf^{*}}$$\scriptstyle{\kappa}$$\scriptstyle{\nabla^{*}}$

where any horizontal arrow is a quasiisomophism. Here
$\Bbbk=\mathbb{Z},\mathbb{Q}$ if both $G$ and $G^{\prime}$ are both connected,
and $\Bbbk=\mathbb{Q}$, otherwise.

###### Proof.

Let $H^{*}(T)\cong\Lambda[u_{1},\dots,u_{r}]$,
$H^{*}(T^{\prime})\cong\Lambda[u^{\prime}_{1},\dots,u^{\prime}_{r}]$ be the
isomorphisms given by Künneth theorem. The formulas

(14) $A^{*}(\psi_{u^{\prime}_{i}})=\sum_{q}a_{q,i}\psi_{u_{q}},\
f^{*}(u^{\prime}_{i})=\sum_{q}a_{q,i}u_{q},$

give (by taking products) the two left vertical arrows in (12). This proves
commutativity of the left square in (12). Choose the generator $u\in
H_{1}(S^{1})$ and let $\varphi_{u}\in\overline{W}_{1}(S^{1})$ be a cycle
representing $u$ in sense of homotopy equivalence from Lemma 3.8. Then
$\overline{W}^{*}(S^{1})\to\operatorname{Hom}(\overline{W}_{*}(S^{1});\
\Bbbk)$ is given by $\psi_{u}\mapsto(\varphi_{u})^{*}$, where
$(\varphi_{u})^{*}$ is the dual to $\varphi_{u}$. Let
$H_{*}(T)\cong\Lambda[w_{1},\dots,w_{r}]$,
$H^{*}(T^{\prime})\cong\Lambda[w^{\prime}_{1},\dots,w^{\prime}_{r}]$ be the
isomorphisms given by Künneth isomorphism. The formulas

(15) $f_{*}(w_{i})=\sum_{q}a_{q,i}w^{\prime}_{q},\
\operatorname{Hom}(A_{*},\operatorname{Id})((\varphi_{w_{i}})^{*})=\sum_{q}a_{q,i}(\varphi_{w^{\prime}_{q}})^{*},$

determine $f_{*}$ and $\operatorname{Hom}(f_{*},\operatorname{Id})$,
respectively. Then the commutativity of the middle square in (12) follows
directly from (14) and (15). Notice that there is the following diagram

(16) ${\overline{W}_{*}(S^{1})^{\otimes
r}}$${\overline{W}_{*}(T)}$${\overline{W}_{*}(S^{1})^{\otimes
r^{\prime}}}$${\overline{W}_{*}(T^{\prime}),}$$\scriptstyle{A_{*}}$$\scriptstyle{\nabla}$$\scriptstyle{f_{*}}$$\scriptstyle{\nabla}$

where $\nabla$ is the shuffle map. The explicit formula for the shuffle map
[ma-92, §29] and (15) imply that the diagram (16) is commutative. This implies
the commutativity of the right square in (12). For the classifying spaces
diagram the proof is similar, see [li-so-22]. Here we use Künneth isomorphism

$H^{*}(BG;\Bbbk)\cong\bigotimes_{i=1}^{r}H^{*}(BG_{i};\Bbbk).$

The proof is complete. ∎

######  Definition 3.11.

For any toric diagram $D\in[C,\operatorname{Top}]$ define the diagrams
$D^{1}=D^{1}(D)$,
$D^{2}=D^{2}(D)$ in $[C^{op},\operatorname{dgm}_{\Bbbk}]$ by

$D^{1}(I):=\overline{W}^{*}(S^{1})^{\otimes\operatorname{rk}D(I)},\ D^{1}(I\to
J):=A(D(I\to J))^{*},$
$D^{2}(I):=\operatorname{Hom}(\overline{W}_{*}(S^{1})^{\otimes\operatorname{rk}D(I)};\
\Bbbk),\ D^{2}(I\to J):=\operatorname{Hom}(A(D(I\to
J))_{*},\operatorname{Id}).$

######  Theorem 3.12.

For any toric diagram $D\in[C,\operatorname{Top}]$ there is the following
zigzag

(17)
${H^{*}(D)}$${D^{1}}$${D^{2}}$${\overline{W}^{*}(D)}$${C^{*}(D),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

of quasiisomorphisms for diagrams in $[C^{op},\operatorname{dgm}_{\Bbbk}]$,
where $\Bbbk=\mathbb{Z},\mathbb{Q}$. In particular, the diagram $D$ is formal.

###### Proof.

The claim follows by composing the chain of quasiisomorphisms from Lemma 3.10
with quasiisomorphisms from Lemma 3.8 on the right. ∎

Fix any quasitoric diagram $D\in[C,\operatorname{Top}]$. If any object of $D$
has $\leqslant 1$ prime-power factor, let $\Bbbk=\mathbb{Z},\mathbb{Q}$, and
let $\Bbbk=\mathbb{Q}$, otherwise. Define the diagrams $BD^{1}$, $BD^{2}$ in
$[C^{op},\operatorname{dgm}_{\Bbbk}]$ (depending on $\alpha$) by

$BD^{1}(I):=\bigotimes_{i=1}^{\operatorname{rk}D(I)}\overline{W}^{*}(BG_{i}(I)),\
D^{1}(I\to J):=BA(D(I\to J))^{*},$
$BD^{2}(I):=\operatorname{Hom}\biggl{(}\bigotimes_{i=1}^{\operatorname{rk}D(I)}\overline{W}_{*}(BG_{i}(I));\
\Bbbk\biggr{)},\ BD^{2}(I\to J):=\operatorname{Hom}(BA(D(I\to
J))_{*},\operatorname{Id}),$

where $D(I)=\prod_{i=1}^{\operatorname{rk}D(I)}D_{i}(I)$ is the decomposition
into direct product of copies of $S^{1}$’s and prime-power subgroups.

######  Theorem 3.13.

There is the following zigzag

(18)
${H^{*}(BD)}$${BD^{1}}$${BD^{2}}$${\overline{W}^{*}(BD)}$${C^{*}(BD),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

of quasiisomorphisms for diagrams in $[C^{op},\operatorname{dgm}_{\Bbbk}]$
(choice of $\Bbbk=\mathbb{Z},\mathbb{Q}$ depends on connectedness of objects
in $D$, as above). In particular, $BD$ is formal.

###### Proof.

The claim follows by composing the chain of quasiisomorphisms from Lemma 3.10
with quasiisomorphisms from Lemma 3.8 on the right. ∎

######  Theorem 3.14.

For any toric diagram $D$ the associated cohomological BKSS over
$\Bbbk=\mathbb{Q},\mathbb{Z}$ collapses at the second page with trivial
extension in $\Bbbk$-modules. One has

$H^{n}(\operatorname{hocolim}D;\
\Bbbk)\cong\bigoplus_{i+j=n}\operatorname{lim}^{i}sH^{j}(D;\ \Bbbk),\
n\geqslant 0.$

###### Proof.

Follows directly from Theorem 2.12 by using Theorem 3.12 and Lemma 2.9. ∎

Let $\alpha\colon D\to\kappa T$ be a morphism of quasitoric diagrams. Recall
that $\Bbbk=\mathbb{Z}$ or $\mathbb{Q}$ if every object of $D$ is connected,
and $\Bbbk=\mathbb{Q}$ otherwise. By using similar arguments one can prove the
following theorem for the diagram $BD$ of classifying spaces.

######  Theorem 3.15.

The cohomological BKSS of $H^{*}(BT)$-modules for $BD$ collapses at the second
page, and the extension is trivial in the category of $\Bbbk$-modules. One has
a group isomorphism

$H^{n}(\operatorname{hocolim}BD;\
\Bbbk)\cong\bigoplus_{i+j=n}\operatorname{lim}^{i}sH^{j}(BD;\ \Bbbk),\
n\geqslant 0.$

##  4\. (Co)homology of (co)sheaves on posets

In this section we recall the basic notions of cellular sheaf theory over
Alexandrov spaces by following [ay-17] (see [cu-14] for the detailed
overview). Let $\operatorname{abcat}_{R}$ be the category of finitely
generated graded modules (or algebras) over a commutative associative ring
$R$.

######  Definition 4.1 ([bj-84]).

A poset $C=C(X)$ is called a CW poset if $C$ is the poset of cells
$\\{\sigma_{c}\\}_{c\in C}$ in a regular finite CW-complex $X$. Let $\dim
C:=\dim X$ be the dimension of $C$.

Given a finite poset $C$, we regard it as a category, whose objects are all
objects of the poset and arrows are the corresponding partial order relations.
The definition of a CW poset given above differs from the one from [bj-84],
where any CW poset is assumed to have the least element corresponding to
$\varnothing$. Any CW poset $C$ admits a grading $\deg\colon C\to\mathbb{Z}$
by $\deg c:=\dim\sigma_{c}\geqslant-1$. For any two elements $p,q\in C(X)$
write $p\leqslant_{i}q$ if $p\leqslant q$ and and $\deg p=\deg q-i$. Given any
pair $p\leqslant_{1}q$, denote by $[q:p]:=[\sigma_{q}:\sigma_{p}]$ the
corresponding index in the regular CW-complex $X$. (The values of such indices
are $-1,0,1$.)

######  Example 4.2.

Consider a cone poset $C=\operatorname{Cone}P$ over a CW poset $P$. The poset
$C$ is obtained from $P$ by adding the least element $\hat{0}$ of degree $-1$.
The CW-complex of $C$ is the cone over $X$, where $X$ is the CW-complex
corresponding to $P$. Let $[p:\hat{0}]:=1$.

######  Definition 4.3 ([cu-14]).

A covariant (contravariant, respectively) functor $\mathcal{A}\colon
C\to\operatorname{abcat}$, is called a _cellular sheaf_ (_cellular cosheaf_ ,
respectively) with values in $\operatorname{abcat}_{R}$. The cellular
cohomology $H^{*}_{cw}(C;\mathcal{A})$ of a cellular sheaf $\mathcal{A}$ on
$C$ is by definition the cohomology $R$-module of the chain complex
$(C^{*}_{cw}(C,\mathcal{A}),d)$ (depending on $C$ and the index numbers of CW-
complex $X$ corresponding to $C$),

$C^{i}_{cw}(C,\mathcal{A}):=\Bbbk\langle c\times y|\ c\in C,\
y\in\mathcal{A}(c),\ \deg c=i\rangle,\ d(c\times
y):=\sum_{c\leqslant_{1}c^{\prime}}[c^{\prime}:c]\cdot
c^{\prime}\times\mathcal{A}(c\leqslant c^{\prime})y.$

Here $\Bbbk\langle S\rangle$ denotes the free $\Bbbk$-module with the basis
$S$. The cellular homology $H_{*}^{cw}(C;\mathcal{B})$ of a cellular cosheaf
$\mathcal{B}$ on $C$ is the homology $R$-module of the chain complex
$(C^{cw}_{*}(C,\mathcal{B}),d)$,

$C_{i}^{cw}(C,\mathcal{B}):=\Bbbk\langle c\times y|\ c\in C,\
y\in\mathcal{B}(c),\ \deg c=i\rangle,\ d(c\times
y):=\sum_{c^{\prime}\geqslant_{1}c}[c^{\prime}:c]\cdot
c^{\prime}\times\mathcal{B}(c^{\prime}\leqslant c)y.$

By a slight abuse of the notation we call a cellular (co)sheaf by a (co)sheaf
in the sequel.

