# On integral cohomology algebra of some oriented Grassmann manifolds

Milica Jovanović<EMAIL_ADDRESS>
###### Abstract

The integral cohomology algebra of $\widetilde{G}_{6,3}$ has been determined
in the recent work of Kalafat and Yalçınkaya. We completely determine the
integral cohomology algebra of $\widetilde{G}_{n,3}$ for $n=8$ and $n=10$. The
main method used to describe these algebras is the Leray-Serre spectral
sequence. We also illustrate this method by determining the integral
cohomology algebra of $\widetilde{G}_{n,2}$ for $n$ odd.

###### keywords:

Grassmann manifold , Leray-Serre spectral sequence , Characteristic class

###### MSC:

[2020] 55R40 , 55R25 , 55T10

††journal: Indagationes Mathematicae

[afil]organization=University of Belgrade, Faculty of
Mathematics,addressline=Studentski trg 16, city=Belgrade, postcode=11000,
country=Serbia

##  1 Introduction

The importance of Grassmann manifolds has been established for a quite some
time. The oriented Grassmann manifold $\widetilde{G}_{n,k}$ is defined as a
space of oriented $k$-dimensional subspaces of $\mathbb{R}^{n}$. As the name
suggests, it is indeed a manifold of dimension $k(n-k)$. Recently, much work
has been done in understanding the $\mathbb{Z}_{2}$ cohomology algebra of
oriented Grassmann manifolds. For $k=2$ it was determined in [5] and in [1]
the authors obtain an almost complete description of this algebra for $k=3$
and $n$ close to a power of two. On the other hand, it seems that very little
is known about the integral cohomology of $\widetilde{G}_{n,k}$, especially
about the algebra structure. In a paper by Lai [6], the cohomology algebra of
$\widetilde{G}_{n,2}$ for $n$ even has been determined and this result was
extended to all $n$ by Vanžura in [9]. We give a different proof of Vanžura’s
result for $n$ odd in Theorem 2.2. For $k\geqslant 3$ far less is known. In
their recent work [4], Kalafat and Yalçınkaya have managed to describe the
integral cohomology algebra of $\widetilde{G}_{6,3}$ by using the special
Lagrangian locus of $\widetilde{G}_{6,3}$. In this paper, using the Leray-
Serre spectral sequence, we give the complete structure of the integral
cohomology algebra of $\widetilde{G}_{n,3}$ for $n=8$ and $n=10$ in Theorems
5.1 and 6.1 respectively. We also give a few general facts about the integral
cohomology algebra of $\widetilde{G}_{n,3}$. Since
$\widetilde{G}_{n,k}\approx\widetilde{G}_{n,n-k}$, we will always assume
$n\geqslant 2k$.

In Section 2 we apply the Leray-Serre spectral sequence in order to obtain a
description of $H^{*}(\widetilde{G}_{n,2};\mathbb{Z})$, for $n$ odd. In
Section 3 we determine the integral cohomology algebra of auxiliary space
$W_{2,1}^{n}$ for even $n$. This space is defined in [1] as

$W_{2,1}^{n}=\left\\{(\widetilde{P},v)\in\widetilde{G}_{n,2}\times
S^{n-1}\mid\text{$\widetilde{P}$ and $v$ are orthogonal}\right\\}.$

We do this by applying the Leray-Serre spectral sequence on the sphere bundle
$S^{n-3}\rightarrow W^{n}_{2,1}\xrightarrow{}\widetilde{G}_{n,2}$. In Section
4 we give some general properties of the integral cohomology of
$\widetilde{G}_{n,3}$. Finally, in Sections 5 and 6 we determine the algebras
$H^{*}(\widetilde{G}_{8,3};\mathbb{Z})$ and
$H^{*}(\widetilde{G}_{10,3};\mathbb{Z})$ with the application of Leray-Serre
spectral sequence on fiber bundles $S^{2}\rightarrow
W_{2,1}^{n}\rightarrow\widetilde{G}_{n,3}$ and $SO(3)\rightarrow
V_{n,3}\rightarrow\widetilde{G}_{n,3}$ as our main tool ($V_{n,k}$ is the
Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^{n}$).

##  2 Cohomology algebra $H^{*}({\widetilde{G}}_{n,2};\mathbb{Z})$

The cohomology algebra $H^{*}(\widetilde{G}_{n,2};\mathbb{Z})$ depends on the
parity of $n$. If $n$ is even, then this algebra is determined in Theorem 2 in
[6]. Specifically, the following result holds.

######  Theorem 2.1 (Lai).

The integral cohomology groups of $\widetilde{G}_{2n,2}$ are isomorphic to
those of $\mathbb{C}P^{n-1}\times S^{2n-2}$, namely, for $0\leqslant
k\leqslant 4n-4$,

$H^{k}(\widetilde{G}_{2n,2};\mathbb{Z})\cong\begin{cases}\mathbb{Z},&\text{if
$k$ is even and $k\neq 2n-2$},\\\ \mathbb{Z}\oplus\mathbb{Z},&\text{if
}k=2n-2,\\\ 0,&\text{if }k\text{ is odd.}\end{cases}$

Let $\widetilde{\gamma}$ be the oriented tautological bundle over
$\widetilde{G}_{2n,2}$ and $e(\widetilde{\gamma})$ its Euler class. As an
algebra, $H^{*}(\widetilde{G}_{2n,2};\mathbb{Z})$ is generated by
$\widetilde{\Omega}=e(\widetilde{\gamma})\in
H^{2}(\widetilde{G}_{2n,2};\mathbb{Z})$ and $\kappa\in
H^{2n-2}(\widetilde{G}_{2n,2};\mathbb{Z})$ with the relation
$\widetilde{\Omega}^{n}=2\kappa\widetilde{\Omega}$. Moreover, if
$\widetilde{\gamma}^{\perp}$ is the orthogonal complement of
$\widetilde{\gamma}$, $\Omega=e(\widetilde{\gamma}^{\perp})\in
H^{2n-2}(\widetilde{G}_{2n,2};\mathbb{Z})$ its Euler class and $\mu\in
H^{4n-4}(\widetilde{G}_{2n,2};\mathbb{Z})$ the fundamental class, then we have
the following relations

$\begin{split}&\Omega+\widetilde{\Omega}^{n-1}=2\kappa,~{}\kappa\widetilde{\Omega}^{n-1}=(-1)^{n}\mu,~{}\kappa\Omega=\mu,\\\
&\kappa^{2}=\frac{1+(-1)^{n-1}}{2}\mu,~{}\widetilde{\Omega}^{2n-2}=2(-1)^{n}\mu,~{}\Omega^{2}=2\mu.\end{split}$

In the case of odd dimension, we have the following result. This was proved in
[9], but here we present a different proof, which relies on the Leray-Serre
spectral sequence.

######  Theorem 2.2.

Let $n\geqslant 2$. The integral cohomology algebra of
$\widetilde{G}_{2n+1,2}$ is

$H^{*}(\widetilde{G}_{2n+1,2};\mathbb{Z})=\frac{\mathbb{Z}[x_{2},x_{2n}]}{\langle
x_{2}^{n}-2x_{2n},x_{2n}^{2}\rangle},$

where $x_{2}\in H^{2}(\widetilde{G}_{2n+1,2};\mathbb{Z})$ and $x_{2n}\in
H^{2n}(\widetilde{G}_{2n+1,2};\mathbb{Z})$. If
$\rho:\mathbb{Z}\rightarrow\mathbb{Z}_{2}$ is the modulo 2 map, then
$\rho_{*}(x_{2})=\widetilde{w}_{2}$ and $\rho_{*}(x_{2n})=a_{2n}$, where
$\widetilde{w}_{2}\in H^{2}(\widetilde{G}_{2n,2};\mathbb{Z}_{2})$ and
$a_{2n}\in H^{2n}(\widetilde{G}_{2n,2};\mathbb{Z}_{2})$ are generators from
Theorem 2.1 in [5].

###### Proof.

In order to determine the cohomology algebra
$H^{*}(\widetilde{G}_{2n+1,2};\mathbb{Z})$, we will use the Leray-Serre
spectral sequence for the sphere bundle $S^{1}\rightarrow
V_{2n+1,2}\xrightarrow{p}\widetilde{G}_{2n+1,2}$ where $p$ maps the ordered
pair of orthogonal vectors to the oriented plane they span. The integral
cohomology ring of Stiefel manifold $V_{2n+1,2}$ is well known (see [7] p. 153
or Theorem 2.3 in [2]). We have

$H^{k}(V_{2n+1,2};\mathbb{Z})\cong\begin{cases}\mathbb{Z},&\text{for
}k=0,4n-1\\\ \mathbb{Z}_{2},&\text{for }k=2n\\\
0,&\text{otherwise}\end{cases}$

and the multiplication is trivial.

The only nontrivial differential in the spectral sequence is $d_{2}$. We will
analyze the $E_{2}$ page from left and right and finish in the middle. Since
$H^{1}(V_{2n+1,2};\mathbb{Z})=0$, we have $E_{2}^{1,0}=0$ and also
$E_{2}^{1,1}=0$. In similar fashion, we get that $E_{2}^{3,0}=E_{2}^{3,1}=0$
as $H^{3}(V_{2n+1,2};\mathbb{Z})=0$. We can continue with same conclusions up
to dimension $2n-1$, i.e., $E_{2}^{k,0}=E_{2}^{k,1}=0$ for $k$ odd between $1$
and $2n-1$. On the other hand, since $E_{2}^{0,1}\cong\mathbb{Z}$ and
$H^{k}(V_{2n+1,2};\mathbb{Z})=0$ for all even $k$ between $2$ and $2n-2$, we
get that each $d_{2}:E_{2}^{k-2,1}\rightarrow E_{2}^{k,0}$ must be an
isomorphism for $2\leqslant k\leqslant 2n-2$, i.e., $E_{2}^{k,0}\cong
E_{2}^{k,1}\cong\mathbb{Z}$ for $k$ even between $0$ and $2n-2$.

Now starting from the right-hand side we first have
$E_{2}^{4n-2,0}\cong\mathbb{Z}$ since
$H^{4n-2}(\widetilde{G}_{2n+1,2};\mathbb{Z})\cong\mathbb{Z}$
($\widetilde{G}_{2n+1,2}$ is an orientable closed connected manifold of
dimension $4n-2$) and then we have a series of isomorphisms

$E_{2}^{2n,0}\cong E_{2}^{2n,1}\xrightarrow{d_{2}}E_{2}^{2n+2,0}\cong
E_{2}^{2n+2,1}\xrightarrow{d_{2}}\cdots\xrightarrow{d_{2}}E_{2}^{4n-2,0}\cong\mathbb{Z},$

so we get $E_{2}^{k,0}=E_{2}^{k,1}=\mathbb{Z}$ for $k$ even between $2n$ and
$4n-2$. Similarly, $E_{2}^{4n-3,1}=0$ since
$H^{4n-2}(V_{2n+1,2};\mathbb{Z})=0$ and we get $E_{2}^{k,0}=E_{2}^{k,1}=0$ for
$k$ odd between $2n+1$ and $4n-3$. The only nontrivial differential which is
not an isomorphism is $d_{2}:E_{2}^{2n-2,1}\rightarrow E_{2}^{2n,0}$ and since
$H^{2n}(V_{2n+1,2};\mathbb{Z})=\mathbb{Z}_{2}$, this differential must be
multiplication by $2$. The $E_{2}$ page of the spectral sequence is presented
in Table 1.

1 |  $\mathbb{Z}$ | 0 | $\mathbb{Z}$ | $\cdots$ |  $\mathbb{Z}$ | 0 |  $\mathbb{Z}$ | 0 | $\mathbb{Z}$ | $\cdots$ | $\mathbb{Z}$
---|---|---|---|---|---|---|---|---|---|---|---
0 | $\mathbb{Z}$ | 0 |  $\mathbb{Z}$ | $\cdots$ | $\mathbb{Z}$ | 0 | $\mathbb{Z}$ | 0 | $\mathbb{Z}$ | $\cdots$ | $\mathbb{Z}$
| 0 | 1 | 2 | $\cdots$ | $2n-2$ | $2n-1$ | $2n$ | $2n+1$ | $2n+2$ | $\cdots$ | $4n-2$
$~{}_{\cdot 2}$

Table 1: The $E_{2}$ page for the sphere bundle $S^{1}\rightarrow
V_{2n+1,2}\rightarrow\widetilde{G}_{2n+1,2}$

So far we have determined cohomology groups of $\widetilde{G}_{2n+1,2}$ to be
$\mathbb{Z}$ in even and $0$ in odd dimensions. Differential
$d_{2}:E_{2}^{k,1}\rightarrow E_{2}^{k+2,0}$ is an isomorphism for $k=0$. If
$s\in H^{1}(S^{1};\mathbb{Z})$ is the generator, let $x_{2}\in
H^{2}(\widetilde{G}_{2n+1,2};\mathbb{Z})$ be the generator such that
$d_{2}(1\otimes s)=x_{2}\otimes 1$. From this relation we can determine all
the other differentials $d_{2}$. If $k$ is odd, $d_{2}:E_{2}^{k,1}\rightarrow
E_{2}^{k+2,0}$ is trivial and if $k$ is even and $x_{k}\in
H^{k}(\widetilde{G}_{2n+1,2};\mathbb{Z})$ then

$d_{2}(x_{k}\otimes s)=d_{2}(x_{k}\otimes 1)(1\otimes s)+(-1)^{k}(x_{k}\otimes
1)d_{2}(1\otimes s)=(x_{k}\otimes 1)(x_{2}\otimes 1)=x_{k}x_{2}\otimes 1.$

Therefore our algebra has two generators, $x_{2}$ and $x_{2n}\in
H^{2n}(\widetilde{G}_{2n+1,2};\mathbb{Z})$ and two relations
$x_{2}^{n}=2x_{2n}$ and $x_{2n}^{2}=0$.

What remains is to prove that modulo two reduction of generators $x_{2}$ and
$x_{2n}$ gives us generators of
$H^{*}(\widetilde{G}_{2n+1,2};\mathbb{Z}_{2})$. From Theorem 2.1 in [5] we
have

$H^{*}(\widetilde{G}_{2n+1,2};\mathbb{Z}_{2})=\mathbb{Z}_{2}[\widetilde{w}_{2}]/\langle\widetilde{w}_{2}^{n}\rangle\otimes\Lambda_{\mathbb{Z}_{2}}(a_{2n}).$

The universal coefficient theorem gives us the following commutative diagram.

${0}$${\mathrm{Ext}(H_{k-1}(\widetilde{G}_{2n+1,2}),\mathbb{Z})}$${H^{k}(\widetilde{G}_{2n+1,2};\mathbb{Z})}$${\mathrm{Hom}(H_{k}(\widetilde{G}_{2n+1,2}),\mathbb{Z})}$${0}$${0}$${\mathrm{Ext}(H_{k-1}(\widetilde{G}_{2n+1,2}),\mathbb{Z}_{2})}$${H^{k}(\widetilde{G}_{2n+1,2};\mathbb{Z}_{2})}$${\mathrm{Hom}(H_{k}(\widetilde{G}_{2n+1,2}),\mathbb{Z}_{2})}$${0}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\rho_{*}}$

For $k=2$, $H_{k-1}(\widetilde{G}_{2n+1,2})=0$, so previous diagram becomes

${0}$${H^{2}(\widetilde{G}_{2n+1,2};\mathbb{Z})}$${\mathrm{Hom}(H_{2}(\widetilde{G}_{2n+1,2}),\mathbb{Z})}$${0}$${0}$${H^{2}(\widetilde{G}_{2n+1,2};\mathbb{Z}_{2})}$${\mathrm{Hom}(H_{2}(\widetilde{G}_{2n+1,2}),\mathbb{Z}_{2})}$${0}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\rho_{*}}$

Poincaré duality gives us that $H_{2}(\widetilde{G}_{2n+1,2})\cong\mathbb{Z}$,
so the right homomorphism $\rho_{*}$ is an epimorphism and since horizontal
homomorphisms are isomorphisms,
$\rho_{*}:H^{2}(\widetilde{G}_{2n+1,2};\mathbb{Z})\rightarrow
H^{2}(\widetilde{G}_{2n+1,2};\mathbb{Z}_{2})$ must also be an epimorphism, so
$\rho_{*}(x_{2})=\widetilde{w}_{2}$.
The situation is same for the case $k=2n$. We have
$H_{2n-1}(\widetilde{G}_{2n+1,2})=0$, so we have smaller diagram as the
previous one from which we conclude that
$\rho_{*}:H^{2n}(\widetilde{G}_{2n+1,2};\mathbb{Z})\rightarrow
H^{2n}(\widetilde{G}_{2n+1,2};\mathbb{Z}_{2})$ is an epimorphism as well, so
$\rho_{*}(x_{2n})=a_{2n}$. ∎

######  Remark 1.

In the previous proof we have seen that differential $d_{2}$ in Table 1 is
multiplication with $x_{2}$, so this must be the Euler class of the fiber
bundle $S^{1}\rightarrow V_{2n+1,2}\rightarrow\widetilde{G}_{2n+1,2}$. But
this bundle is easily identified with the sphere bundle associated to the
tautological bundle $\widetilde{\gamma}$, so
$x_{2}=\widetilde{\Omega}=e(\widetilde{\gamma})$, as in Theorem 2.1. On the
other hand, $\Omega=e(\widetilde{\gamma}^{\perp})=0$, since $\Omega\in
H^{2n-1}(\widetilde{G}_{2n+1,2},\mathbb{Z})=0$.

##  3 Cohomology algebra $H^{*}(W^{2n}_{2,1};\mathbb{Z})$

The space $W^{n}_{2,1}$ is defined in [1] as

$W_{2,1}^{n}=\left\\{(\widetilde{P},v)\in\widetilde{G}_{n,2}\times
S^{n-1}\mid\text{$\widetilde{P}$ and $v$ are orthogonal}\right\\}.$

For $n$ even, we will determine the integral cohomology algebra of this space
using the Leray-Serre spectral sequence for the sphere bundle
$S^{n-3}\rightarrow W^{n}_{2,1}\xrightarrow{p_{1}}\widetilde{G}_{n,2}$ where
$p_{1}$ is the projection onto the first coordinate. Note that this is
actually the sphere bundle associated to the orthogonal complement
$\widetilde{\gamma}^{\perp}$ of the oriented tautological bundle over
$\widetilde{G}_{n,2}$.

######  Theorem 3.1.

Let $n\geqslant 3$. The integral cohomology algebra of $W^{2n}_{2,1}$ is

$H^{*}(W^{2n}_{2,1};\mathbb{Z})=\frac{\mathbb{Z}[\bar{x}_{2},\bar{x}_{2n-2},\bar{x}_{2n-1}]}{\langle\bar{x}_{2}^{n-1}-2\bar{x}_{2n-2},\bar{x}_{2n-2}^{2},\bar{x}_{2n-1}^{2}\rangle},$

where $\bar{x}_{2}\in H^{2}(W^{2n}_{2,1};\mathbb{Z})$, $\bar{x}_{2n-2}\in
H^{2n-2}(W^{2n}_{2,1};\mathbb{Z})$ and $\bar{x}_{2n-1}\in
H^{2n-1}(W^{2n}_{2,1};\mathbb{Z})$.

###### Proof.

Theorem 2.1 gives us the $E_{2}$ page of the Leray-Serre spectral sequence for
the sphere bundle $S^{2n-3}\rightarrow
W_{2,1}^{2n}\rightarrow\widetilde{G}_{2n,2}$. In each cell of Table 2 are
generators of this $E_{2}$ page, where $\kappa$ and $\widetilde{\Omega}$ are
those generators from Theorem 2.1. For instance, $\widetilde{\Omega}^{2}$ in
cell $(4,0)$ means
$E_{2}^{4,0}\cong\mathbb{Z}\langle\widetilde{\Omega}^{2}\rangle$ and
$\widetilde{\Omega}^{n-1},~{}\kappa$ in cell $(2n-2,0)$ means
$E_{2}^{2n-2,0}\cong\mathbb{Z}\langle\widetilde{\Omega}^{n-1}\rangle\oplus\mathbb{Z}\langle\kappa\rangle$.

$2n-3$ | $1$ |  | $\widetilde{\Omega}$ |  | $\widetilde{\Omega}^{2}$ | $\cdots$ | $\widetilde{\Omega}^{n-2}$ |  | $\widetilde{\Omega}^{n-1},~{}\kappa$ |  | $\kappa\widetilde{\Omega}$ | $\cdots$ | $\kappa\widetilde{\Omega}^{n-2}$ |  | $\kappa\widetilde{\Omega}^{n-1}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\vdots$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$0$ | $1$ |  | $\widetilde{\Omega}$ |  | $\widetilde{\Omega}^{2}$ | $\cdots$ | $\widetilde{\Omega}^{n-2}$ |  | $\widetilde{\Omega}^{n-1},~{}\kappa$ |  | $\kappa\widetilde{\Omega}$ | $\cdots$ | $\kappa\widetilde{\Omega}^{n-2}$ |  | $\kappa\widetilde{\Omega}^{n-1}$
| 0 |  | 2 |  | 4 | $\cdots$ | $2n-4$ |  | $2n-2$ |  | $2n$ | $\cdots$ | $4n-6$ |  | $4n-4$

Table 2: The $E_{2}$ page for the sphere bundle $S^{2n-3}\rightarrow
W_{2,1}^{2n}\rightarrow\widetilde{G}_{2n,2}$

The only nontrivial differential is $d_{2n-2}$ and we know that it is
multiplication by $\Omega$, since $\Omega$ is the Euler class of this sphere
bundle. By multiplication we mean the following. Generators of
$E_{2n-2}^{k,2n-3}=E_{2}^{k,2n-3}$ are tensor products of elements in
$H^{k}(\widetilde{G}_{2n,2};\mathbb{Z})$ and a generator $s\in
H^{2n-3}(S^{2n-3};\mathbb{Z})$. For instance, by $\widetilde{\Omega}\in
E_{2}^{2,2n-3}$ we mean $\widetilde{\Omega}\otimes s$. Similarly,
$\widetilde{\Omega}\in E_{2}^{2,0}$ is actually $\widetilde{\Omega}\otimes 1$.
Claim that $d_{2n-2}$ is multiplication by $\Omega$ only means that for $x\in
H^{k}(\widetilde{G}_{2n,2};\mathbb{Z})$ we have $d_{2n-2}(x\otimes
s)=x\Omega\otimes 1$. Since $d_{2n-2}$ is the only nontrivial differential, we
have $E_{2}=E_{3}=\cdots=E_{2n-2}$ and $E_{2n-1}=E_{2n}=\cdots=E_{\infty}$.
Since $\widetilde{\Omega}\Omega=0$, we get
$d_{2n-2}:E_{2n-2}^{k,2n-3}\rightarrow E_{2n-2}^{k+2n-2,0}$ is zero except at
most for $k=0$ and $k=2n-2$. We know from Theorem 2.1 that
$\Omega=2\kappa-\widetilde{\Omega}^{n-1}$, so
$\mathrm{im}\\{d_{2n-2}:E_{2n-2}^{0,2n-3}\rightarrow
E_{2n-2}^{2n-2,0}\\}=\mathbb{Z}\langle
2\kappa-\widetilde{\Omega}^{n-1}\rangle$ and so we get

$E_{\infty}^{2n-2,0}=E_{2n-2}^{2n-2,0}\big{/}\mathrm{im}~{}d_{2n-2}=\frac{\mathbb{Z}\langle\kappa\rangle\oplus\mathbb{Z}\langle\widetilde{\Omega}^{n-1}\rangle}{\mathbb{Z}\langle
2\kappa-\widetilde{\Omega}^{n-1}\rangle}\cong\mathbb{Z}\langle\kappa\rangle$

and $E_{\infty}^{0,2n-3}=0$. Also from the same Theorem 2.1 we get that
$d_{2n-2}:E_{2n-2}^{2n-2,2n-3}\rightarrow E_{2n-2}^{4n-4,0}$ is given on
generators by

$d_{2n-2}\widetilde{\Omega}^{n-1}=0,~{}d_{2n-2}\kappa=(-1)^{n}\kappa\widetilde{\Omega}^{n-1},$

from which we get
$E_{\infty}^{2n-2,2n-3}\cong\mathbb{Z}\langle\widetilde{\Omega}^{n-1}\rangle$
and $E_{\infty}^{4n-4,0}=0$.