######  Example 4.4.

For any diagram $D\in[C,\operatorname{Top}]$, the functor $H^{*}(D)$ is a
cosheaf of graded $\Bbbk$-algebras on $C$.

For any elements $c<_{2}d$ of a CW-poset $C$, the interval $(c,d)$ in $C$
consists precisely of $\\{c^{\prime},d^{\prime}\\}$, and the identity

(19) $[c,c^{\prime}][c^{\prime},d]+[c,d^{\prime}][d^{\prime},d]=0,$

holds [bj-84]. This implies that $d^{2}=0$ holds for cosheaf homology and
sheaf cohomology.

######  Definition 4.5 ([ay-17]).

For a cosheaf $\mathcal{A}\colon C\to\operatorname{abcat}_{R}$, the refinement
sheaf $s\mathcal{A}\colon\operatorname{ord}C\to\operatorname{abcat}_{R}$ is
the sheaf given by the formula

$s\mathcal{A}(c_{0}>c_{1}>\cdots>c_{s}):=\mathcal{A}(c_{s}),$
$s\mathcal{A}((c_{0}>c_{1}>\cdots>c_{s})\to(c^{\prime}_{0}>c^{\prime}_{1}>\cdots>c^{\prime}_{t})):=\mathcal{A}(c^{\prime}_{t}\to
c_{s}),$

For a morphism of cosheaves $\alpha\colon\mathcal{A}\to\mathcal{B}$ (that is,
a natural transfromation of the corresponding functors) define the induced
morphism $s\alpha\colon s\mathcal{A}\to s\mathcal{B}$ of sheaves by the
formula

$s\mathcal{A}(c_{0}>c_{1}>\dots>c_{s})=\mathcal{A}(c_{s})\overset{\alpha}{\to}\mathcal{B}(c_{s})=s\mathcal{B}(c_{0}>c_{1}>\dots>c_{s}).$

######  Remark 4.6.

The choice of this notation for refinement is explained as follows. The sheaf
$s\mathcal{A}$ may be regarded as the restriction of the simplicial
replacement
$\operatorname{srep}\mathcal{A}\in[\Delta^{op},\operatorname{abcat}_{R}]$ of
$\mathcal{A}$ (e.g. see proof of Lemma 2.9) to the collection of nondegenerate
simplices in the nerve $N_{\bullet}(C)$.

######  Definition 4.7.

For any poset $C$ the order complex $\operatorname{ord}C$ of $C$ is the
simplicial complex of of all strictly increasing chains in $C$. Notice that
$\operatorname{ord}C$ is a cellular poset with the index numbers

$[\sigma:\partial_{i}\sigma]=(-1)^{s-i},\ \sigma=(c_{0}>c_{1}>\cdots>c_{s}),\
\partial_{i}\sigma:=(c_{0}>\cdots>\widehat{c_{i}}>\cdots>c_{s}),$

where hat symbol means omitting the corresponding element in the chain.

######  Definition 4.8 ([cu-14]).

By definition, for a sheaf
$\mathcal{A}\in[\operatorname{ord}C,\operatorname{abcat}_{R}]$ the
corresponding cohomology $H^{*}(C;\mathcal{A})$ are the cohomology $R$-modules
of the cochain complex $C^{*}_{cw}(\operatorname{ord}C;\mathcal{A})$. The
sheaf $\mathcal{A}\in[\operatorname{ord}C,\operatorname{abcat}_{R}]$ is called
acyclic if $H^{i}(C;\mathcal{A})$ vanishes unless $i=0$.

The proof of the following proposition is well known.

######  Proposition 4.9 ([cu-14]).

For any cellular sheaf $\mathcal{A}$ on $\operatorname{ord}C$ with values in
$\operatorname{abcat}_{R}$, one has isomorphism of groups

$\operatorname{lim}^{*}s\mathcal{A}=H^{*}(C;\ \mathcal{A}).$

##  5\. Zeeman-McCrory spectral sequence for homology manifolds

In the present section we prove the the cohomological Zeeman-McCrory spectral
sequence (ZMSS, for short) collapse for (co)sheaves over oriented homology
manifolds. This implies the Poincaré-Lefschetz type duality for sheaf
cohomology. Our approach is similar to that from [mc-79, ay-17]. It implies
the collapse of the spectral sequence of the coskeletal filtration on
$\operatorname{hocolim}D$ for a toric diagram $D$ (or, more generally, a
formal diagram) over oriented homology manifold. The ZMSS collapse can be
alternatively obtained by the collapse of the dihomology bicomplex spectral
sequence, see [ch-23].

######  Definition 5.1 ([mc-79]).

Given a cellular poset $C$ of dimension $d$, define the pseudocell $G_{c}$ and
its boundary $\partial G_{c}$, $c\in C$ as the subsets of
$\operatorname{ord}C$ given by

$G_{c}=\\{c\leqslant c_{1}\leqslant\cdots\\},\ \partial
G_{c}=\\{c<c_{1}\leqslant\cdots\\}.$

Define the codimension $\operatorname{codim}c$ of $c\in C$ as the maximal
length of chain in $G_{c}$. The coskeletal filtration on $\operatorname{ord}C$
is by definition the filtration

(20) $\varnothing=C^{\prime}_{-1}\subseteq
C^{\prime}_{0}\subseteq\cdots\subseteq C^{\prime}_{d}=C^{\prime},\
C^{\prime}_{i}=\bigcup_{\operatorname{codim}G_{c}\geqslant d-i}G_{c},\
-1\leqslant i\leqslant d,$

on $C^{\prime}=\operatorname{ord}C$.

######  Remark 5.2.

Notice that $\operatorname{codim}c$ may differ from $d-\dim c$ (e.g. for
$c=\hat{0}$ in a cone poset), in general.

######  Definition 5.3 ([mc-79]).

The cosheaf $\mathcal{U}^{q}$ of local cohomology for $C$ is given by

$\mathcal{U}^{q}(c):=H^{q}(C,C\setminus\operatorname{star}^{\circ}_{C}c),$

and the map $\mathcal{U}^{q}(c<c^{\prime})$ is defined by the embedding
$\operatorname{star}^{\circ}_{C}c^{\prime}\to\operatorname{star}^{\circ}_{C}c$.
Here the open star $\operatorname{star}^{\circ}_{C}c=\\{c^{\prime}\in C\colon
c^{\prime}\geqslant c\\}$ of $c$ in $C$ denotes the subset (without partial
order relation) corresponding to cells contained in the cell of $c$ in the CW
complex $X$ of $C$. The set $C\setminus\operatorname{star}^{\circ}_{C}c$ is a
subposet of $C$. Define the cosheaf $\mathcal{H}^{q}$ on $C$ by

$\mathcal{H}^{q}(c):=H^{q+\operatorname{codim}c}(G_{c},\partial G_{c}).$

The map $\mathcal{H}^{q}(c<c^{\prime})$ is defined to be
$[c^{\prime}:c]m^{q}_{c<c^{\prime}}$, where $m^{q}_{c<c^{\prime}}$ is the
following composition

(21) $m^{q}_{c<c^{\prime}}\colon
H^{i+\operatorname{codim}c^{\prime}}(G_{c^{\prime}},\partial
G_{c^{\prime}})\cong H^{i+(\operatorname{codim}c-1)}(\partial G_{c},\partial
G_{c}\setminus G^{\circ}_{c^{\prime}})\to
H^{i+(\operatorname{codim}c-1)}(\partial G_{c})\to\\\ \to
H^{i+\operatorname{codim}c}(G_{c},\partial G_{c}),$

if $c^{\prime}<_{1}c$, i.e. $\deg c=\deg c^{\prime}-1$, and

$\mathcal{H}^{q}(c_{0}<c_{1})\circ\cdots\circ\mathcal{H}^{q}(c_{n-1}<c_{n}),$

for any saturated chain $c=c_{0}<_{1}\cdots<_{1}c_{n}=c^{\prime}$ in $C$,
otherwise. Here, $G^{\circ}_{c}:=G_{c}\setminus\partial G_{c}$.

The proof of the following lemma is similar to [ay-17]. We provide it in the
setting of CW posets.

######  Lemma 5.4 ([ay-17]).

The cosheaf $\mathcal{H}^{q}$ is well defined. One has

(22) $\mathcal{H}^{q}\cong\mathcal{U}^{d+q}.$

###### Proof.

For any elements $c,d\in C$ such that $c<_{2}d$ holds, there are precisely two
different elements $c<c^{\prime},d^{\prime}<d$ in $C$. (Such property holds
equally well for any interval $[a,b]$ in $C$.) For any chain
$\sigma\in\operatorname{ord}C$ containing a subchain
$\tau=(c\leqslant_{1}c^{\prime}\leqslant_{1}d)$, consider the chain
$\sigma^{\prime}$ obtained from $\sigma$ by replacing $\tau$ with
$\tau^{\prime}=(c\leqslant_{1}d^{\prime}\leqslant_{1}d)$. An operation of the
form $\sigma\to\sigma$ is called an elementary flip.

The consition $d^{2}=0$ for the first page of the spectral sequence of the
filtration (20) with coefficients in the constant sheaf $\kappa\mathbb{Z}$
coincides with

$m^{q}_{c<c^{\prime}}\circ m^{q}_{c^{\prime}<d}+m^{q}_{c<d^{\prime}}\circ
m^{q}_{d^{\prime}<d}=0.$

This identity together with (19) implies the claim for any two saturated
chains related to each other by a sequence of elementary flips. For any
$c\leqslant d\in C$, let $k,k^{\prime}$ be any two saturated chains from $c$
to $d$. The interval $[c,d]$ is a link in a sphere $[\hat{0},d]$ (considered
in the poset $\operatorname{Cone}C$). Therefore, $[c,d]$ is a pseudomanifold
[bj-84]. Hence, there exists a sequence of saturated chains

$k=m_{0},\ m_{1},\ \dots,\ m_{n}=k^{\prime},$

such that $m_{i+1}$ is obtained from $m_{i}$ by an elementary flip,
$i=0,\dots,n-1$. This proves that the definitions of arrows for
$\mathcal{H}(c\leqslant d)$ in terms of $k$ and $k^{\prime}$ coincide. The
isomorphism (22) follows by the argument from [mc-79]. ∎

Consider the spectral sequence of the coskeletal filtration on
$\operatorname{ord}C$ with coefficients in the sheaf $s\mathcal{A}$ (see
[mc-79]):

${}^{\prime}E^{i,j}_{1}=H^{i+j}(C_{j},C_{j-1};\ s\mathcal{A})\Rightarrow
H^{i+j}(C;\ \mathcal{A}).$

######  Lemma 5.5 ([ay-17]).

One has an isomorphism

$H^{i}(C_{j},C_{j-1};\
s\mathcal{A})\cong\bigoplus_{c\colon\operatorname{codim}c=d-j}H^{i}(G_{c},\partial
G_{c};\ s\mathcal{A}).$

If any stalk of $\mathcal{A}$ is torsion-free, then one has

$H^{i}(G_{c},\partial G_{c};\ s\mathcal{A})\cong H^{i}(G_{c},\partial
G_{c})\otimes\mathcal{A}(c),$

and

${}^{\prime}E_{2}^{i,j}\cong H_{d-i}^{cw}(C;\
\mathcal{H}^{j}\otimes\mathcal{A}).$

######  Corollary 5.6.

Let $\mathcal{A}$ be a cosheaf on a CW poset $C=C(X)$ of dimension $d$ such
that every stalk of $\mathcal{A}$ is torsion-free. Then ZMSS from the second
page takes the following form:

${}^{\prime}E^{i,j}_{2}=H_{d-i}^{cw}(C;\
\mathcal{H}^{j}\otimes\mathcal{A})\Rightarrow H^{i+j}(C;\ \mathcal{A}).$

Notice that if $\mathcal{A}$ is a cosheaf of $R$-modules (over a commutative,
associative ring $R$) then ZMSS is a spectral sequence of $R$-modules.

######  Definition 5.7.

A CW poset $C$ of dimension $d$ is called an orientable homology manifold
(over $\mathbb{Z}$) if

$\mathcal{U}^{d+i}=\begin{cases}\kappa\mathbb{Z},\ i=0,\\\ 0,\
\mbox{otherwise}.\end{cases}$

If, in addition, $H^{*}(\operatorname{ord}C)\cong H^{*}(S^{d})$, then $C$ is
called a homology sphere of dimension $d$.

######  Example 5.8.

The cone $C=\operatorname{Cone}P$ is a homology manifold of dimension $d$ iff
$P$ is a homology sphere of dimension $d-1$.

Corollary 5.6 directly implies the following theorem.

######  Theorem 5.9 ([mc-79]).

For any cosheaf $\mathcal{A}$ with torsion-free stalks over any homology
manifold $C$ of dimension $d$, there is an isomorphism of groups

(23) $H^{i}(C;\ \mathcal{A})\cong H_{d-i}^{cw}(C;\ \mathcal{A}).$

######  Definition 5.10 ([ay-17]).

Let $D\in[C,\operatorname{Top}]$ be any diagram over a CW poset $C$ of
dimension $d$. Define the coskeletal filtration on $C$

$\varnothing=C_{\geqslant d+1}\subset C_{\geqslant d}\subset\cdots\subset
C_{\geqslant-1}=C,\ C_{\geqslant i}:=\\{c\in C|\ \dim c\geqslant i\\},$

where $C_{\geqslant i}$ is the $i$-coskeleton subposet in $C$. It induces the
coskeletal filtration on $\operatorname{hocolim}D$

(24)
$\varnothing=\operatorname{hocolim}\tau^{*}_{d+1}D\subset\operatorname{hocolim}\tau^{*}_{d}D\subset\cdots\subset\operatorname{hocolim}D,$

where $\tau_{i}\colon C_{\geqslant i}\to C$ is the natural embedding functor.

The spectral sequence of the filtration (24) is given as follows.

######  Lemma 5.11 ([ay-17]).

For any toric diagram $D\in[C,\operatorname{Top}]$ the spectral sequence of
the coskeletal filtration on $\operatorname{hocolim}D$ is given as follows:

$C^{2}_{s,t}=\bigoplus_{i+j=d+t}H^{cw}_{d-s}(C;\ \mathcal{U}^{i}\otimes
H^{j}(D))\Rightarrow H^{s+t}(\operatorname{hocolim}D),$

######  Theorem 5.12.

Suppose that $C$ is an oriented homology manifold of dimension $d$. Then the
spectral sequence of the coskeletal filtration on $\operatorname{hocolim}D$
for a toric diagram $D$ collapses at the second page. In particular, one has

$H^{k}(\operatorname{hocolim}D)\cong\bigoplus_{s+t=k}H^{cw}_{d-i}(C;\
H^{t}(D)).$

###### Proof.

By Theorem 3.14 and Proposition 4.9, $H^{k}(\operatorname{hocolim}D)$ is a
direct sum of the corresponding cohomology groups of the refinement sheaf
$sH^{*}(D)$. By Theorem 5.9, these are isomorphic to the cellular homology of
the cosheaf $H^{t}(D)$. The proof is complete. ∎

##  6\. Faces and orbits for $T$-diagrams

In this section, we compare the coskeletal and orbit filtrations, as well as
compare the indexing category and orbit poset for homotopy colimits of
$T$-diagrams (Definition 6.3, Proposition 6.7). We characterize $T$-spaces
having orbit CW posets arising as homotopy colimits of $T$-diagrams (Theorem
6.12). Then the proof of Theorem 1 is given.

We recall the definitions of the orbit filtration and faces for a $T$-space
below. Let $X$ be a finite regular $T$-CW complex. Then $X$ is a Hausdorff
compact $T$-space. This implies that the orbit space $Q=X/T$ is a regular CW
complex, and that there are only finitely many distinct orbit types in $X$.

######  Definition 6.1.

Consider the orbit type filtration [br-72]

(25) $\varnothing=X_{-1}\subset X_{0}\subset\cdots\subset X_{k}=X,$

where $k$ is the rank of $T=(S^{1})^{k}$. It induces the filtration of the
orbit space $Q=X/T$

$\varnothing=Q_{-1}\subset Q_{0}\subset\cdots\subset Q_{k}=Q,$

by putting $Q_{i}:=\pi(X_{i})$, where $\pi\colon X\to Q$ is the quotient map.
Any connected component of $Q_{i}\setminus Q_{i-1}$ is called an open face of
$X$. The indexing set $C$ for the collection $\\{Q^{\circ}_{c}\\}_{c\in C}$ of
all open faces in $X$ is a finite (due to finitely many orbit types) partially
ordered set. The partial order is defined by $c\leqslant d$ iff
$Q_{d}\subseteq Q_{c}$, where $Q_{c}:=\overline{Q^{\circ}_{c}}$ and the bar
denotes the closure in $Q$. The set $Q_{c}$ is called a closed face of the
$T$-space $X$. The poset $C$ is called the orbit poset of the $T$-space $X$.

######  Remark 6.2.

Notice that $Q$ is a face of $X$ iff the orbit poset has the least element.
The orbit poset of a complete toric variety of complex dimension $n$ has the
least element (because there is a dense $(\mathbb{C}^{\times})^{n}$-orbit, and
the corresponding fan has the least element). For any smooth compact
$T$-manifold such a condition is also satisfied by the principal orbit theorem
[br-72, Theorem 3.1].

Let $C$ be any CW poset.

######  Definition 6.3.

We call a morphism of diagrams $\alpha\colon\kappa T\to D$ over $C$ a
$T$-diagram, if any arrow in $\alpha$ is an epimorphism of groups.

Any $T$-diagram $D\in[C,\operatorname{Top}]$ is a diagram with values in the
category of $T$-spaces. The $T$-action is transitive on each object of $D$.
Therefore, the stabilizers of the $T$-action
$S_{c}=\operatorname{Stab}_{x}D(c)$ on $D(c)$ depend only on $c\in C$. The
collection $\\{S_{c}|\ c\in C\\}$ with the natural embeddings of the isotropy
subgroups gives a well-defined quasitoric diagram $S=S(D)$ over $C$.

######  Definition 6.4.

We call $S$ the stabilizer diagram of a $T$-diagram $D$.

######  Remark 6.5.

It follows directly from the definition of a $T$-diagram $D$ that
$S_{c}\subseteq S_{d}$ holds for any $c\leqslant d\in C$, where $S$ is the
stabilizer diagram of $D$. In particular, a closure of any orbit stratum in
$\operatorname{hocolim}D$ is a union of orbit types of non-increasing
dimension.

By the orbit-stabilizer correspondence, there is a short objectwise-exact
sequence of diagrams over $C$ with values in groups

(26) ${\kappa 1}$${S}$${\kappa T}$${D}$${\kappa 1.}$

######  Definition 6.6.

We call a $T$-diagram $D$ over $C$ a reduced diagram if the inequality
$S_{c}\neq S_{d}$ holds for any $c<d\in C$. We call a $T$-diagram $D$ a
strongly reduced diagram if one has $\dim S_{c}\neq\dim S_{d}$ for any $c<d\in
C$.

The proof of the following proposition is straightforward.

######  Proposition 6.7.

Let $D$ be a $T$-diagram over $C$. Then $C$ is isomorphic to the orbit poset
of $\operatorname{hocolim}D$ iff $D$ is reduced. If $D$ is strongly reduced
then the coskeletal and orbit filtrations on $\operatorname{hocolim}D$
coincide.

For any reduced $T$-diagram $D$ over any CW poset $C$ the cells of the CW-
complex
$Q=(\operatorname{hocolim}D)/T$ for the $T$-space $\operatorname{hocolim}D$
are in bijection with the elements of the orbit poset $C$.