Apart from these four, every other cell in $E_{\infty}$ page is the same as in
$E_{2}$ page.

Now we can determine the cohomology algebra $H^{*}(W_{2,1}^{2n};\mathbb{Z})$.
We have an algebra isomorphism

$H^{*}(W_{2,1}^{2n};\mathbb{Z})\cong\mathrm{total}(E_{\infty}^{*,*}),$

where $\mathrm{total}(E_{\infty}^{*,*})$ is the total complex (for definition
see [7] p. 24). From $E_{\infty}$ we can conclude that
$H^{*}(W_{2,1}^{2n};\mathbb{Z})$ will have three generators
$\bar{x}_{2},~{}\bar{x}_{2n-2}$ and $\bar{x}_{2n-1}$ corresponding to
$\widetilde{\Omega}\otimes 1$, $\kappa\otimes 1$ and
$\widetilde{\Omega}\otimes s$ respectively and from relations in
$H^{*}(\widetilde{G}_{2n,2};\mathbb{Z})$ we get relations in
$H^{*}(W_{2,1}^{2n};\mathbb{Z})$. For instance, generator of
$H^{4n-3}(W_{2,1}^{2n};\mathbb{Z})$ is in correspondence with
$\kappa\widetilde{\Omega}\otimes s=(\kappa\otimes 1)(\widetilde{\Omega}\otimes
s)$ so it must be $\bar{x}_{2n-1}\bar{x}_{2n-2}$ and so on. After simple
calculations we get the specified algebra structure. ∎

##  4 Some general properties of $H^{*}(\widetilde{G}_{n,3};\mathbb{Z})$

######  Lemma 4.1.

Let $n\geqslant 7$. For the first four and the last four integral cohomology
groups of $\widetilde{G}_{n,3}$ the following relations hold:

$\begin{matrix}\begin{split}&H^{0}(\widetilde{G}_{n,3};\mathbb{Z})\cong\mathbb{Z},\\\
&H^{1}(\widetilde{G}_{n,3};\mathbb{Z})=0,\\\
&H^{2}(\widetilde{G}_{n,3};\mathbb{Z})=0,\\\
&H^{3}(\widetilde{G}_{n,3};\mathbb{Z})\cong\mathbb{Z}_{2},\end{split}&\begin{split}&H^{3(n-3)-3}(\widetilde{G}_{n,3};\mathbb{Z})=0,\\\
&H^{3(n-3)-2}(\widetilde{G}_{n,3};\mathbb{Z})\cong\mathbb{Z}_{2},\\\
&H^{3(n-3)-1}(\widetilde{G}_{n,3};\mathbb{Z})=0,\\\
&H^{3(n-3)}(\widetilde{G}_{n,3};\mathbb{Z})\cong\mathbb{Z}.\end{split}\end{matrix}$

Also, the group $H^{4}(\widetilde{G}_{n,3};\mathbb{Z})$ is free.

###### Proof.

Consider the homotopy exact sequence for the fiber bundle $SO(3)\rightarrow
V_{n,3}\xrightarrow{p}\widetilde{G}_{n,3}$ where $p$ is given by the oriented
span of the three vectors.

$\begin{split}\cdots&\rightarrow\pi_{3}(V_{n,3})\rightarrow\pi_{3}(\widetilde{G}_{n,3})\rightarrow\pi_{2}(SO(3))\rightarrow\pi_{2}(V_{n,3})\rightarrow\pi_{2}(\widetilde{G}_{n,3})\\\
&\rightarrow\pi_{1}(SO(3))\rightarrow\pi_{1}(V_{n,3})\rightarrow\pi_{1}(\widetilde{G}_{n,3})\rightarrow
0\end{split}$

The Stiefel manifold $V_{n,3}$ is $(n-4)$-connected. Since, $n\geqslant 7$ the
first three homotopy groups of $V_{n,3}$ are zero. Also,
$\pi_{1}(SO(3))\cong\mathbb{Z}_{2}$ and $\pi_{2}(SO(3))=0$, so our sequence
becomes

$\cdots\rightarrow 0\rightarrow\pi_{3}(\widetilde{G}_{n,3})\rightarrow
0\rightarrow
0\rightarrow\pi_{2}(\widetilde{G}_{n,3})\rightarrow\mathbb{Z}_{2}\rightarrow
0\rightarrow\pi_{1}(\widetilde{G}_{n,3})\rightarrow 0.$

It is now obvious that

$\pi_{1}(\widetilde{G}_{n,3})=0,~{}~{}\pi_{2}(\widetilde{G}_{n,3})=\mathbb{Z}_{2},~{}~{}\pi_{3}(\widetilde{G}_{n,3})=0.$

Since $\widetilde{G}_{n,3}$ is $1$-connected, we have that Hurewicz
homomorphism $h:\pi_{k}(\widetilde{G}_{n,3})\rightarrow
H_{k}(\widetilde{G}_{n,3})$ is an isomorphism for $k\in\\{1,2\\}$ and an
epimorphism for $k=3$ (see Theorem 10.25 in [8]). This gives us that the
first, the second and the third homology groups of $\widetilde{G}_{n,3}$ are
$0$, $\mathbb{Z}_{2}$ and $0$ respectively.

The universal coefficient theorem claims that

$H^{k}(\widetilde{G}_{n,3};\mathbb{Z})\cong\mathrm{Ext}(H_{k-1}(\widetilde{G}_{n,3}),\mathbb{Z})\oplus\mathrm{Hom}(H_{k}(\widetilde{G}_{n,3}),\mathbb{Z}).$

From this relation we get the first four cohomology groups and also the fact
that $H^{4}(\widetilde{G}_{n,3};\mathbb{Z})$ is free. On the other hand, the
Poincaré duality

$H^{3(n-3)-k}(\widetilde{G}_{n,3};\mathbb{Z})\cong H_{k}(\widetilde{G}_{n,3})$

gives us the last four groups. ∎

######  Remark 2.

The case $n=6$ is covered in [4], so the previous Lemma holds for all
$n\geqslant 6$.

Let us label the free and the torsion part of $H_{k}(\widetilde{G}_{n,3})$ as
$F_{k}$ and $T_{k}$, i.e., $H_{k}(\widetilde{G}_{n,3})\cong F_{k}\oplus
T_{k}$. The universal coefficient theorem gives us

$H^{k}(\widetilde{G}_{n,3};\mathbb{Z})\cong F_{k}\oplus T_{k-1}.$

Since $\widetilde{G}_{n,3}\approx SO(n)/SO(3)\times SO(n-3)$, we can find the
Poincaré polynomials for $\widetilde{G}_{n,3}$ in [3] (p. 494-496), so all
groups $F_{k}$ are known.

Combining this with the Poincaré duality, which gives us

$T_{k}\cong T_{3(n-3)-k-1},$

the problem of determining integral cohomology groups of $\widetilde{G}_{n,3}$
becomes determining torsion parts $T_{k}$ for $4\leqslant
k\leqslant\left[\frac{3(n-3)}{2}\right]$.

##  5 Cohomology algebra $H^{*}(\widetilde{G}_{8,3};\mathbb{Z})$

######  Theorem 5.1.

The integral cohomology algebra of $\widetilde{G}_{8,3}$ is

$H^{*}(\widetilde{G}_{8,3};\mathbb{Z})\cong\frac{\mathbb{Z}[y_{3},x_{4},x_{7}]}{\langle
2y_{3},y_{3}x_{4},y_{3}^{3},x_{4}^{3},x_{7}^{2}\rangle},$

where $y_{3}\in H^{3}(\widetilde{G}_{8,3};\mathbb{Z})$, $x_{4}\in
H^{4}(\widetilde{G}_{8,3};\mathbb{Z})$ and $x_{7}\in
H^{7}(\widetilde{G}_{8,3};\mathbb{Z})$.

###### Proof.

From [3] (p. 496), the Poincaré polynomial for $\widetilde{G}_{8,3}$ is

$p(\widetilde{G}_{8,3})=1+x^{4}+x^{7}+x^{8}+x^{11}+x^{15},$

so we know the free part of the cohomology groups of $\widetilde{G}_{8,3}$.
When we apply results from Section 4, we see that integral cohomology groups
of $\widetilde{G}_{8,3}$ are:

$k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$H^{k}(\widetilde{G}_{8,3};\mathbb{Z})$ | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $T_{4}$ | $T_{5}$ | $\mathbb{Z}\oplus T_{6}$ | $\mathbb{Z}\oplus T_{7}$ | $T_{6}$ | $T_{5}$ | $\mathbb{Z}\oplus T_{4}$ | 0 | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}$

Table 3: Integral cohomology groups of $\widetilde{G}_{8,3}$

What remains is to determine these four torsion groups.

Recall from Theorem 3.1 the cohomology algebra of $W_{2,1}^{8}$:

$H^{*}(W^{8}_{2,1};\mathbb{Z})\cong\frac{\mathbb{Z}[\bar{x}_{2},\bar{x}_{6},\bar{x}_{7}]}{\langle\bar{x}_{2}^{3}-2\bar{x}_{6},\bar{x}_{6}^{2},\bar{x}_{7}^{2}\rangle}.$
(1)

Consider the Leray-Serre spectral sequence for the sphere bundle
$S^{2}\rightarrow W_{2,1}^{8}\xrightarrow{p}\widetilde{G}_{8,3}$ where $p$
maps the ordered pair of an oriented plane and a vector to the oriented
3-space they span. Its $E_{2}$ page is given in Table 4.

2 | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $T_{4}$ | $T_{5}$ | $\mathbb{Z}\oplus T_{6}$ | $\mathbb{Z}\oplus T_{7}$ | $T_{6}$ | $T_{5}$ | $\mathbb{Z}\oplus T_{4}$ | 0 | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0 | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $T_{4}$ | $T_{5}$ | $\mathbb{Z}\oplus T_{6}$ | $\mathbb{Z}\oplus T_{7}$ | $T_{6}$ | $T_{5}$ | $\mathbb{Z}\oplus T_{4}$ | 0 | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}$
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15

Table 4: The $E_{2}$ page for the sphere bundle $S^{2}\rightarrow
W_{2,1}^{8}\rightarrow\widetilde{G}_{8,3}$

The only nontrivial differential is $d_{3}$ so we have $E_{2}=E_{3}$ and
$E_{4}=E_{\infty}$. Since $H^{5}(W_{2,1}^{8};\mathbb{Z})=0$, differential
$d_{3}:E_{3}^{2,2}\rightarrow E_{3}^{5,0}$ is onto, so $T_{4}=0$. Also, for
the same reason we have $d_{3}:E_{3}^{3,2}\rightarrow E_{3}^{6,0}$ is one-to-
one so $T_{5}$ contains an element of order two. On the other hand,
$H^{12}(W_{2,1}^{8};\mathbb{Z})=0$ so $d_{3}:E_{3}^{10,2}\rightarrow
E_{3}^{13,0}$ is one-to-one and $T_{5}$ is isomorphic to a subgroup of
$\mathbb{Z}_{2}$. This gives us $T_{5}\cong\mathbb{Z}_{2}$. Since
$E_{3}^{5,2}=T_{4}=0$, differential $d_{3}:E_{3}^{5,2}\rightarrow E_{3}^{8,0}$
is trivial, so $E_{\infty}^{8,0}\cong\mathbb{Z}\oplus T_{7}$, but from (1) we
see that this group must be free, as $E_{\infty}^{8,0}$ is a subgroup of the
free group $H^{8}(W_{2,1}^{8};\mathbb{Z})$, so $T_{7}=0$. We have
$E_{\infty}^{9,0}=0$, since it must be free as a subgroup of the free group
$H^{9}(W_{2,1}^{8};\mathbb{Z})$, so $d_{3}:E_{3}^{6,2}\rightarrow E_{3}^{9,0}$
is onto and since we know $T_{5}=\mathbb{Z}_{2}$, this gives us that $T_{6}$
is either $0$ or $\mathbb{Z}_{2}$.

$k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$H^{k}(\widetilde{G}_{8,3};\mathbb{Z})$ | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}\oplus T_{6}$ | $\mathbb{Z}$ | $T_{6}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | 0 | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}$
$H^{k}(\widetilde{G}_{8,3};\mathbb{Z}_{2})$ | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}\oplus T_{6}$ | $\mathbb{Z}_{2}\oplus T_{6}$ | $\mathbb{Z}_{2}\oplus T_{6}$ | $\mathbb{Z}_{2}\oplus T_{6}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}_{2}$

Table 5:  Cohomology groups of $\widetilde{G}_{8,3}$ with coefficients in
$\mathbb{Z}$ and $\mathbb{Z}_{2}$

Cohomology groups of $\widetilde{G}_{8,3}$ with coefficients in $\mathbb{Z}$
and $\mathbb{Z}_{2}$ are given in Table 5. Proposition 3.1 in [5] gives us
that $H^{6}(\widetilde{G}_{8,3};\mathbb{Z}_{2})\cong\mathbb{Z}_{2}$, but from
Table 5 we see that this group is isomorphic to $\mathbb{Z}_{2}\oplus T_{6}$.
This means $T_{6}=0$.

We have proved so far that the integral cohomology groups of
$\widetilde{G}_{8,3}$ are those given in Table 6.

$k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$H^{k}(\widetilde{G}_{8,3};\mathbb{Z})$ | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | 0 | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}$

Table 6:  Integral cohomology groups of $\widetilde{G}_{8,3}$

Let us label generators of these groups as
$y_{3},x_{4},y_{6},x_{7},x_{8},y_{10},x_{11},y_{13},x_{15}$ where the index of
a generator represents its degree. Some relations are obvious due to
dimensions, namely, $y_{3}^{3}=0$, $x_{4}^{3}=0$, $x_{7}^{2}=0$, because these
elements are in trivial cohomology groups.

2 | $1$ |  |  | $y_{3}$ | $x_{4}$ |  | $y_{6}$ | $x_{7}$ | $x_{8}$ |  | $y_{10}$ | $x_{11}$ |  | $y_{13}$ |  | $x_{15}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0 | $1$ |  |  | $y_{3}$ | $x_{4}$ |  | $y_{6}$ | $x_{7}$ | $x_{8}$ |  | $y_{10}$ | $x_{11}$ |  | $y_{13}$ |  | $x_{15}$
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15

Table 7:  Generators in the $E_{2}$ page for the sphere bundle
$S^{2}\rightarrow W_{2,1}^{8}\rightarrow\widetilde{G}_{8,3}$

We return now to the spectral sequence for the sphere bundle $S^{2}\rightarrow
W_{2,1}^{8}\rightarrow\widetilde{G}_{8,3}$. Since
$H^{3}(W_{2,1}^{8};\mathbb{Z})=0$, differential $d_{3}:E_{3}^{0,2}\rightarrow
E_{3}^{3,0}$ is onto. If we label a generator of $H^{2}(S^{2};\mathbb{Z})$
with $s$, we have $d_{3}(1\otimes s)=y_{3}\otimes 1$ which then gives us that
for any $x\otimes s\in E_{3}^{k,2}$

$d_{3}(x\otimes s)=xy_{3}\otimes 1.$

Further, $d_{3}:E_{3}^{3,2}\rightarrow E_{3}^{6,0}$ is an isomorphism and so
we get $y_{6}=y_{3}^{2}$. Similarly, $y_{10}=y_{3}x_{7}$ and
$y_{13}=y_{3}y_{10}=y_{3}^{2}x_{7}$.

The differential $d_{3}:E_{3}^{4,2}\rightarrow E_{3}^{7,0}$ is multiplication
by some integer, but since both groups $H^{6}(W_{2,1}^{8};\mathbb{Z})$ and
$H^{7}(W_{2,1}^{8};\mathbb{Z})$ are isomorphic to $\mathbb{Z}$, this integer
must be zero, i.e., this differential is trivial and so we get

$y_{3}x_{4}=0.$ (2)

There is a correspondence between elements of
$H^{*}(\widetilde{G}_{8,3};\mathbb{Z})$ and $H^{*}(W_{2,1}^{8};\mathbb{Z})$
which will give us relations between generators of
$H^{*}(\widetilde{G}_{8,3};\mathbb{Z})$ in dimensions $4,7,8,11$ and $15$,
where the free part of the cohomology group is nontrivial. Let us first notice
that all differentials $d_{3}:E_{3}^{k-3,2}\rightarrow E_{3}^{k,0}$ are
trivial for $k\in\\{4,7,8,11,15\\}$. We already explained the case $k=7$ when
proving the relation (2). The same argument works for $k=11$ and the other
cases are obvious due to the fact that $E_{3}^{k-3,2}=0$ for
$k\in\\{4,8,15\\}$. This means that $E_{2}^{k,0}=E_{\infty}^{k,0}$, for
$k\in\\{4,7,8,11,15\\}$.
Theorem 5.9 in [7] (p. 147) states that the composition

$H^{k}(\widetilde{G}_{8,3};\mathbb{Z})=E_{2}^{k,0}\twoheadrightarrow
E_{3}^{k,0}\twoheadrightarrow\cdots\twoheadrightarrow
E_{k+1}^{k,0}\twoheadrightarrow E_{\infty}^{k,0}\subseteq
H^{k}(W_{2,1}^{8};\mathbb{Z})$

is actually the map $p^{*}:H^{k}(\widetilde{G}_{8,3};\mathbb{Z})\rightarrow
H^{k}(W_{2,1}^{8};\mathbb{Z})$. In the case $k\in\\{4,7,8,11,15\\}$,
$E_{2}^{k,0}=E_{\infty}^{k,0}$, so we have that in these dimensions $p^{*}$ is
an injection.
The Leray-Serre spectral sequence comes with the filtration of the cohomology
groups of the total space

$0\subseteq F^{p}H^{p}\subseteq F^{p-1}H^{p}\subseteq\cdots\subseteq
F^{1}H^{p}\subseteq F^{0}H^{p}=H^{p}(W_{2,1}^{8};\mathbb{Z}),$

and $E_{\infty}^{r,p-r}\cong F^{r}H^{p}/F^{r+1}H^{p}$.
Let $k\in\\{4,7,11\\}$. We have
$E_{\infty}^{k,0}=F^{k}H^{k}=F^{0}H^{k}=H^{k}(W_{2,1}^{8};\mathbb{Z})$, which
implies that $p^{*}:H^{k}(\widetilde{G}_{8,3};\mathbb{Z})\rightarrow
H^{k}(W_{2,1}^{8};\mathbb{Z})$ is an isomorphism, and so the generators
$x_{4},x_{7}$ and $x_{11}$ can be chosen in such a way that

$p^{*}(x_{4})=\bar{x}_{2}^{2},~{}p^{*}(x_{7})=\bar{x}_{7},~{}p^{*}(x_{11})=\bar{x}_{2}^{2}\bar{x}_{7}.$
(3)

On the other hand, if $k\in\\{8,15\\}$ we have

$E_{\infty}^{k,0}=F^{k}H^{k}(W_{2,1}^{8};\mathbb{Z})=F^{k-1}H^{k}=2F^{k-2}H^{k}=2F^{0}H^{k}=2H^{k}(W_{2,1}^{8};\mathbb{Z}),$

since $E_{\infty}^{k-2,2}\cong F^{k-2}H^{k}/F^{k-1}H^{k}\cong\mathbb{Z}_{2}$.
The generators $x_{8}$ and $x_{15}$ can now be chosen with properties

$p^{*}(x_{8})=2\bar{x}_{2}\bar{x}_{6},~{}p^{*}(x_{15})=2\bar{x}_{2}\bar{x}_{6}\bar{x}_{7}.$
(4)

From relations in the algebra (1) and equations (3) and (4), we get

$\begin{split}p^{*}(x_{8})&=2\bar{x}_{2}\bar{x}_{6}=\bar{x}_{2}^{4}=p^{*}(x_{4}^{2}),\\\
p^{*}(x_{11})&=\bar{x}_{2}^{2}\bar{x}_{7}=p^{*}(x_{4}x_{7}),\\\
p^{*}(x_{15})&=2\bar{x}_{2}\bar{x}_{6}\bar{x}_{7}=\bar{x}_{2}^{4}\bar{x}_{7}=p^{*}(x_{4}^{2}x_{7}).\end{split}$

Since $p^{*}$ is an injection in dimensions 8, 11 and 15, we have the
following relations in $H^{*}(\widetilde{G}_{8,3};\mathbb{Z})$:

$x_{8}=x_{4}^{2},~{}x_{11}=x_{4}x_{7},~{}x_{15}=x_{4}^{2}x_{7},$

which, in addition to previous relations, give the specified structure of the
algebra $H^{*}(\widetilde{G}_{8,3};\mathbb{Z})$. ∎

######  Remark 3.

If $\rho:\mathbb{Z}\rightarrow\mathbb{Z}_{2}$ is the modulo 2 map, then

$\rho_{*}(y_{3})=\widetilde{w}_{3},~{}\rho_{*}(x_{4})=\widetilde{w}_{2}^{2},~{}\rho_{*}(x_{7})=a_{7},$

where $\widetilde{w}_{2},\widetilde{w}_{3}$ and $a_{7}$ are generators of
$H^{*}(\widetilde{G}_{8,3};\mathbb{Z}_{2})$ from Proposition 3.1 in [5]. This
is proven by using the universal coefficient theorem in the same way as in the
proof of Theorem 2.2.

##  6 Cohomology algebra $H^{*}(\widetilde{G}_{10,3};\mathbb{Z})$

In the following theorem we will prove that the integral cohomology algebra of
$\widetilde{G}_{10,3}$ is isomorphic to the quotient of the polynomial algebra
in five variables with coefficients in $\mathbb{Z}$ by the sum of ideals
$\mathcal{I}$ and $\mathcal{J}$. Since the dimension of $\widetilde{G}_{10,3}$
is 21, we know that $H^{k}(\widetilde{G}_{10,3};\mathbb{Z})=0$ for $k>21$, so
the ideal $\mathcal{J}$ will contain all monomials of (cohomological) degree
larger than 21. The ideal $\mathcal{I}$ will be generated by all homogeneous
polynomials corresponding to the relations in cohomology groups up to
dimension 21.

######  Theorem 6.1.

The integral cohomology algebra of $\widetilde{G}_{10,3}$ is given by

$H^{*}(\widetilde{G}_{10,3};\mathbb{Z})\cong\frac{\mathbb{Z}[y_{3},x_{4},x_{9},x_{12},x_{13}]}{\mathcal{I}+\mathcal{J}},$

where $y_{3}\in H^{3}(\widetilde{G}_{10,3};\mathbb{Z})$, $x_{i}\in
H^{i}(\widetilde{G}_{10,3};\mathbb{Z})$, for $i\in\\{4,9,12,13\\}$, the ideal
$\mathcal{I}$ is generated by polynomials:

$\begin{split}&2y_{3},~{}y_{3}^{3},~{}y_{3}^{2}x_{4},~{}y_{3}x_{4}^{2},~{}x_{4}^{3}-2x_{12},~{}y_{3}x_{9},~{}x_{4}x_{9}-2x_{13},\\\
&y_{3}x_{13}-x_{4}x_{12},~{}x_{9}^{2},~{}y_{3}^{2}x_{12},~{}x_{4}^{2}x_{12},~{}x_{9}x_{12}-x_{4}^{2}x_{13};\end{split}$

and the ideal $\mathcal{J}$ is generated by monomials:

$y_{3}^{a}x_{4}^{b}x_{9}^{c}x_{12}^{d}x_{13}^{e}~{}\text{such
that}~{}3a+4b+9c+12d+13e>21.$

###### Proof.

From [3] (p. 496), the Poincaré polynomial for $\widetilde{G}_{10,3}$ is

$p(\widetilde{G}_{10,3})=1+x^{4}+x^{8}+x^{9}+x^{12}+x^{13}+x^{17}+x^{21},$

so we know the free part of the cohomology groups of $\widetilde{G}_{10,3}$.