######  Remark 6.8.

Let $D\in[C,\operatorname{Top}]$ be a non-reduced $T$-diagram. Consider the
quotient morphism
$q\colon C\to C/\sim^{\prime}$ of posets induced by the equivalence relation

$c\sim^{\prime}d\Leftrightarrow S_{c}=S_{d}\ \&\ (c\geqslant d\vee c\leqslant
d).$

Then the homotopy pushdown diagram $q_{*}D$ [we-zi-zi-99] over the poset
$q(C)$ has object

$\lvert\operatorname{ord}q^{-1}(q(c))\rvert\times D(q(c)),\ c\in C,$

over $q(c)\in q(C)$, with the transitive $T$-action. The diagram $q_{*}D$ is
reduced (in the appropriate sense). However, $q_{*}D$ may not be a
$T$-diagram, because it has non-acyclic faces
$\lvert\operatorname{ord}q^{-1}(r)\rvert$, $r\in q(C)$, in general. Another
issue is that the poset $q(C)$ is not a graded poset, in general. The homotopy
pushdown with respect to

$c\sim^{\prime\prime}d\Leftrightarrow\dim S_{c}=\dim S_{d}\ \&\ (c\geqslant
d\vee c\leqslant d),$

gives a diagram with a non-transitive $T$-action on its objects, in general.

######  Remark 6.9.

It follows directly from the definition that for any strongly reduced
$T$-diagram there is the inequality $\dim C\leqslant\operatorname{rk}T$.

######  Definition 6.10 ([ha-ha-10]).

A $T$-space $X$ is called split if the projection $\pi\colon X\to Q=X/T$ onto
the orbit space of $X$ admits a $T$-equivariant section.

######  Lemma 6.11 ([ha-ha-10]).

A split $T$-space $X$ is $T$-equivariantly homeomorphic to

(27) $(Q\times T)/\sim,$ $(q,t)\sim(q^{\prime},t^{\prime})\Leftrightarrow\
q=q^{\prime}\in Q^{\circ}_{c}\ \&\ t^{-1}t^{\prime}\in S_{c}.$

######  Theorem 6.12.

Let $X$ be a finite $T$-CW complex such that the orbit poset $C$ of $X$ is a
CW-poset. Then $X$ is $T$-equivariantly homeomorphic to
$\operatorname{hocolim}D$ for some reduced $T$-diagram $D$ over $C$ iff the
$T$-space $X$ is split. In particular, every face of $X$ is contractible.

###### Proof.

Suppose that $X$ is split. Let $C$ be the orbit poset of the $T$-space $X$. By
Lemma 6.11, the space $X$ is $T$-equivariantly homotopy equivalent to (27).
Consider the reduced $T$-diagram $D$ over $C$ given by $D(c):=T/S_{c}$, and
the arrows in $D$ are the natural projections. The space (27) is equivariantly
homeomorphic to $\operatorname{hocolim}D$, see (3). The corresponding
homeomorphism is induced by the homeomorphism
$Q\simeq\lvert\operatorname{ord}C\rvert$ for the regular CW poset $C$ [bj-84,
Proposition 3.1]. This proves the claim in one direction.

Let $D$ be a reduced $T$-diagram $D$ over any finite poset $C$. Then
$\operatorname{hocolim}D$ is a $T$-space, because the realization functor for
diagrams in $T$-spaces is the same as in $\operatorname{Top}$. There is a
bijection between the orbit poset and $C$ by the reduced assumption. The face
of $\operatorname{hocolim}D$ corresponding to $c\in C$ is equal to the
contractible space $\lvert\operatorname{ord}C_{\geqslant c}\rvert$. Here,
$\operatorname{ord}(C_{\geqslant c})$ denotes the order complex of the
category $C_{\geqslant c}=c\downarrow C$. The projection $\pi\colon X\to
Q=\lvert\operatorname{ord}C\rvert$ has an equivariant section, given in terms
of (3) by the formula

$s\colon Q\to\operatorname{hocolim}D,\ s(x):=[(x,e_{D(c_{i})})],$

where $e_{D(c_{i})}$ denotes the identity of the group $D(c_{i})$. This proves
the claim in the opposite direction. ∎

We conclude this section by deducing the orbit spectral sequence collapse for
compact toric varieties.

######  Definition 6.13 ([fu-93]).

A normal reduced irreducible variety $X$ over $\mathbb{C}$,
$\dim_{\mathbb{C}}X=n$, is called a toric variety if the left action of
$(\mathbb{C}^{\times})^{n}$ on itself extends to an action on $X$.

Let $X_{\Sigma}$ be any compact toric variety with the corresponding complete
rational polyhedral polyhedral fan $\Sigma$ in
$N_{\mathbb{Q}}=N\otimes\mathbb{Q}$, $N=\operatorname{Hom}(S^{1},T^{d})$.
(Here it is required that any cone $\sigma\in\Sigma$ is salient, i.e. contains
no lines from $N_{\mathbb{Q}}$.) The variety $X_{\Sigma}$ is equipped with the
complex-analytical topology. Consider the diagram
$D_{\Sigma}\in[\Sigma,\operatorname{Top}]$, where $D_{\Sigma}(\sigma)$ is the
compact form of the algebraic torus corresponding to the orbit $X_{\sigma}$.
The arrow $D_{\Sigma}(\sigma\leqslant\tau)$ is defined by taking quotient by a
larger isotropy subgroup, $\sigma,\tau\in\Sigma$. The $T$-equivariant
homeomorphism of the underlying topological space for $X_{\Sigma}$ with
$\operatorname{hocolim}D_{\Sigma}$ follows by reduction to a quotient
construction over the order complex of $\Sigma$ [fr-10]. The latter is a
homotopy colimit of a diagram $D_{\Sigma}$.

######  Remark 6.14.

For any non-complete rational polyhedral fan $\Sigma$ there is a
$T$-equivariant homotopy equivalence of the quotient construction with
$\operatorname{hocolim}D_{\Sigma}$ from [fr-10].

The orbit poset of $X$ is identified with the set of cones $\Sigma$ ordered by
inclusion relation. Notice that $\Sigma$ is a homology manifold of dimension
$d$, namely, the cone over a homology sphere of dimension $d-1$. One can show
that $D_{\Sigma}$ is a strongly reduced $T$-diagram.

The next theorem is a direct corollary from Theorem 5.12 and Proposition 6.7.
It generalizes the result of [da-78, §12] to arbitrary complete rational
polyhedral fans (consisting of salient cones), with integral cohomology
coefficients. Collapse of the orbit spectral sequence for any complete toric
variety was proved with rational coefficients in [to-14]. The orbit spectral
sequence collapse for any nonsingular toric variety was obtained in [fr-06]
with any coefficients.

######  Theorem 6.15.

$H^{k}(X_{\Sigma};\ \mathbb{Z})\cong\bigoplus_{s+t=i}H^{cw}_{d-i}(\Sigma;\
H^{t}(D_{\Sigma})).$

######  Remark 6.16.

Notice that for an arbitrary rational polyhedral fan $\Sigma$ in
$N_{\mathbb{Q}}$ the orbit poset of $X_{\Sigma}$ has dimension different from
$d$, in general. For example, consider the fan of
$\mathbb{P}^{1}\times\mathbb{C}^{\times}$. Theorem 6.15 generalizes
straightforwardly to the case of an arbitrary complete rational polyhedral fan
$\Sigma$ such that the orbit poset is an oriented homology manifold.

######  Remark 6.17.

For a CW poset $C$ different from a homology manifold, the orbit spectral
sequence for a strongly reduced $T$-diagram does not collapse at the second
page, in general. All differentials of the homological orbit spectral sequence
for a Buchsbaum simplicial poset $C$ (in the case of complexity zero, i.e.
$\dim C=\operatorname{rk}T$) were explicitly described in [ay-16]; the
cohomological spectral sequence can be described in a similar way. In the
setting of this note, nontrivial higher differentials occur for the first
coskeleton $D_{\geqslant 1}$ for a $T$-diagram $D$ over a cone CW poset $C$,
i.e. for the spectral sequence of the truncated orbit filtration

$\varnothing=X_{-1}\subset X_{0}\subset\cdots
X_{k-1}=\operatorname{cosk}^{1}X,$

compare with (25).

##  7\. Borel space of a homotopy colimit

In this short section we describe the homotopy decomposition for the Borel
fibration of $\operatorname{hocolim}D$ for any $T$-diagram $D$. Then we recall
the definition of the Eilenberg-Moore spectral sequence in the case of the
Borel fibration.

Let $C$ be any connected (i.e. $\lvert\operatorname{ord}C\rvert$ is connected)
finite CW-poset. By (26), for any $T$-diagram $D\in[C,\operatorname{Top}]$
there is the respective stabilizer diagram $S$. Let
$\Bbbk=\mathbb{Q},\mathbb{Z}$.

######  Lemma 7.1.

For any $T$-diagram $D$ over a connected CW poset $C$, there is a commutative
diagram

(28)
${B_{T}\operatorname{hocolim}D}$${\operatorname{hocolim}B_{T}D}$${\operatorname{hocolim}BS}$${BT}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the horizontal arrows are homotopy equivalences.

###### Proof.

Consider $D\to 1$ as a morphism of diagrams over the category $C\times T$.
There is a commutative diagram

(29)
${\operatorname{hocolim}B_{T}D}$${\operatorname{hocolim}_{C}\operatorname{hocolim}_{T}D}$${\operatorname{hocolim}_{C}\operatorname{hocolim}_{T}\kappa
1}$${B_{T}\operatorname{hocolim}D}$${\operatorname{hocolim}_{T}\operatorname{hocolim}_{C}D}$${\operatorname{hocolim}_{T}\operatorname{hocolim}_{C}\kappa
1}$

By Fubini theorem for homotopy colimits [ch-sc-02, Theorem 24.9] (or by
applying commutation of colimits to a respective cofibrant replacement), both
vertical arrows in (29) are homotopy equivalences. The right vertical arrow in
(29) is the identity of $BT\times|\operatorname{ord}C|$. The post-composition
of these arrows with the projection gives the Borel fibration over $BT$. This
proves the claims about the left triangle in (28). The map $B_{T}D\to BS$ is a
homotopy equivalence of diagrams, because the $T$-action is transitive on
objects of $D$ with kernel $S$. (This is the fibration associated with a short
exact sequence of Lie groups.) This implies the claim about the right triangle
in (28). The proof is complete. ∎

Puppe’s theorem together with the above two lemmas directly implies the
following (see [li-so-22]).

######  Proposition 7.2 ([li-so-22]).

Let $D$ be any $T$-diagram over a connected CW poset $C$. Then the Borel
construction for the $T$-space $\operatorname{hocolim}D$ takes form

(30) $\operatorname{hocolim}D\to\operatorname{hocolim}BS\to BT.$

For a Serre fibration $p\colon E\to B$ with a connected fiber $F$ the
Eilenberg-Moore spectral sequence $(E^{*,*}_{*},d)$ of the fiber inclusion is
[mcc-01, p.233]

$E^{-i,j}_{2}=\operatorname{Tor}^{-i,j}_{H^{*}(B)}(H^{*}(E),\Bbbk)\Rightarrow
H^{j-i}(F;\ \Bbbk),$

where the first grading is cohomological (i.e. corresponds to element of the
resolution of $H^{*}(B)$-module $H^{*}(E)$) and the second is inner (i.e.
coincides with the degree of $H^{*}(E)$). (We follow the negative grading
notation that was used in [bu-pa-15], for example.) If $B$ is simply-
connected, then $(E^{*,*}_{*},d)$ converges strongly to $H^{*}(F)$, see
[mcc-01, p.233].