When we apply results from Section 4, we see that integral cohomology groups
of $\widetilde{G}_{10,3}$ are:

$\begin{split}H^{*}(\widetilde{G}_{10,3};\mathbb{Z})=(&\mathbb{Z},0,0,\mathbb{Z}_{2},\mathbb{Z},T_{4},T_{5},T_{6},\mathbb{Z}\oplus
T_{7},\mathbb{Z}\oplus T_{8},T_{9},T_{10},\mathbb{Z}\oplus T_{9},\\\
&\mathbb{Z}\oplus T_{8},T_{7},T_{6},T_{5},\mathbb{Z}\oplus
T_{4},0,\mathbb{Z}_{2},0,\mathbb{Z}).\end{split}$

What remains is to determine these seven torsion groups.

Recall from Theorem 3.1 the cohomology algebra of $W_{2,1}^{10}$

$H^{*}(W^{10}_{2,1};\mathbb{Z})\cong\frac{\mathbb{Z}[\bar{x}_{2},\bar{x}_{8},\bar{x}_{9}]}{\langle\bar{x}_{2}^{4}-2\bar{x}_{8},\bar{x}_{8}^{2},\bar{x}_{9}^{2}\rangle}.$
(5)

Consider the Leray-Serre spectral sequences for the sphere bundle
$S^{2}\rightarrow W_{2,1}^{10}\xrightarrow{p}\widetilde{G}_{10,3}$ and for the
fiber bundle $SO(3)\rightarrow V_{10,3}\rightarrow\widetilde{G}_{10,3}$. The
cohomology ring $H^{*}(V_{10,3};\mathbb{Z})$ is known from Theorem 2.3 in [2].

In order to differentiate these spectral sequences, we will label the first
one as $E$ and the second one as $\bar{E}$. Arguments similar as in the proof
of Theorem 5.1 give us $T_{4}=0$, $T_{5}=\mathbb{Z}_{2}$ and $T_{7}=0$. Since
$H^{9}(W_{2,1}^{10};\mathbb{Z})\cong\mathbb{Z}$, $E_{\infty}^{9,0}$ must be
free, so $d_{3}:E_{3}^{6,2}\rightarrow E_{3}^{9,0}$ from Table 8 maps
$\mathbb{Z}_{2}$ onto $T_{8}$, i.e., $T_{8}$ is either $\mathbb{Z}_{2}$ or
trivial. From Table 8 we also see that
$H^{8}(\widetilde{G}_{10,3};\mathbb{Z}_{2})\cong\mathbb{Z}_{2}\oplus T_{8}$,
but from Proposition 3.1 in [5] this group is isomorphic to $\mathbb{Z}_{2}$,
so we get $T_{8}=0$.

2 | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $T_{6}$ | $\mathbb{Z}$ | $\mathbb{Z}\oplus T_{8}$ | $T_{9}$ | $T_{10}$ | $\mathbb{Z}\oplus T_{9}$ | $\mathbb{Z}\oplus T_{8}$ | $0$ | $T_{6}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $0$ | $\mathbb{Z}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0 | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $T_{6}$ | $\mathbb{Z}$ | $\mathbb{Z}\oplus T_{8}$ | $T_{9}$ | $T_{10}$ | $\mathbb{Z}\oplus T_{9}$ | $\mathbb{Z}\oplus T_{8}$ | $0$ | $T_{6}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $0$ | $\mathbb{Z}$
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21

Table 8: The $E_{2}$ page for the sphere bundle $S^{2}\rightarrow
W_{2,1}^{10}\rightarrow\widetilde{G}_{10,3}$

Since $H^{18}(V_{10,3};\mathbb{Z})=0$ [2], Table 9 gives us that differential
$d_{3}:\bar{E}_{3}^{16,2}\rightarrow\bar{E}_{3}^{19,0}$ is an isomorphism and
$\bar{E}_{4}^{19,0}=0$. This means that
$d_{2}:\bar{E}_{2}^{15,3}\rightarrow\bar{E}_{2}^{17,2}$ is a monomorphism, as
this is the only nontrivial differential with domain $E_{k}^{15,3}$. On the
other hand, since $H^{19}(V_{10,3};\mathbb{Z})=0$ [2], this differential must
also be an epimorphism. So we get $T_{6}\cong\mathbb{Z}_{2}$.

3 | $0$ | $T_{6}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | 0
---|---|---|---|---|---|---|---
2 | $\mathrm{Tor}(T_{6},\mathbb{Z}_{2})$ | $T_{6}\otimes\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | 0
1 |  |  |  |  |  |  |
0 | $0$ | $T_{6}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | 0
| 14 | 15 | 16 | 17 | 18 | 19 | 20

Table 9: The part of the $\bar{E}_{2}$ page for the fiber bundle
$SO(3)\rightarrow V_{10,3}\rightarrow\widetilde{G}_{10,3}$

If we return to Table 8, differential $d_{3}:E_{3}^{7,2}\rightarrow
E_{3}^{10,0}$ must be onto since $H^{10}(W_{2,1}^{10};\mathbb{Z})$ is free.
This means that $T_{9}$ is either $\mathbb{Z}_{2}$ or trivial, as now we know
that $E_{3}^{7,2}\cong T_{6}\cong\mathbb{Z}_{2}$. From Table 8 we get that
$H^{9}(\widetilde{G}_{10,3};\mathbb{Z}_{2})\cong\mathbb{Z}_{2}\oplus T_{9}$,
but from Proposition 3.1 in [5] this group is isomorphic to $\mathbb{Z}_{2}$,
so it must be $T_{9}=0$.

What remains is to determine the torsion group $T_{10}$. In Table 10 the other
part of the second spectral sequence is given. Since
$H^{14}(V_{10,3};\mathbb{Z})=0$ [2], differential
$d_{3}:\bar{E}_{3}^{12,2}\rightarrow\bar{E}_{3}^{15,0}$ is an isomorphism, so
the only possible nontrivial differential with domain $\bar{E}_{k}^{11,3}$ is
$d_{2}:\bar{E}_{2}^{11,3}\rightarrow\bar{E}_{2}^{13,2}$ and it must be a
monomorphism because of the same argument $H^{14}(V_{10,3};\mathbb{Z})=0$.
This gives us that $T_{10}$ is isomorphic to a subgroup of $\mathbb{Z}_{2}$.
When we combine this result with the fact that
$H^{10}(\widetilde{G}_{10,3};\mathbb{Z}_{2})=0$ from Proposition 3.1 in [5],
we get $T_{10}=0$.

3 | $0$ | $T_{10}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$
---|---|---|---|---|---|---
2 | $\mathrm{Tor}(T_{10},\mathbb{Z}_{2})$ | $T_{10}\otimes\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$
1 |  |  |  |  |  |
0 | $0$ | $T_{10}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$
| 10 | 11 | 12 | 13 | 14 | 15

Table 10: The part of the $\bar{E}_{2}$ page for the fiber bundle
$SO(3)\rightarrow V_{10,3}\rightarrow\widetilde{G}_{10,3}$

Finally, we have integral cohomology groups of $\widetilde{G}_{10,3}$:

$k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$H^{k}(\widetilde{G}_{10,3};\mathbb{Z})$ | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $0$ | $\mathbb{Z}$

Table 11: Integral cohomology groups of $\widetilde{G}_{10,3}$

Let us label generators of these groups as
$y_{3},x_{4},y_{6},y_{7},x_{8},x_{9},x_{12},x_{13},$
$y_{15},y_{16},x_{17},y_{19},x_{21}$ where the index of a generator represents
its degree. In the same way as in the proof of Theorem 5.1, we conclude that
the only nontrivial differential $d_{3}:E_{3}^{k,2}\rightarrow E_{3}^{k+3,0}$
is given by

$d_{3}(x\otimes s)=xy_{3}\otimes 1,$

for $x\in H^{k}(\widetilde{G}_{10,3};\mathbb{Z})$ and that the following
relations hold:

$y_{6}=y_{3}^{2},y_{7}=y_{3}x_{4},y_{15}=y_{3}x_{12},y_{16}=y_{3}x_{13},y_{19}=y_{3}^{2}x_{13}.$

We will now use Theorem 5.9 in [7] (p. 147) in the same way as in the proof of
Theorem 5.1 in order to determine relations between generators.

2 |  $\mathbb{Z}$ | 0 | 0 |  $\mathbb{Z}_{2}$ |  $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $0$ |  $\mathbb{Z}$ |  $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ |  $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $0$ | $\mathbb{Z}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0 | $\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $0$ | $\mathbb{Z}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $0$ | $\mathbb{Z}_{2}$ | $0$ | $\mathbb{Z}$
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21

Table 12: The $E_{3}$ page for the sphere bundle $S^{2}\rightarrow
W_{2,1}^{10}\rightarrow\widetilde{G}_{10,3}$

In the case $k\in\\{4,8,9,12,13,17,21\\}$ we see from Table 12 (where all but
trivial differentials are represented by arrows and dimensions of nontrivial
cohomology groups of $W_{2,1}^{10}$ are in bold) that
$E_{2}^{k,0}=E_{\infty}^{k,0}$, so we have that in these dimensions $p^{*}$ is
an injection.
Let $k\in\\{4,12,13\\}$. We have
$E_{\infty}^{k,0}=F^{k}H^{k}=F^{0}H^{k}=H^{k}(W_{2,1}^{10};\mathbb{Z})$, so
$p^{*}$ maps a generator of $H^{k}(\widetilde{G}_{10,3};\mathbb{Z})$ to the
specified (in (5)) generator of $H^{k}(W_{2,1}^{10};\mathbb{Z})$, i.e., we can
choose generators $x_{4},x_{12}$ and $x_{13}$ to have

$p^{*}(x_{4})=\bar{x}_{2}^{2},~{}p^{*}(x_{12})=\bar{x}_{2}^{2}\bar{x}_{8},~{}p^{*}(x_{13})=\bar{x}_{2}^{2}\bar{x}_{9}.$
(6)

On the other hand, if $k\in\\{8,9,17,21\\}$ we have

$E_{\infty}^{k,0}=F^{k}H^{k}(W_{2,1}^{10};\mathbb{Z})=F^{k-1}H^{k}=2F^{k-2}H^{k}=2F^{0}H^{k}=2H^{k}(W_{2,1}^{10};\mathbb{Z}),$

since $E_{\infty}^{k-2,2}\cong F^{k-2}H^{k}/F^{k-1}H^{k}\cong\mathbb{Z}_{2}$.
So we choose $x_{8},x_{9},x_{17}$ and $x_{21}$ with properties

$p^{*}(x_{8})=2\bar{x}_{8},~{}p^{*}(x_{9})=2\bar{x}_{9},~{}p^{*}(x_{17})=2\bar{x}_{8}\bar{x}_{9},~{}p^{*}(x_{21})=2\bar{x}_{2}^{2}\bar{x}_{8}\bar{x}_{9}.$
(7)

From relations in the algebra (5) and equations (6) and (7), we get

$\begin{split}p^{*}(x_{8})&=2\bar{x}_{8}=\bar{x}_{2}^{4}=p^{*}(x_{4}^{2}),\\\
p^{*}(2x_{12})&=2\bar{x}_{2}^{2}\bar{x}_{8}=\bar{x}_{2}^{6}=p^{*}(x_{4}^{3}),\\\
p^{*}(2x_{13})&=2\bar{x}_{2}^{2}\bar{x}_{9}=p^{*}(x_{4}x_{9}),\\\
p^{*}(x_{17})&=2\bar{x}_{8}\bar{x}_{9}=\bar{x}_{2}^{2}\bar{x}_{2}^{2}\bar{x}_{9}=p^{*}(x_{4}x_{13}),\\\
p^{*}(x_{21})&=2\bar{x}_{2}^{2}\bar{x}_{8}\bar{x}_{9}=\bar{x}_{2}^{4}\bar{x}_{2}^{2}\bar{x}_{9}=p^{*}(x_{4}^{2}x_{13})=p^{*}(x_{9}x_{12}).\end{split}$

Since $p^{*}$ is an injection in these dimensions, we have the following
relations in $H^{*}(\widetilde{G}_{10,3};\mathbb{Z})$:

$x_{8}=x_{4}^{2},~{}2x_{12}=x_{4}^{3},~{}2x_{13}=x_{4}x_{9},~{}x_{17}=x_{4}x_{13},~{}x_{21}=x_{4}^{2}x_{13}=x_{9}x_{12}.$

This means that cohomology algebra $H^{*}(\widetilde{G}_{10,3};\mathbb{Z})$ is
generated by five generators $y_{3},x_{4},x_{9},x_{12}$ and $x_{13}$.
Some relations are obvious due to dimension. We have $y_{3}^{2}x_{4}=0$,
$y_{3}x_{4}^{2}=0$, $x_{9}^{2}=0$, $y_{3}^{2}x_{12}=0$,
$x_{4}^{2}x_{12}=y_{3}x_{4}x_{13}=0$ since these elements are in trivial
cohomology groups. Other relations are obvious since cup product of $y_{3}$
and any other element must be trivial in torsion free groups, so $y_{3}^{3}=0$
and $y_{3}x_{9}=0$. Also, $y_{3}^{a}x_{4}^{b}x_{9}^{c}x_{12}^{d}x_{13}^{e}=0$
whenever $3a+4b+9c+12d+13e>21$ since the dimension of $\widetilde{G}_{10,3}$
is 21.
What remains is to prove the relation $y_{3}x_{13}=x_{4}x_{12}$. We will use
naturality of the universal coefficient theorem and Proposition 3.1 in [5].
This proposition gives us

$H^{4}(\widetilde{G}_{10,3};\mathbb{Z}_{2})=\mathbb{Z}_{2}\langle\widetilde{w}_{2}^{2}\rangle,H^{12}(\widetilde{G}_{10,3};\mathbb{Z}_{2})=\mathbb{Z}_{2}\langle
a_{12}\rangle,H^{16}(\widetilde{G}_{10,3};\mathbb{Z}_{2})=\mathbb{Z}_{2}\langle\widetilde{w}_{2}^{2}a_{12}\rangle.$

We have the following commutative diagram:

${0}$${\mathrm{Ext}(H_{k-1}(\widetilde{G}_{10,3}),\mathbb{Z})}$${H^{k}(\widetilde{G}_{10,3};\mathbb{Z})}$${\mathrm{Hom}(H_{k}(\widetilde{G}_{10,3}),\mathbb{Z})}$${0}$${0}$${\mathrm{Ext}(H_{k-1}(\widetilde{G}_{10,3}),\mathbb{Z}_{2})}$${H^{k}(\widetilde{G}_{10,3};\mathbb{Z}_{2})}$${\mathrm{Hom}(H_{k}(\widetilde{G}_{10,3}),\mathbb{Z}_{2})}$${0}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\rho_{*}}$

For $k=4$ both $\mathrm{Ext}$-groups are zero and
$\rho_{*}:\mathrm{Hom}(H_{4}(\widetilde{G}_{10,3}),\mathbb{Z})\rightarrow\mathrm{Hom}(H_{4}(\widetilde{G}_{10,3}),\mathbb{Z}_{2})$
is an epimorphism, so
$\rho_{*}:H^{4}(\widetilde{G}_{10,3};\mathbb{Z})\rightarrow
H^{4}(\widetilde{G}_{10,3};\mathbb{Z}_{2})$ is an epimorphism, i.e.,
$\rho_{*}(x_{4})=\widetilde{w}_{2}^{2}$. The situation is the same for $k=12$,
so $\rho_{*}(x_{12})=a_{12}$. Now we have
$\rho_{*}(x_{4}x_{12})=\widetilde{w}_{2}^{2}a_{12}\neq 0$, so $x_{4}x_{12}\neq
0$, i.e., $x_{4}x_{12}=y_{3}x_{13}$. ∎

######  Remark 4.

Since $\mathbb{Z}[y_{3},x_{4},x_{9},x_{12},x_{13}]$ is a Noetherian ring, it
is possible to find a finite generating set for the ideal
$\mathcal{I}+\mathcal{J}$ from Theorem 6.1. One can do this by starting with
the generators of $\mathcal{I}$, then going through all elements
$y_{3}^{a}x_{4}^{b}x_{9}^{c}x_{12}^{d}x_{13}^{e}$ with $3a+4b+9c+12d+13e=k$
for $k=22,23,24,\ldots$ to see which elements are already in $\mathcal{I}$,
and then adding those that are not. It suffices to do this checking up to
dimension $k=34$ since all the monomials of degree greater than $34$ are
multiples of those in degrees $22\leqslant k\leqslant 34$ (this is because the
maximal degree of generators $y_{3},x_{4},x_{9},x_{12}$ and $x_{13}$ is $13$).
After going through this procedure, we obtain that there are only a few
elements that need to be added to the generating set of $\mathcal{I}$ in order
to get the whole ideal $\mathcal{I}+\mathcal{J}$, namely
$x_{9}x_{13},x_{12}^{2},x_{12}x_{13}$ and $x_{13}^{2}$. Finally, we can write
the integral cohomology algebra of $\widetilde{G}_{10,3}$ as

$H^{*}(\widetilde{G}_{10,3};\mathbb{Z})\cong\frac{\mathbb{Z}[y_{3},x_{4},x_{9},x_{12},x_{13}]}{\mathcal{K}},$

where $\mathcal{K}$ is generated by polynomials:

$\begin{split}&2y_{3},~{}y_{3}^{3},~{}y_{3}^{2}x_{4},~{}y_{3}x_{4}^{2},~{}x_{4}^{3}-2x_{12},~{}y_{3}x_{9},~{}x_{4}x_{9}-2x_{13},\\\
&y_{3}x_{13}-x_{4}x_{12},~{}x_{9}^{2},~{}y_{3}^{2}x_{12},~{}x_{4}^{2}x_{12},~{}x_{9}x_{12}-x_{4}^{2}x_{13},\\\
&x_{9}x_{13},~{}x_{12}^{2},~{}x_{12}x_{13},~{}x_{13}^{2}.\end{split}$

######  Remark 5.

If $\rho:\mathbb{Z}\rightarrow\mathbb{Z}_{2}$ is the modulo 2 map, then

$\rho_{*}(y_{3})=\widetilde{w}_{3},~{}\rho_{*}(x_{4})=\widetilde{w}_{2}^{2},~{}\rho_{*}(x_{9})=\widetilde{w}_{2}^{3}\widetilde{w}_{3},~{}\rho_{*}(x_{12})=a_{12},~{}\rho_{*}(x_{13})=a_{13},$

where $\widetilde{w}_{2},\widetilde{w}_{3},a_{12}$ and $a_{13}$ are generators
of $H^{*}(\widetilde{G}_{10,3};\mathbb{Z}_{2})$ from Proposition 3.1 in [5].
Two of these relations we have proved in the proof of Theorem 6.1 and the
other three are proven in the same way by using the universal coefficient
theorem and Poincaré duality.

##  7 Acknowledgements

This research was supported by the Science Fund of the Republic of Serbia,
Grant No. 7749891, Graphical Languages \- GWORDS.

## References

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  * Čadek, Mimura, Vanžura [2003] M. Čadek, M. Mimura, J. Vanžura, The cohomology rings of real Stiefel manifolds with integer coefficients, J. Math. Kyoto Univ. 43-2 (2003), 411–428
  * Greub, Halperin, Vanstone [1976] W. Greub, S. Halperin, R. Vanstone, Connections, curvature, and cohomology, Volume III: Cohomology of principal bundles and homogeneous spaces, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, (1976)
  * Kalafat, Yalçınkaya [2021] M. Kalafat, E. Yalçınkaya, Algebraic topology of special Lagrangian manifolds, Indag. Math. 32 (2021) 579–597
  * Korbaš, Rusin [2016] J. Korbaš, T. Rusin, A note on the $\mathbb{Z}_{2}$-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo 2(65) (2016), 507–517
  * Lai [1974] H.-F. Lai, On the topology of the even-dimensional complex quadrics, Proc. Am. Math. Soc 46-3 (1974), 419–425
  * McCleary [2001] J. McCleary, A user’s guide to spectral sequences, Second Edition, Cambridge University Press, (1976)
  * Switzer [1975] R. M. Switzer, Algebraic topology – homotopy and homology, Classics in Mathematics, Springer-Verlag, Berlin, 2002. Reprint of the 1975 original.
  * Vanzura [1999] J. Vanžura, The cohomology of $\widetilde{G}_{m,2}$ with integer coefficients, The 18th Winter School ”Geometry and Physics” (Srní, 1998). Rend. Circ. Mat. Palermo 2(Suppl. 59), (1999), 201-208


11institutetext:  IRAP, Université de Toulouse, CNRS, UPS, CNES 9, Avenue du
Colonel Roche, BP 44346, F-31028, Toulouse Cedex 4, France, 11email:
<EMAIL_ADDRESS>22institutetext: INAF – Osservatorio Astrofisico di
Arcetri, Largo Enrico Fermi 5, I-50125 Firenze, Italy 33institutetext: Centre
National d’Études Spatiales (CNES), 18 Avenue Edouard Belin, 31400 Toulouse
Cedex 4, France 44institutetext: Department of Physics and Astronomy,
University of Leicester, Leicester, UK 55institutetext: Dr. Karl Remeis-
Observatory and Erlangen Centre for Astroparticle Physics, Sternwartstr. 7,
D-96049 Bamberg, Germany 66institutetext: ESA European Space Research and
Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ, Noordwĳk, The Netherlands
77institutetext: Department of Astronomy, University of Michigan, 1085 South
University Avenue, Ann Arbor, MI 48109, USA

# The defocused observations of bright sources with Athena/X-IFU

E. S. Kammoun The defocused observations of bright sources with
Athena/X-IFUThe defocused observations of bright sources with Athena/X-IFUThe
defocused observations of bright sources with Athena/X-IFUThe defocused
observations of bright sources with Athena/X-IFU D. Barret The defocused
observations of bright sources with Athena/X-IFUThe defocused observations of
bright sources with Athena/X-IFU P. Peille The defocused observations of
bright sources with Athena/X-IFUThe defocused observations of bright sources
with Athena/X-IFU R. Willingale The defocused observations of bright sources
with Athena/X-IFUThe defocused observations of bright sources with
Athena/X-IFU T. Dauser The defocused observations of bright sources with
Athena/X-IFUThe defocused observations of bright sources with Athena/X-IFU J.
Wilms The defocused observations of bright sources with Athena/X-IFUThe
defocused observations of bright sources with Athena/X-IFU M. Guainazzi The
defocused observations of bright sources with Athena/X-IFUThe defocused
observations of bright sources with Athena/X-IFU J. M. Miller The defocused
observations of bright sources with Athena/X-IFUThe defocused observations of
bright sources with Athena/X-IFU

(Received ; accepted )

###### Abstract

Context. The X-ray Integral Field Unit (X-IFU) is the high resolution X-ray
spectrometer of ESA’s Athena X-ray observatory. It will deliver X-ray data in
the $0.2-12$ keV band with an unprecedented spectral resolution of 2.5 eV up
to 7 keV. During the observation of very bright X-ray sources, the X-IFU
detectors will receive high photon rates. The count rate capability of the
X-IFU will be improved by using the defocusing option, which will enable the
observations of extremely bright sources with fluxes up to $\simeq 1$ Crab. In
the defocused mode, the point spread function (PSF) of the telescope will be
spread over a large number of pixels. In this case, each pixel receives a
small fraction of the overall flux. Due to the energy dependence of the PSF,
this mode will generate energy dependent artefacts increasing with count rate
if not analysed properly. To account for the degradation of the energy
resolution with pulse separation in a pixel, a grading scheme (here four
grades) will be defined to affect the proper energy response to each event.
This will create selection effects preventing the use of the nominal Auxiliary
Response File (ARF) for all events.