By applying Eilenberg-Moore spectral sequence to the fibration (30) one
obtains the following.

######  Corollary 7.3.

For any $T$-diagram $D$ over any connected CW poset $C$, the corresponding
Eilenberg-Moore spectral sequence for the Borel fibration of the $T$-space
$\operatorname{hocolim}D$ is isomorphic to

$\operatorname{Tor}^{-i,t}_{H^{*}(BT)}(H^{*}(\operatorname{hocolim}BS),\Bbbk)\Rightarrow
H^{t-i}(\operatorname{hocolim}D;\ \Bbbk).$

##  8\. EMSS and BKSS

In this section, we introduce a comparison spectral sequence (CSS, for short),
relating the Eilenberg-Moore (Corollary 7.3) and Bousfield-Kan (Theorem 3.14)
spectral sequences for any $T$-diagram $D$ (see Theorem 8.5). If the
stabilizer diagram $S$ associated to $D$ is toric and the sheaf $sH^{*}(BS)$
is acyclic, then CSS collapses at the second page and gives isomorphism of
EMSS and BKSS second pages, up to a change of the corresponding indices.

Let $D$ be any $T$-diagram over a CW poset $C$. Throughout this section, we
assume that $\lvert\operatorname{ord}C\rvert$ is connected, and
$\Bbbk=\mathbb{Q}$, or $\Bbbk=\mathbb{Z},\mathbb{Q}$ if all stabilizers of the
$T$-action on $\operatorname{hocolim}D$ are connected (i.e. the diagram $S$
for $D$ is toric). We need the following simple lemma.

######  Lemma 8.1.

Let $T\to T^{\prime}$ be a monomorphism of tori. Then

$\operatorname{Tor}^{-p,2q}_{H^{*}(BT^{\prime})}(H^{*}(BT),\Bbbk)\cong\begin{cases}H^{q}(T^{\prime}/T),p=q;\\\
0,\ \mbox{ else}.\end{cases}$

###### Proof.

Consider the Koszul resolution [we-97]

${0}$${H^{q}(T^{\prime}/T)}$${\cdots}$${H^{2q}(BT^{\prime})}$${H^{2q}(BT)}$${0,}$

for the $H^{*}(BT^{\prime})$-module $H^{*}(BT)$ in degree $2q$. Applying the
functor $-\otimes_{H^{*}(BT^{\prime})}\Bbbk$ one obtains trivial cohomology
except the boundary term in the cohomological degree $-q$. This gives the
required formula by the definition of $\operatorname{Tor}$ as a left derived
functor of tensor product [we-97]. ∎

######  Definition 8.2.

Denote the $(-i)$-th group of the Koszul resolution

(31) ${1}$${\Lambda^{t}(T)\otimes H^{*}(BT)}$${\Lambda^{t-1}(T)\otimes
H^{*}(BT)}$${\cdots}$${H^{*}(BT)}$${\Bbbk}$${1,}$$\scriptstyle{f_{t,1}}$$\scriptstyle{f_{t,t}}$

of the trivial $H^{*}(BT)$-module $\Bbbk$ by $R^{i}$. Let $D$ be a
$T$-diagram, with the associated stabilizer diagram $S$ (see (26)). Define the
$t$-graded bicomplex $(C^{i,j}_{t},d_{h},d_{v})$ (depending on $D$), where
(see Definition 4.3)

$C^{i,j}_{t}:=C^{j}(C;\ s(R^{2t-i}\otimes H^{*}(BS))^{2t}),$

power $2t$ denotes taking the respective grading of the $H^{*}(BT)$-module,
$d_{h}$ is the differential for the sheaf cohomology (increasing $j$ by $1$)
and $d_{v}$ is induced by the morphism from the Koszul resolution (increasing
$i$ by $1$).

######  Lemma 8.3.

Let $\mathcal{F}$, $\mathcal{G}$ be two sheaves on $C$. Suppose that
$\mathcal{G}=s\kappa V$ is a constant torsion-free sheaf for some flat
$\Bbbk$-module $V$. Then

$H^{i}(C;\mathcal{F}\otimes s\kappa V)\cong H^{i}(C;\mathcal{F})\otimes V.$

###### Proof.

Follows directly from the definitions by flatness of the $\Bbbk$-module $V$. ∎

######  Lemma 8.4.

One has

$H^{i}_{v}(H^{j}_{h}(C^{*,*}_{t}))=\operatorname{Tor}^{-(2t-i),2t}_{H^{*}(BT)}(\operatorname{lim}^{j}sH^{*}(BS),\Bbbk);$
$H^{j}_{h}(H^{i}_{v}(C^{*,*}_{t}))=\begin{cases}\lim^{j}sH^{t}(D),\ i=t,\\\
0,\ i\neq t.\end{cases}$

###### Proof.

We make the following computation.

(32) $H^{j}_{h}(C^{i,*}_{t})=H^{j}(C;\ s(R^{2t-i}\otimes
H^{*}(BS))^{2t})\cong\biggl{(}R^{2t-i}\otimes H^{j}(C;\
sH^{*}(BS))\biggr{)}^{2t}\cong\\\
\cong\biggl{(}R^{2t-i}\otimes\operatorname{lim}^{j}H^{*}(BS)\biggr{)}^{2t}.$

The last isomorphism is given by Lemma 8.3, because the components $R^{*}$ of
the Koszul resolution (31) have no $\Bbbk$-torsion. The above isomorphisms
respect the vertical differential whose action on the right-hand side of (32)
is induced by $f_{t,*}\otimes\operatorname{Id}$, see (31). Therefore, the
first equality of the lemma follows by the definition of
$\operatorname{Tor}$-groups using the resolution $(P^{*},f_{t,*})$. By the
definition of the $\operatorname{Tor}$-groups (using the resolution (31)), we
find

$H^{j}_{h}(H^{i}_{v}(C^{*,*}_{t}))=H^{j}(C;\
\operatorname{Tor}^{-i,2t}_{H^{*}(BT)}(H^{*}(BS),\Bbbk)).$

The second equality of the lemma now follows from this computation by Lemma
8.1. ∎

######  Theorem 8.5.

For any $T$-diagram $D$ there exists the $t$-graded spectral sequence
($t\in\mathbb{Z}$) over $\Bbbk=Q$ or $\Bbbk=\mathbb{Z}$ (if the stabilizers
are connected)

(33)
$\operatorname{Tor}^{-(2t-i),2t}_{H^{*}(BT)}(\operatorname{lim}^{j}H^{*}(BS),\Bbbk)\Rightarrow\operatorname{lim}^{i+j-t}sH^{t}(D).$

###### Proof.

Follows directly by considering the bicomplex spectral sequences corresponding
to $(C^{i,j}_{t},d_{h},d_{v})$ and using Lemma 8.4. ∎

######  Corollary 8.6.

Suppose that the sheaf $sH^{*}(BS)$ is acyclic. Then

(34)
$\operatorname{Tor}^{-i,2t}_{H^{*}(BT)}(H^{*}(\operatorname{colim}BS),\Bbbk)\cong\operatorname{lim}^{-i+t}sH^{t}(D),$
(35) $\operatorname{lim}^{i}sH^{j}(D)=0,\ i>j,$

and BKSS and EMSS for $D$ are isomorphic from the second pages up to a linear
change of indices. In particular, EMSS for $D$ collapses at the second page.

###### Proof.

By Theorem 3.15, the BKSS for the diagram $BS$ collapses. By the acyclicity
assumption this implies the isomorphism

(36) $H^{*}(\operatorname{colim}BS)\cong\operatorname{lim}sH^{*}(BS),$

of $H^{*}(BT)$-modules (for a different proof see [no-ra-05]). Furthermore,
under the same assumption the spectral sequence (33) collapses. This implies
(34). The identity (35) follows by using bar-resolution to compute
$\operatorname{Tor}$-groups from pigeonhole principle. We prove the last
claim. We observe that EMSS and BKSS converge to the same object:

$\operatorname{Tor}^{-i,2t}_{H^{*}(BT)}(\operatorname{lim}sH^{*}(BS),\Bbbk)\Rightarrow
H^{2t-i}(\operatorname{hocolim}D),$
$\operatorname{lim}^{-i+t}sH^{t}(D)\Rightarrow
H^{2t-i}(\operatorname{hocolim}D).$

Now the claim follows by comparison from the collapse of BKSS (Theorem 3.14)
and EMSS (see (34)). ∎

######  Remark 8.7.

For an acyclic sheaf $sH^{*}(BS)$, one has $sH^{*}(BS)=\kappa\Bbbk$. Hence,
$\lvert\operatorname{ord}C\rvert$ is $\Bbbk$-acyclic, because

$H^{i}(C;\ \kappa\Bbbk)\cong H^{i}(\lvert\operatorname{ord}C\rvert;\ \Bbbk).$

######  Remark 8.8.

The comparison spectral sequence (33) measures commutativity of
$\operatorname{Tor}$ and $\lim$, because one has

$\operatorname{lim}^{j}s\operatorname{Tor}^{-i,2t}_{H^{*}(BT)}(H^{*}(BS),\Bbbk)=\begin{cases}\operatorname{lim}^{j}sH^{t}(D),\
i=t;\\\ 0,\ \mbox{ else}.\end{cases}$

The functors $-\otimes_{H^{*}(BT)}\Bbbk$ and $\lim$ are right- and left-exact,
respectively. Therefore, there is no associated Grothendieck spectral sequence
[we-97, pp.150–153] and the respective bicomplex has to be inspected on its
own.

##  9\. Equivariant formality for toric diagrams

In this section we recall the notion and criteria of equivariant formality for
a $T$-space $X$. After that, we characterize when a $T$-diagram $D$ over an
oriented homology manifold is equivariantly formal (Theorem 9.7).

######  Theorem 9.1 ([fr-pu-06, al-fr-pu-14]).

Let $X$ be a $T$-space, where $T$ has rank $k$. Let
$\Bbbk=\mathbb{Z},\mathbb{Q}$. Then the following conditions (over $\Bbbk$)
are equivalent:

  1. (i)

The inclusion of the fibre $\iota\colon X\to B_{T}X$ induces a surjection
$\iota^{*}\colon H_{T}^{*}(X)\to H^{*}(X)$. Equivalently, the map
$H^{*}_{T}(X)\otimes_{H^{*}(BT)}\Bbbk\to H^{*}(X)$ induced by $\iota^{*}$ is
an isomorphism.

  2. (ii)

The Serre spectral sequence of the fibration $X\hookrightarrow B_{T}X\to BT$
collapses at $E_{2}$ level, i.e. $E_{2}=E_{\infty}$.

  3. (iii)

$\operatorname{Tor}^{j}_{H^{*}(BT)}(H_{T}^{*}(X),\Bbbk)=0$ for all $j>0$.

  4. (iv)

$\operatorname{Tor}^{1}_{H^{*}(BT)}(H_{T}^{*}(X),\Bbbk)=0$.

The above conditions are implied by the next one

  1. (v)

The following sequence of $H^{*}(BT)$-modules is exact

(37)
${0}$${H^{*}_{T}(X)}$${H^{*}_{T}(X_{0})}$${H^{*+1}_{T}(X_{1},X_{0})}$${\cdots}$${H^{*+k}_{T}(X_{k},X_{k-1})}$${0.}$

Let $\Bbbk=\mathbb{Q}$, or let $\Bbbk=\mathbb{Z},\mathbb{Q}$ if any stabilizer
of the $T$-space $X$ is connected. Then the condition $(v)$ is equivalent to
$(i)$—$(iv)$, and $H^{*}_{T}(X)$ is a free $H^{*}(BT)$-module.

######  Definition 9.2.

A $T$-space $X$ satisfying conditions $(i)-(iv)$ of Theorem 9.1 is called
equivariantly formal. The sequence (37) is called Atiyah-Bredon-Franz-Puppe
sequence, or ABFP sequence, for short.

The second page of the comparison SS collapses in the case of equivariant
formality for $\operatorname{hocolim}D$, yielding the following.

######  Corollary 9.3.

Let $\Bbbk=\mathbb{Q}$, or $\Bbbk=\mathbb{Z},\mathbb{Q}$ if all stabilizers of
the $T$-action on $\operatorname{hocolim}D$ are connected. Suppose that the
sheaf $sH^{*}(BS)$ is acyclic and $\operatorname{hocolim}D$ is equivariantly
formal. Then one has

$\operatorname{lim}^{i}sH^{j}(D)\cong\begin{cases}\operatorname{Tor}^{0,i}_{H^{*}(BT)}(\operatorname{lim}^{i}sH^{*}(BS),\Bbbk),\
i=j,\\\ 0,\ \mbox{otherwise}.\end{cases}$

###### Proof.

Follows directly from Theorem 8.5 by using the condition $(iii)$ of Theorem
9.1. ∎

The proof of the following fact is similar to the proof of [ay-ma-23,
Proposition 2.4], because all faces of $\operatorname{hocolim}D$ are acyclic
(see Section 6).

######  Lemma 9.4.

Let $C$ be an oriented homology manifold of dimension $d$, and let
$D\in[C,\operatorname{Top}]$ be a $T$-diagram. Then the $(i,j)$-th element of
the second page for the equivariant cohomology spectral sequence of the
coskeletal filtration on $\operatorname{hocolim}D$ is isomorphic to the group

$H^{cw}_{d-i}(C;\ H^{j}(BS)).$

######  Remark 9.5.

The isomorphism of Lemma 9.4 is in the category of groups, not
$H^{*}(BT)$-modules. In other words, we have shown that the ABFP-sequence is
exact in $\Bbbk$-modules over oriented homology manifolds. Acyclicity of
$sH^{*}(BS)$ does not imply neither equivariant formality, nor
$H^{*}(BT)$-exactness of ABFP (37), in general. For example, consider the
moment-angle complex $\mathcal{Z}_{P}$ over any convex simplicial polytope
$P$, equipped with the action of torus $T$ of rank equal to the number of
vertices in $P$ (for the definition see e.g. [bu-pa-15]). The space
$\mathcal{P}$ is homotopy equivalent to the homotopy colimit of a $T$-diagram
$D$. The sheaf $sH^{*}(BS)$ (where $S$ is the stabilizer diagram for $D$) is
acyclic (e.g. see [no-ra-05]). However, it is known that
$H^{odd}(\mathcal{Z}_{P})\neq 0$ for some $P$ [bo-me-06].

######  Lemma 9.6.

Let $C$ is an oriented homology manifold of dimension $d$ and let
$D\in[C,\operatorname{Top}]$ be a $T$-diagram. If $\operatorname{hocolim}D$ is
equivariantly formal then the sheaf $sH^{*}(BS;\Bbbk)$ is acyclic. In
particular, $|\operatorname{ord}C|$ is $\Bbbk$-acyclic.

###### Proof.

By Theorem 5.9, there is the isomorphism of $\Bbbk$-modules

$H^{cw}_{d-i}(C;\ H^{*}(BS))\cong H^{i}(C;\ sH^{*}(BS)).$

However, by Theorem 9.1 and Lemma 9.4, equivariant formality implies that

(38) $H^{cw}_{d-i}(C;\ H^{j}(BS))=0,$

holds for any $i\neq 0$. Therefore, $sH^{*}(BS)$ is acyclic. The last claim
follows trivially. ∎

######  Theorem 9.7.

Let $C$ is an oriented homology manifold of dimension $d$ and let
$D\in[C,\operatorname{Top}]$ be a $T$-diagram. Then $\operatorname{hocolim}D$
is equivariantly formal iff $H^{odd}(\operatorname{hocolim}D;\Bbbk)=0$. In the
case when these conditions are satisfied, one has the isomorphism
$H^{*}(BT)$-modules

(39) $H^{*}_{T}(\operatorname{hocolim}D)\cong\lim sH^{*}(BS).$

and the isomorphism of abelian groups

(40) $H^{2i}(\operatorname{hocolim}D)\cong\operatorname{lim}^{i}sH^{i}(D),\
i\geqslant 0.$

Furthermore, if $\Bbbk=\mathbb{Z}$ and the stabilizers are connected, then
$H^{*}(\operatorname{hocolim}D;\ \mathbb{Z})$ is torsion-free.

###### Proof.

The condition $H^{odd}(\operatorname{hocolim}D;\Bbbk)=0$ implies degeneration
of Serre SS of the Borel fibration by dimensional reasons, i.e. equivariant
formality of $\operatorname{hocolim}D$. Suppose that $\operatorname{hocolim}D$
is equivariantly formal. Then the sheaf $sH^{*}(BS)$ is acyclic by Lemma 9.6.
This together with equivariant formality (see Theorem 9.1, implies
$H^{odd}(\operatorname{hocolim}D;\Bbbk)=0$ by Corollary 8.6. The identity (39)
follows by Theorem 3.15 using the acyclicity of $sH^{*}(BS)$. Therefore, (40)
follows directly by Theorem 3.14 and Corollary 9.3. Torsion-freeness follows
directly from Theorem 9.1 by freeness criterion for equivariant cohomology
$H^{*}(BT)$-module [al-fr-pu-14]. The acyclicity of
$\lvert\operatorname{ord}C\rvert$ follows from acyclicity of $sH^{*}(BS)$. ∎

######  Remark 9.8.

Let $X$ be an oriented real $T$-manifold of complexity $0$ (i.e. $2n=\dim
X=2\dim T$) with non-empty fixed point set. By [ma-pa-06],
$H^{odd}(X;\mathbb{Z})$ is equivalent to acyclicity of proper (i.e. $\neq Q$)
faces in $Q=X/T$ and local standardness of the $T$-action on $X$. The local
standardness condition implies that $Q$ is a manifold with corners, and
therefore a homology manifold (with orientation descending from $X$ to
$X/T=Q$). Suppose in addition that the orbit poset of $X$ is a CW poset. In
such particular case, we obtain the odd cohomology vanishing for equivariantly
formal $X$, compare with [ma-pa-06].

######  Remark 9.9.

Any compact toric variety $X_{\Sigma}$ has connected isotropy groups. It is
well known that the equivariant formality of $X_{\Sigma}$ is equivalent to
vanishing of odd cohomology, e.g. see [fr-10, Lemma 5.1].

Let $C$ is an oriented homology manifold of dimension $d$. Let $D$ be a
reduced $T$-diagram over $C$. Let $D_{\leqslant c}:=\iota_{c}^{*}D$, where
$\iota_{c}\colon C_{\geqslant c}\to C$ is the natural embedding of the filter
corresponding to $c\in C$. The space $\operatorname{hocolim}D_{\leqslant c}$
is the closure of orbit corresponding to $c\in C$ (see Proposition 6.7). The
following statement is known for torus manifolds [ma-pa-06] and for complete
toric varieties [fr-10, Proposition 1.5].

######  Corollary 9.10.

If $\operatorname{hocolim}D$ is equivariantly formal then any orbit closure in
$\operatorname{hocolim}D$ is equivariantly formal, has vanishing odd
cohomology and is torsion-free.

###### Proof.

Notice that $C_{\geqslant c}$ is an oriented homology manifold. The condition
of connected stabilizers is preserved under taking quotient by a subtorus in
$T$. Then by Theorem 9.7 the claim follows from the identity
$H^{odd}(\operatorname{hocolim}D_{\leqslant c})=0$. This identity is proved by
a similar argument to [fr-10, Proposition 6.2]. ∎

##  10\. Betti numbers for some $T$-diagrams

Let $C$ be an oriented homology manifold of dimension $d$. Let $D$ be a
(reduced) $T$-diagram, and $\Bbbk=\mathbb{Z},\mathbb{Q}$ depending on the
stabilizers of the $T$-action on $\operatorname{hocolim}D$, as above. Suppose
that $\operatorname{hocolim}D$ is equivariantly formal (hence, $H^{*}(BS)$ is
acyclic). Then by Theorem 9.7 the groups $H^{*}(\operatorname{hocolim}D;\
\Bbbk)$ are completely determined by the their ranks (called Betti numbers).
In this section we determine the even Betti numbers below (the odd Betti
numbers vanish by Theorem 9.7), see Proposition 10.1. Notice that a similarly
looking formula was obtained in [ch-23] for tropical toric orbifolds. Then we
proceed by the description of cohomology groups of skeleta for a
$T$-characteristic diagram $D$ (Theorem 10.12). There we use the Taylor-type
complex from [ay-17]. The explicit formula for the respective bigraded Betti
numbers is given in the case of equivariantly formal $D$ (Corollary 10.13). We
show that it agrees with the computations for the skeleta of
$\mathbb{C}P^{m-1}$ following from [fu-19] and [gr-th-07].