Aims. We present a new method for the reconstruction of the spectra obtained
from observations performed with a PSF that varies as a function of energy. We
apply our method to the case of the X-IFU spectra obtained during the
defocused observations.

Methods. We use the end-to-end SIXTE simulator to model defocused X-IFU
observations. Then we estimate new ARF for each of the grades by calculating
the effective area at the level of each pixel.

Results. Our method allows us to successfully reconstruct the spectra of
bright sources when employed in the defocused mode, without any bias. Finally,
we address how various sources of uncertainty related to our knowledge of the
PSF as a function of energy affect our results.

###### Key Words.:

Instrumentation: detectors – Techniques: spectroscopic – X-rays: general

##  1 Introduction

The X-ray Integral Field Unit (X-IFU; Barret et al., 2016, 2018) is the high
resolution X-ray spectrometer that is planned for launch on-board the European
Space Agency Athena X-ray observatory (Nandra et al., 2013). The instrument
will deliver X-ray spectra in the $0.2-12$ keV range with a spectral
resolution of 2.5 eV up to 7 keV. The overall field of view of 5′ equivalent
diameter will be subdivided into several thousand of ${\sim}5$″ pixels111The
exact size of the pixels has not yet been decided but this will not impact the
results reported in this work.. The main scientific goals of the instruments
are 1) to study the dynamical, physical, and chemical properties of hot
plasmas, notably those found in galaxy clusters, and 2) to study the extreme
environment around galactic stellar-mass and supermassive black holes,
accretion discs, jets, outflows and winds. In addition, the unprecedented
capabilities of the X-IFU will enable the study of many Athena observatory
science targets, such as planets, stars, supernovæ, compact objects, and
interstellar medium. The throughput of the X-IFU decreases at high count rate.
In fact, the count rate capability of the X-IFU is limited by the pixel speed,
the record length required to achieve the spectral resolution, and the
crosstalk level (see Peille et al., 2018). This capability will be improved by
using the defocusing option offered by the Movable Mirror Assembly (MMA). A
defocusing of 35 mm (with respect to the nominal focal length of 12 m) will
enable the observations of extremely bright galactic sources with fluxes up to
${\sim}1$ Crab (Peille et al., 2018), with only limited spectral resolution
degradation ($\lesssim 10$ eV). In this case, the point spread function (PSF)
of the telescope will be spread over a large number of pixels so that each
pixel receives a small fraction of the overall flux. Due to the energy
dependence of the PSF shape, this mode will generate energy dependent
artefacts increasing with count rate if not analysed properly. Peille et al.
(2018) focused on the ability to obtain a good enough throughput in the
defocused PSF mode, without addressing the ability to reconstruct broadband
energy spectra. Here, we continue this work. We present a new method that
allows the analysis of X-IFU observations of bright sources, in the defocused
configuration. Whereas we focus here on the particular case of the X-IFU, we
emphasize that the analysis presented below applies to any observatory with an
energy dependent PSF and count rate dependent selection effects at the level
of individual pixels (grading, dead time, etc…).

Figure 1: The defocused PSF of X-IFU at 13 energies. The bottom rightmost
panel shows the in-focus PSF at 1 keV ($\rm HEW=5$″), for comparison. The grid
in this pannel shows the X-IFU pixels. We note that the in-focus PSF is shown
on a $20\arcsec\times 20\arcsec$ image while the defocused PSF is shown on a
$200\arcsec\times 200\arcsec$ image. The full field of view of the X-IFU is of
the order of 300″ (in diameter) as shown in Fig. 2. All the PSFs are shown in
the linear scale.

In Sect. 2, we present the Athena defocused PSF. In Sect. 3, we introduce the
method we developed to analyse the simulated spectra. In Sect. 4, we study the
various sources of uncertainties that may affect the performance of the
instrument. In Sect. 5, we show an example of the ability of X-IFU to
constrain the wind absorption in bright X-ray binaries thanks to its
defocusing mode. Finally, we summarise our results in Sect. 6.

##  2 Defocused PSF

Figure 1 shows the model defocused Athena PSF at different energies222The PSF
can be obtained from the following link
https://www.cosmos.esa.int/web/athena/resources-by-esa. The PSF is extended up
to $\simeq 140$″ in diameter in the soft X-rays. For comparison, we show the
in-focus PSF at 1 keV assuming a half-energy width (HEW) of 5″, comparable to
the X-IFU pixel size. In Fig. 2, we show the defocused PSF at 1 keV together
with the X-IFU field of view.

Figure 2: A zoom-out on the PSF at 1 keV, showing the full X-IFU field of
view. We note that some detailed features and the spatial variations in the
PSF are happening at a level that is comparable to the X-IFU pixel size. The
same color bar as in Fig. 1 applies here.

The model defocused PSF is generated using a detailed physical model of the
complete Athena mirror system. The model includes the full X-ray energy
dependence of the PSF of each of the ${\sim}600$ mirror modules that populate
the full aperture. The position of each module within the aperture is set such
that the individual PSFs align in the focal plane to produce the in-focus PSF
with HEW 5″. The telescope is defocused by shifting the mirror along the
optical axis by 35 mm away from the in-focus position. In this configuration,
the defocused PSF comprises a mapping of the aperture layout of the modules,
including the module support structure, convolved with the PSF of the
individual modules. The defocused PSF has a very strong energy dependence
because the effective area of the modules depends on the radial position of
each module within the aperture. Low energy X-rays ($<1.5$ keV) are focused by
all the modules but as the energy increases, the outer modules have reduced
reflectivity/efficiency and at $>10$ keV only the innermost modules produce
any significant effective area.

##  3 Method

###  3.1 Simulations

We use the SImulation of X-ray TElescopes333http://www.sternwarte.uni-
erlangen.de/research/sixte/ (SIXTE) software package (Dauser et al., 2019) to
simulate X-IFU observations. SIXTE is designed to perform end-to-end
simulations of various X-ray observatories, including Athena. Photons are
generated in a Monte-Carlo framework and then followed through the full
imaging and detection process. The package uses a set of calibration files
allowing to properly distribute the photon impacts, with relevant energy and
timing properties, onto the focal plane. In particular, it allows us to
simulate energy dependent mirror defocusing of Athena, which is the subject of
this work. We note that SIXTE adopts a linear interpolation scheme between the
different energies at which the PSF is defined for lack of a more physically
motivated interpolation scheme. At the end of the process, the readout energy
of each event is computed through a set of response matrices and coupling
mechanisms, taking into account the degradation of the instrument performance
with count rate (e.g., the crosstalk and the event grading; see Peille et al.,
2018). These are determined using SIXTE’s tool xifusim (Kirsch et al., 2020;
Lorenz et al., 2020) which provides a representative simulation of the full
detection pipeline of the X-IFU including all relevant detector physics and
the behavior of the readout chain.

The X-IFU pixels are microcalorimeters: when an X-ray photon impacts the
pixel, it will thermalize in an absorber thermally linked to a Transition Edge
Sensor (TES; Smith et al., 2016) microcalorimeter (originally operating at a
temperature of ${\sim}90$ mK) whose temperature will increase, leading to a
rapid increase in resistance which produces a corresponding rapid decrease in
current passing through the TES. This signal will be then used to estimate the
energy of the event, through the application of an optimal filter (e.g.,
Szymkowiak et al., 1993; Peille et al., 2016). At high count rates, the
current pulses will get packed together in the pixels’ timelines. Thus,
considering an example of a pulse triplet, if the preceding event is too close
to the middle event, the pulse tail from the preceding pulse leaks signal into
the middle one and biases the energy estimation (see Peille et al., 2018).
When the succeeding event is too close to the middle event, it limits the
length for reconstructing the energy of the middle event, leading to a
degradation of the energy resolution. To characterize these effects, four
grades - namely: high resolution (HR), medium resolution (MR), limited
resolution (LimR), and low resolution (LowR) - are defined in the simulator
depending on the time separation of the pulses. It is worth noting that this
grading scheme is applied at the level of each X-IFU pixel.

In this work, we focus only on the effect of the grading on the broadband
energy spectrum, without addressing other degradation factors that may affect
the instrument performance such as pure event pile-up444The pile-up effect in
the X-IFU should at a much lower level than the grading impact studied here.
In all observation scenarios, the pileup fraction remains below 1% and the
undetected fraction below 0.1% (see Cobo et al., 2018).  (i.e, two photons
being reconstructed as one), and crosstalk. Table 1 summarizes the assumptions
of time separation that we use in this work. The definition of the event
grades and the responses of SIXTE are informed by detailed simulations of the
detection process (see e.g., Doriese et al., 2009; Peille et al., 2018; Kirsch
et al., 2020). These definitions represent realistic assumptions, but are not
the ultimate configuration that will be adopted for full operation of the
instrument. In fact, the detailed X-IFU grading rules are still being
optimized and are constantly evolving. As a consequence, the values reported
here constitute a snapshot at the time of the simulations, explaining why they
differ slightly from the earlier ones presented by Peille et al. (2018).
However, we note that any changes in the definition of the grades will not
affect the results of our analysis.

Table 1: A typical set of parameters used for the event grade selection in this work. Grade | Time since | Time until | Energy
---|---|---|---
| previous pulse | next pulse | resolution
High resolution | 7.9 ms | 52.4 ms | 2.5 eV
Medium resolution | 7.9 ms | 3.3 ms | 3 eV
Limited resolution | 7.9 ms | 1.6 ms | 7 eV
Low resolution | 7.9 ms | 51.2 $\rm\mu s$ |  $\sim 30$ eV

The spectral analysis in this work is performed using the X-ray spectral
fitting package XSPEC v12.11.1 (Arnaud, 1996). We use the Cash statistics
(C-stat; Cash, 1979) to evaluate the best fits. In this work, the simulations
are performed by assuming the following definition of the Crab555We note that
we do not simulate the actual Crab source, which is known to be an extended
source. Instead, the Crab is used to define a standard spectral shape and flux
level. All the simulations used in this work assume a point-like source., in
the XSPEC parlance:

${\tt Model=TBabs\times powerlaw,}$

where TBabs (Wilms et al., 2000) represents the Galactic absorption assumed to
be $\hbox{$N_{\rm H}$}=4\times 10^{21}\leavevmode\nobreak\ \rm cm^{-2}$, and
the power-law photon index $\Gamma=2.1$. In this work, we consider two flux
levels at 200 mCrab and 1 Crab corresponding to a normalization of 1.9 and
$9.5\leavevmode\nobreak\ \rm photon\leavevmode\nobreak\
s^{-1}\leavevmode\nobreak\ cm^{-2}\leavevmode\nobreak\ keV^{-1}$ at 1 keV,
respectively. These models are equivalent to an observed 2–10 keV flux of
$0.41\times 10^{-8}$ and $2.05\times 10^{-8}\leavevmode\nobreak\ \rm
erg\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ cm^{-2}$, respectively. In
the case of a 1 Crab flux level, we assume a $100\leavevmode\nobreak\ \rm\mu
m$ thick beryllium filter which suppresses $\sim 96\%$ and $33\%$ of the
events at ${\sim}1.5$ keV and 3 keV, respectively (Barret et al., 2018; Peille
et al., 2018). This helps maximizing the throughput in the 5–8 keV range,
where disc/wind features are expected in accreting objects. We also assume the
wilm abundance and vern cross section (Verner et al., 1996). We ran the
simulations with an exposure time of 2 ks and 10 ks, for each of the flux
levels, assuming that the source is constant during the given exposure.