Let $D$ be a $T$-diagram, $k=\operatorname{rk}T$,
$s_{c}=\operatorname{rk}S_{c}$, where $S$ is the stabilizer diagram
corresponding to $D$.

######  Proposition 10.1.

Suppose that $\operatorname{hocolim}D$ is equivariantly formal and that the
sheaf $sH^{*}(BS)$ is acyclic. Then one has

(41) $b_{2i}(\operatorname{hocolim}D)=(-1)^{i+1}\sum_{c\in
C}\tilde{\chi}(\operatorname{ord}C_{>c})\binom{k-s_{c}}{i},\ i\geqslant 0.$

###### Proof.

Consider the cochain complex $(C^{*},d)=(C^{i}(C,sH^{*}(D)),d)$ with the
standard differential. By the standard property of the reduced Euler
characteristic of a cochain complex, one has

$\tilde{\chi}(C^{i}(C,sH^{*}(D)))=\tilde{\chi}(H^{i}(C,sH^{*}(D))).$

By the definition of the cochain complex and simplicial homology of the chain
complex $C_{\geqslant c}$, one has

$\tilde{\chi}(C^{*}(C,sH^{*}(D)))=\sum_{i=-1}^{d}(-1)^{i}\sum_{c\in
C}\operatorname{rk}\tilde{C}^{i}(\operatorname{ord}C_{>c})\cdot\operatorname{rk}D(c)=\sum_{c\in
C}\tilde{\chi}(\operatorname{ord}C_{>c})\binom{k-s_{c}}{i}$

In the last equality we use $\operatorname{rk}H^{i}(D(c))=\binom{k-s_{c}}{i}$.
By Corollary 9.3,

$\tilde{\chi}(H^{i}(C,sH^{*}(D)))=(-1)^{i+1}\cdot
b_{2i}(\operatorname{hocolim}D).$

(The degree shift by $1$ is because we treat zero degree of the corresponding
cochain complex as augmentation in the definition of Euler characteristic.)
The identity (41) now follows directly from the last three identities. ∎

######  Remark 10.2.

By Lemma 9.6, the acyclicity condition on the sheaf $sH^{*}(BS)$ can be
omitted, provided that $C$ is an oriented homology manifold.

######  Example 10.3.

Let $\Sigma$ be the fan of any compact nonsingular toric variety $X$ of
dimension $d$. Then

$\operatorname{ord}\operatorname{lk}_{\Sigma}I\simeq S^{d-1-|I|},\
s_{I}=d-|I|,$

where

$\operatorname{lk}_{\Sigma}I=\Sigma_{>I}=\\{J\in\Sigma\colon I\sqcup
J\in\Sigma\\},$

is the link subposet of $I$ in $\Sigma$. Therefore, the right-hand side of
(41) coincides with

$(-1)^{i+1}\sum_{j=0}^{d}(-1)^{d-1-j}f_{i-1}(\Sigma)\binom{d-j}{i}=h_{d-i}(\Sigma),$

by the definition of an $h$-number (e.g. see [bu-pa-15]). Here,
$f_{i}(\Sigma)$ denotes the number of $(i-1)$-dimensional elements in
$\Sigma$, $f_{-1}(\Sigma):=1$. This agrees with the known description of Betti
numbers for complete toric varieties [fu-93]. A similar computation holds for
toric manifolds, by considering the face poset of a polytope dual to simple
polytope [da-ja-91].

######  Definition 10.4.

A poset $C$ is called a simplicial poset if $C$ has the least element
$\hat{0}$ and the ideal $C_{\leqslant c}$ is isomorphic to the Boolean lattice
(i.e. the face lattice of a simplex).

Any simplicial poset $C$ is a CW poset. The CW-complex $X$ of $C$ is the cone
$\lvert\operatorname{ord}C\rvert$, where to $c\in C$ corresponds the simplex
$\Delta_{\sigma_{c}}$ for a unique saturated chain $\sigma_{c}$ in
$[\hat{0},c]$. Let $C$ be a simplicial poset of dimension $d$. We may identify
an element $c$ of $C$ with the set of its atoms $I=a(c)$ by writing $I\in C$.

######  Definition 10.5.

Let $D$ be any $T$-diagram over any simplicial poset $C$. We call $D$ a
$T$-characteristic diagram (say, over $\mathbb{Z}$) if the stabilizer diagram
$S$ of $D$ satisfies the conditions

(42) $S(c)=\prod_{i\in a(c)}S(i)\subseteq T,\ \underset{i\in
a(C)}{\operatorname{span}}S(i)=T,$

where $a(C)$ denotes the set of dimension $0$ elements in $C$, $\Pi$ denotes
the direct product of groups and $\operatorname{span}$ denotes the group
generated in $T$ by the respective subgroups. (Notice that $\hat{0}$ is not an
atom, because it has dimension $-1$.)

Let $\mathcal{J}_{*}$ be the (graded) sheaf defined by the ideals
$\mathcal{J}_{*}(c)=H_{*}(T)\cdot H_{>0}(S(c))$ in the exterior algebra
$H_{*}(T)$, and the arrows of $\mathcal{J}_{*}$ are induced by $H_{*}(S)$.
There is the exact sequence of sheaves

(43) ${0}$${\mathcal{J}_{*}}$${\kappa H_{*}(T)}$${H_{*}(D)}$${0.}$

Let $\mathcal{J}^{*}=\operatorname{Hom}(\mathcal{J}_{*},\ \mathbb{Z})$ be the
cosheaf over $C$. It follows from the assumption on the stabilizers that (43)
splits objectwisely. This implies that the dual to (43) sequence

(44) ${0}$${H^{*}(D)}$${\kappa H^{*}(T)}$${\mathcal{J}^{*}}$${0,}$

of cosheaves is exact.

######  Lemma 10.6.

One has

(45) $H^{i}(C;\ sH^{*}(D))\cong\begin{cases}0,\ i=0,\\\ H^{i-1}(C;\
s\mathcal{J}^{*}),\ i\geqslant 2.\end{cases}$

The following sequence is exact

(46) ${0}$${H^{*}(T)}$${\lim s\mathcal{J}^{*}}$${\lim^{1}sH^{*}(D)}$${0.}$

###### Proof.

Notice that any arrow in the cosheaf $H^{*}(D)$ is a monomorphism of groups.
Therefore,

$\lim H^{*}(D)=\bigcap_{i\in a(C)}H^{*}(D(i))\subset H^{*}(D(\hat{0})),$

where the intersection is taken over the set $a(C)$ of all atoms in $C$. The
nondegeneracy condition of $S$ from Definition 10.5 implies that such an
intersection is trivial. By Lemma 8.3 one has

$H^{i}(C;\ s\kappa H^{*}(T))\cong H^{i}(\operatorname{ord}C;\
\mathbb{Z})\otimes H^{*}(T).$

Therefore, $s\kappa H^{*}(T)$ is acyclic because $C$ has the least element.
Now the claim follows by the long exact sequence in sheaf cohomology of the
refinement (i.e. applying the functor $s$) for (44). ∎

The following lemma is related to the dual version of [ay-17, Corollary 2];
however, we state it for an arbitrary simplicial poset (whereas in [ay-17] the
isomorphism is provided only for homological manifolds) and use a slightly
different proof.

######  Construction 10.7 ([ay-17]).

Let $\mathcal{P}_{*}$ be the sheaf given by

$\mathcal{P}_{*}(c):=H_{*}(T)\cdot H_{top}(S(c)),$

and the arrows of $\mathcal{P}_{*}$ are defined by division of top degree
exterior forms. Let $\mathcal{P}^{*}=\operatorname{Hom}(\mathcal{P}_{*},\
\mathbb{Z})$ be the cosheaf over $C$. Consider the sequence of cosheaf
morphisms

(47)
${0}$${\mathcal{J}^{*}}$${\mathcal{R}_{0}}$${\cdots}$${\mathcal{R}_{d}}$${0,}$$\scriptstyle{\eta^{*}}$$\scriptstyle{\xi^{*}}$$\scriptstyle{\xi^{*}}$
$\mathcal{R}_{i}(c):=\bigoplus_{\dim
c^{\prime}=i}[\mathcal{P}^{*}(c^{\prime})]_{\leqslant c^{\prime}},$

where $[V]_{\leqslant c}$ denotes the pushforward cosheaf with respect to
filter inclusion for any abelian group $V$:

$[V]_{\leqslant c}(c^{\prime})=\begin{cases}V,\ \mbox{if }c^{\prime}\leqslant
c,\\\ 0,\ \mbox{otherwise}.\end{cases}$

The morphism $\eta^{*}(c,x)$ is given by the embeddings corresponding to the
factors of $x\in J^{*}(c)$. The $(d^{\prime},d)$-component of the morphism
$\xi^{*}(c)$, $\dim d^{\prime}=\dim d-1$, is the map
$\mathcal{P}^{*}(d^{\prime})\to\mathcal{P}^{*}(d)$ that is dual to the natural
embedding $\mathcal{P}_{*}(d)\to\mathcal{P}_{*}(d^{\prime})$ of ideals.

######  Lemma 10.8 ([ay-17, §4]).

For any $T$-characteristic diagram $D$ there is the isomorphism

(48) $\operatorname{lim}^{i}s\mathcal{J}^{*}\cong\lim s\mathcal{R}_{i}$

respecting the (co)sheaf grading.

###### Proof.

The sequence (47) of cosheaves is exact. This is verified at each stalk by the
argument from [ay-17] based on inclusion-exclusion formula. The sheaf
$s\mathcal{R}_{i}$ is acyclic, because $C$ has the least element (being a
simplicial set). (In fact, $s\mathcal{R}_{i}$ is a direct sum of injective
sheaves [cu-14]. Therefore, the refinement of (47) is an injective resolution
for $s\mathcal{J}^{*}$.) This implies the claim by considering the associated
spectral sequence in cohomology with coefficients in refinement of (47). ∎

######  Definition 10.9.

Let $C$ be a simplicial poset of dimension $d$. Let

$\iota_{q}\colon\mathrm{sk}^{q}C=\\{c\in C\colon\dim c\leqslant q\\}\to C,$

be the natural embedding of the $q$-skeleton of $C$. Let $D$ be a
$T$-characteristic diagram over $C$. We call
$\operatorname{hocolim}\iota_{q}^{*}D$ a $q$-skeleton of
$\operatorname{hocolim}D$.

######  Corollary 10.10.

$\operatorname{lim}^{i}s\iota_{q}^{*}\mathcal{J}^{*}\cong\operatorname{lim}^{i}s\mathcal{J}^{*},\
i<q.$

###### Proof.

By Lemma 10.8, the group $\operatorname{lim}\mathcal{R}_{i}$ depends only on
the restriction of the cosheaf $\mathcal{P}^{*}$ to $\mathrm{sk}^{i+1}C$ in
$C$. Such restrictions coincide for $C$ and $\mathrm{sk}^{q}C$, $i<q$. This
implies the claim. ∎

######  Remark 10.11.

The argument in [ay-17] was given by considering the bicomplex with
coefficients in (47). In particular, it depends on the topology of the
simplicial set $C$ (namely, the local homology of $C$). The dual version of it
implies that one has

$\lim s\mathcal{R}_{i}\cong H^{cw}_{i}(C,\mathcal{P}^{*}),$

provided that $C$ is an oriented homology manifold. Such an argument is not
applicable for skeleta of homology manifolds. The $q$-skeleton
$\mathrm{sk}^{q}C$ of the homology manifold is Buchsbaum, i.e. the cosheaf
$\mathcal{H}^{j}$ on $\mathrm{sk}^{q}C$ vanishes unless $j=0$. This follows by
identifying the groups of the cohomological long exact sequence of the triple
$(\operatorname{ord}C,\operatorname{ord}\mathrm{sk}^{q}C,(\operatorname{ord}\mathrm{sk}^{q}C)\setminus\operatorname{star}^{\circ}_{\mathrm{sk}^{q}C}c)$,
where $c\in\mathrm{sk}^{q}C$ with the values of the cosheaf
$\mathcal{U}^{j+d}$. By combining Lemma 10.8 with the Poincaré-Lefschetz type
duality for $C$ and $\mathrm{sk}^{q}C$ (see Corollary 5.6) one obtains the
isomorphism

$H^{cw}_{i}(\mathrm{sk}^{q}C,\mathcal{H}^{0}\otimes\mathcal{P}^{*})\cong
H^{cw}_{i+(d-q)}(C,\mathcal{P}^{*}).$

(Here by abuse of the notation we denote the pullback of $\mathcal{P}^{*}$
with respect to $\iota^{q}$ by $\mathcal{P}^{*}$.)

Let $k=\operatorname{rk}T$.

######  Theorem 10.12.

(49)
$\operatorname{lim}^{i}s\iota^{*}_{q}H^{j}(D)\cong\begin{cases}\lim^{i}sH^{j}(D),\
i<q\\\ 0,\ i>q\ \mbox{or}\ i>j.\end{cases}$

There is the identity

$\dim\operatorname{lim}^{q}s\iota^{*}_{q}H^{j}(D)=(-1)^{q+1}\biggl{(}\sum_{c\in\mathrm{sk}^{q}C}\tilde{\chi}(\operatorname{lk}_{\mathrm{sk}^{q}C}c)\binom{k-1-|I|}{j}-\sum_{i=0}^{q-1}(-1)^{i}\dim\operatorname{lim}^{i}sH^{j}(D)\biggr{)}.$

###### Proof.

The isomorphism (49) in degrees $0,1$ follows by Lemma 10.6 (namely, in degree
$1$ from exactness of (46) by $5$-lemma). In degrees $\geqslant 2$ it follows
directly from Corollary 10.10. The vanishing below diagonal is due to (35).
This proves the first identity. The second identity then follows by a routine
computation of the reduced Euler characteristics for the cochain complex
$C^{*}(C;\ s\iota^{*}_{q}H^{j}(D))$. ∎

Let $C$ be a cone over a homology sphere, $\dim C=d$. Let $D$ be a
$T$-characteristic diagram, where $\operatorname{rk}T=k$ such that
$\operatorname{hocolim}D$ is equivariantly formal. Let

$\beta^{-i,2j}(D):=\dim\operatorname{Tor}^{-i,2j}_{H^{*}(BT)}(\lim
H^{*}(BS),\Bbbk),$

be the bigraded Betti number of $\operatorname{hocolim}D$.

######  Corollary 10.13.

$\beta^{-i,2j}(\iota^{*}_{q}D)=\begin{cases}\beta^{0,2j}(D),i=0,j<q\\\
\beta^{-i,2(i+q)}(\iota^{*}_{q}D),\ j=i+q,\\\ 0,\ i>q.\end{cases}$

There is the identity

$\beta^{-i,2(i+q)}(\iota^{*}_{q}D)=(-1)^{q+1}\sum_{c\in\mathrm{sk}^{q}C}\tilde{\chi}(\operatorname{lk}_{\mathrm{sk}^{q}C}c)\binom{k-1-|I|}{j}.$

###### Proof.

Follows directly from Theorems 10.12, 3.14 and Corollary 9.3. ∎

######  Remark 10.14.

The formulas of Corollary 10.13 give the Betti numbers of skeleta for any
toric manifold and any compact nonsingular toric variety, because one has

$\beta^{q}(\operatorname{hocolim}\iota^{*}_{q}D)=\sum_{-i+2j=q}\beta^{-i,2j}(\iota^{*}_{q}D),$

where $D$ is the corresponding diagram to such a space, as explained above.

We conclude this section with a comparison of Betti numbers formulas in the
case of skeleta for $\mathbb{C}P^{m-1}$ implied by [gr-th-07, fu-19] versus
the above.

######  Example 10.15.

Let $C=\operatorname{Cone}\partial\Delta_{m}$, $T=T^{m}/S^{1}_{d}\cong
T^{m-1}$, where $S^{1}_{d}$ is the big diagonal circle, and let $D$ be the
$T$-characteristic diagram

$D(I)=T^{m}/(T^{I}\cdot S^{1}_{d}),$

$T^{I}$ denotes the product in $T^{m}=(S^{1})^{m}$ of coordinate circles
$S^{1}_{i}$, $i\in I$, and the arrows of $D$ are induced by the embedding of
$T^{I}\cdot S^{1}_{d}\to T^{J}\cdot S^{1}_{d}$. One has
$\mathbb{C}P^{m-1}\cong\operatorname{hocolim}D$ [we-zi-zi-99]. One has

$\dim\operatorname{lim}^{i}sH^{j}(D)=\begin{cases}1,\mbox{ if }i=j\leqslant
m,\\\ 0,\ \mbox{otherwise}.\end{cases}$

Consider $\tilde{C}=\mathrm{sk}^{q+1}C=\operatorname{Cone}\Delta^{q}_{m}$ (the
cone over the $q$-dimensional skeleton of the simplex $\Delta_{m}$ on $m$
vertices). Then

$\dim\operatorname{lim}^{i}s\iota^{*}_{q+1}H^{j}(D)=\begin{cases}1,\
i=j\leqslant q,\\\ 0,\ i,j\leqslant q,i\neq j.\end{cases}$

Furthermore,

$\dim\operatorname{lim}^{i}s\iota^{*}_{q+1}H^{j}(D)=0,\ i>j.$

It remains to compute dimensions for $i=q+1$ and $j\geqslant q+1$. The number
of $i$-dimensional faces in $\tilde{C}$ is equal to $\binom{m}{i+1}$,
$-1\leqslant i\leqslant q$. Any such face $I$, $\dim I=i$, has the link

$\operatorname{lk}_{\tilde{C}}I\cong\Delta_{m-1-i}^{q-1-i}.$

One has

$\Delta_{m}^{l}\simeq(S^{l})^{\vee\binom{m-1}{l+1}}.$

Therefore,

$\tilde{\chi}(\operatorname{lk}_{\tilde{C}}I)=(-1)^{q-1-i}\binom{m-2-i}{q-i}.$

We compute

(50)
$\dim\operatorname{lim}^{q+1}\iota^{*}_{q+1}H^{j}(D)=(-1)^{q}\sum_{i=-1}^{q}\binom{m}{i+1}(-1)^{q-1-i}\binom{m-2-i}{q-i}\binom{m-2-i}{j}.$

On the other hand, by [gr-th-07, fu-19] it follows that

$\operatorname{hocolim}D\simeq\mathbb{C}P^{q+1}\vee\bigvee_{i=0}^{q}\bigvee_{b=q-i+2}^{m-1-i}(S^{q+b+i+1})^{\vee\binom{m-1-i}{b}\binom{b-1}{q-i-1}}\vee\bigvee_{i=1}^{m-q-2}(S^{2q+2+i})^{\vee\binom{m-q-2}{i}}.$

This implies by a routine computation that

(51)
$b_{q+1+j}(\operatorname{hocolim}D)=\dim\operatorname{lim}^{q+1}\iota^{*}_{q+1}H^{j}(D)=\sum_{i=0}^{q+1}\binom{m-1-i}{j-i}\binom{j-i-1}{q-i+1}.$

We compare (50) with (51) as follows. The expressions (50), (51) are
identified respectively with

$\binom{m-1}{j}\binom{m-1}{q+1}{}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-q-1\mskip
8.0mu-m\mskip 8.0muj-m+1}{1-m\mskip 8.0mu1-m};1\biggr{]},$
$\binom{m-1}{j}\binom{j-1}{q+1}{}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-q-1\mskip
8.0mu1\mskip 8.0mu-j}{1-m\mskip 8.0mu1-j};1\biggr{]},$

where

${}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mskip 8.0mub\mskip
8.0muc}{d\mskip
8.0mue};1\biggr{]}:=\sum_{i=0}^{\infty}\frac{1}{i!}\frac{(a)_{i}(b)_{i}(c)_{i}}{(d)_{i}(e)_{i}},\
a,b,c,d,e\in\mathbb{Z},$

denotes the value of the generalized hypergeometric function at unit (if the
series converge), and

$(d)_{n}=d\cdot(d+1)\cdots(d+n-1),$

denotes the Pochhammer symbol. As expected, we obtain the equality of these
two expressions directly from the identity [sr-je-ra-92, Appendix, $(II)$]:

${}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mskip 8.0mub\mskip
8.0muc}{d\mskip
8.0mue};1\biggr{]}=\frac{(d+e-b-c)_{n}}{(d)_{n}}{}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n\mskip
8.0mue-b\mskip 8.0mue-c}{e\mskip 8.0mud+e-b-c};1\biggr{]},\ n\in\mathbb{N},$

by substituting

$n=q+1,\ b=1,\ c=-j,\ e=1-m,\ d=1-j.$

######  Remark 10.16.

The skeletal filtration on $\operatorname{hocolim}D$ corresponds to a
“stabilizer” filtration, whereas coskeletal filtration on
$\operatorname{hocolim}D$ corresponds to the orbit filtration. The first and
second pages of the “stabilizer” filtration do not have a similar description.
Reason for this is that the construction of the homotopy colimit is based on
the notion of filters, not ideals.

## References

  * [99] biblio_eng