###  3.2 Reconstruction

Figure 3: The spectra extracted (in units of $\rm count\leavevmode\nobreak\
s^{-1}$) from two adjacent pixels (shown in red and blue) for the HR and MR
event grades (left and right, respectively). The spectra are rebinned for
clarity reasons. Figure 4: Simulated MR spectrum assuming a flux level of 200
mCrab (with an exposure of 10 ks), using the nominal ARF (i.e., without
applying any reconstruction) in red and the reconstructed ARF in green. The
black dashed line corresponds to the model obtained by fitting the spectrum
using the nominal ARF, fixing $N_{\rm H}$ and $\Gamma$ to the input values,
letting only the normalization free. We show the corresponding residuals in
the bottom panel (red points). In the case of the reconstructed ARF, we show
the residuals using the input parameters used in the simulations, without
performing any fit. The spectra are rebinned for clarity reasons. Figure 5:
Upper panel: The reconstructed ARFs for the various event grades. The thick
solid line shows the nominal ARF for comparison. Lower panel: The ratio of the
ARF at each grade divided by the nominal ARF.

The output of SIXTE is a set of four spectra (one spectrum per event grade).
As a result of the defocusing of the optics and the event grade scheme, the
ARF of each grade will differ from the nominal ARF ($\rm ARF_{nom}$) of the
instrument. The fraction of each grade in the pixels depends on the count rate
the pixel detects. This depends on the shape of the PSF, and on the spectrum
of the source. Thus, for each observation, a new ARF-per-grade should be
calculated based on the spectrum and the flux-level of the source, which
requires estimating the ARF at the pixel level. Figure 3 shows the HR and MR
spectra of two adjacent pixels, simulated assuming a flux level of 200 mCrab.
This clearly demonstrates that the event grading affects each pixel
differently, as it receives a different count rate. The differences between
the spectra of the two pixels clearly depend on the energy, and cannot be
connected by a simple normalization change.

The red spectrum in Fig. 4 corresponds to the total (i.e., the final output
using all the pixels) MR spectrum obtained from the SIXTE simulation at 200
mCrab, assuming the nominal ARF. We fitted this spectrum by fixing $N_{\rm H}$
and $\Gamma$ to the input values, and letting the normalization free to vary.
The fit is not statistically accepted ($\mathrm{C-stat/dof=4.4}$), and results
in a normalization that is $\sim 2.4$ times smaller than the input value. The
bottom panel of Fig. 4 shows clear energy-dependent residuals that result from
the use of the nominal ARF. Similar erroneous fits and strong residuals can be
seen for all of the event grades.

In the following, we present a new, generic technique that allows to estimate
the ARF for each grade, taking into consideration all the aforementioned
effects. We stress that this method can be applied to the observations of any
instrument operated out of focus and/or whose PSF depends on the energy. This
method is thus independent of shape of the PSF. In this work, we apply this
method to the particular case of the X-IFU. The simulations performed using
SIXTE are intended to replicate a real observation by the X-IFU. We note that
in the absence of a physical, parametric, model of the PSF, we assume in the
following a perfect knowledge of the PSF based on the description provided in
Sect. 2. Such physical model of the PSF will be based on measurements obtained
from the calibration of the Athena mirror. Here are the steps we follow to
reconstruct the spectrum at each of the grades:

Figure 6: The reconstructed simulated spectra (normalised by the effective
area) assuming flux levels of 200 mCrab (left) and 1 Crab (right) for the
different event grades (HR: blue, MR: red, LimR: green, LowR: grey), assuming
an exposure of 10 ks. In the latter case, we use a $100\leavevmode\nobreak\
\rm\mu m$ Be filter. The bottom panels show the residuals for each event
grade. All the spectra are rebinned for clarity reasons.

  * 1\. Estimating the PSF and ARF at each of the pixels. As mentioned before, the grading is applied at the pixel level. Thus, we need to estimate the PSF at a pixel ($i$), at all energies, $\rm PSF_{i}(E)$. To do so, we integrate the PSF within each of the X-IFU pixel at each of the energies where the PSF is defined.

As mentioned earlier, we assume a perfect knowledge of the instrument PSF.
Thus, in order to estimate $\mathrm{PSF_{i}}$ in the energy ranges where the
model PSF is not actually defined (see Sect. 2), we linearly interpolate over
the full energy range to replicate the particular way in which SIXTE runs the
simulations.

Then, we estimate the ARF at each of the pixels by multiplying the nominal ARF
by the PSF at each pixel:

$\rm ARF_{i}(E)=PSF_{i}(E)\times ARF_{nom}(E).$

  * 2\. Estimating the total number of counts per pixel ($\mathrm{CR}_{\rm tot,i}$). The event grading applied at each pixel depends on the total count rate that is detected in each pixel. In the case of SIXTE, the simulator provides the number of valid events ($\mathrm{CR}_{\rm val,i}$), which corresponds to the number of generated photons that are within the energy band $0.2-12$ keV. To estimate $\mathrm{CR}_{\rm tot,i}$, we numerically solved the following equation:

$\mathrm{CR}_{\rm val,i}=\mathrm{CR}_{\rm tot,i}\exp\left(-\mathrm{CR}_{\rm
tot,i}\times\delta t\right),$

where $\delta t$ is the minimum time separation with respect to a previous
pulse for an event to be valid as defined in Table 1 ($\delta
t=7.9\leavevmode\nobreak\ $ms, which corresponds to 10 times the fall time of
the X-ray pulses. ).

  * 3\. Estimating the fraction of events of each grade in each pixel. As the arrival time of X-ray events follow Poisson statistics, we adopt the following top down exclusion scheme (see e.g., Seta et al., 2012; Peille et al., 2018):

$\displaystyle w_{\rm HR,i}$ $\displaystyle=e^{-\Delta T_{\rm
HR}\times\mathrm{CR}_{\rm tot,i}}$ (1) $\displaystyle w_{\rm MR,i}$
$\displaystyle=e^{-\Delta T_{\rm MR}\times\mathrm{CR}_{\rm tot,i}}-e^{-\Delta
T_{\rm HR}\times\mathrm{CR}_{\rm tot,i}}$ (2) $\displaystyle w_{\rm LimR,i}$
$\displaystyle=e^{-\Delta T_{\rm LimR}\times\mathrm{CR}_{\rm
tot,i}}-e^{-\Delta T_{\rm MR}\times\mathrm{CR}_{\rm tot,i}}$ (3)
$\displaystyle w_{\rm LowR,i}$ $\displaystyle=e^{-\Delta T_{\rm
LowR}\times\mathrm{CR}_{\rm tot,i}}-e^{-\Delta T_{\rm
LimR}\times\mathrm{CR}_{\rm tot,i}}$ (4)

where $\Delta T_{\rm HR}$, $\Delta T_{\rm MR}$, $\Delta T_{\rm LimR}$, and
$\Delta T_{\rm LowR}$ are the sum of the grading criteria in both columns of
Table 1, for each of the event grades.

  * 4\. Calculating the total ARF for each of the event grades ($\rm ARF_{g}$), being the sum of all $\rm ARF_{i}$ weighted by the fraction of events detected at each grade,

${\rm ARF_{g}(E)}=\sum_{i}w_{\rm g,i}\times{\rm ARF_{i}(E)}.$

We calculate the sum using the FTOOLS command addarf.

The end result of this approach are four ARFs, one for each of the event
grades. The different ARFs calculated for a simulation of 200 mCrab are shown
in Fig. 5.

Figure 7: The best-fit parameters obtained by fitting simultaneously all the
event grades for a flux level of 200 mCrab using the nominal PSF (left) and
the modified PSF (right, as explained is Sect. 4.2), for $t_{\rm exp}=2$ ks
and 10 ks (blue circles and red diamonds, respectively). The error bars
correspond to the 90% confidence level. The bottom panel shows the deviation
of the best-fit C-stat from the expected value estimated by following the
analytic prescription presented by Kaastra (2017).

###  3.3 Spectral fitting

The green spectrum in Fig. 4 represents the MR spectrum using the
reconstructed ARF in green. The green residuals shown in the bottom panel of
this figure correspond to the input model parameters, without fitting.
Contrary to the red spectrum, where the nominal ARF is used, the residuals are
flat (consistent with zero) not showing any systematic. This figure
demonstrates that the differences between the two spectra cannot be explained
by a simple normalization factor. Instead, strong energy-dependent features
can be seen in the non-reconstructed spectrum.

In Fig. 6, we show the spectra of all the grades for the two flux levels, with
an exposure time of 10 ks. We also show the input model, without any fitting.
The model is statistically accepted and does not show any systematic residuals
which arose when using the nominal ARF. In the bottom panels of Fig. 6, we
show the residuals for each of the event grade. We note that in the case of
the 200 mCrab spectrum, we increased the number of energy bins of the model to
be able to account for all the features in the spectrum, using the XSPEC
command energies. The left panel of Fig. 7 shows the best-fit parameters for
the 200-mCrab observations, with $t_{\rm exp}=2$ ks an 10 ks. The best-fit
parameters are all consistent with the input values. In order to test the
goodness of the fit, we compare the best-fit C-stat to the expected C-stat and
its corresponding standard deviation, following the analytic prescription by
Kaastra (2017). For both cases, the observed C-stat is consistent with the
expected one within less than 1$\sigma$.

##  4 Impact of the PSF calibration uncertainties

The reconstruction method defined above assumes a perfect knowledge of the
PSF. However, the final performance of the instrument, and the effectiveness
of the reconstruction method, both heavily rely on the degree of knowledge of
the PSF. In this section, we explore the different sources of uncertainties
that may affect the reconstruction of the ARFs. Notably, we address the
uncertainties that are related to the knowledge of the PSF and its energy
dependency, within the current framework in which an analytical model of the
PSF is used. The tests performed in this section use a toy model for the PSF
uncertainties, in the absence of a parametric model of the PSF.

###  4.1 PSF calibration as a function of energy

Figure 8: The percent difference between the ARF obtained by removing an
energy during the reconstruction and the ARF using all energies as a function
of the removed energy, for the different event grades. Figure 9: Left: Nominal
defocused PSF at 1 keV. Middle: The modified PSF at 1 keV, estimated by
including an uncertainty of with a standard deviation of 10% on the value of
the PSF in bins of 10″. Right: The ratio of the modified to nominal PSF value.
The colorbar corresponds to the ratio in the right panel.

In this section, we investigate the uncertainties that may be introduced in
the ARF reconstruction due to the lack of knowledge of the PSF at a given
energy. In this case, we use the simulations presented in Sect. 3.2, for the
200 mCrab level, that are performed assuming a knowledge of the PSF at all 13
energies. Then, during the reconstruction we assume that in Step 1, the PSF is
known at 12 energies only. In other terms, the interpolation of the PSF as
function of energy is performed by omitting one of the energies in the range
$[0.75-7]$ keV.

Figure 8 shows the percent difference in ARF that is introduced by removing
one of the energies during the interpolation for the various event grades. The
difference is significant when we removed energies in the ${\sim}1.25-3$ keV
range, in all event grades (reaching $\sim 12\%$ at 2.5 keV for the HR). This
is due to the fact that, in this energy range, the PSF significantly changes
shape by going from one energy to another. Thus, by omitting one intermediate
energy, the interpolation scheme (in Step 1) will not be able to account for
the change in the PSF shape, which will result in a wrong estimate of the
effective area around the omitted energy. We present in Fig. 12 the residuals
obtained by removing each of those energies during the reconstruction, for all
the event grades. We also show in the bottom panel of the same figure, how the
C-stat changes by removing each of those energies. We note that the residuals
and the C-stat values reported in this figure are estimated by comparing the
spectra to the input model, without fitting. Strong systematic residuals (in
all event grades) are observed in the ${\sim}1.25-3$ keV range, where the
difference in the ARF is the largest. These residuals are not random, but
appear to be a systematic excess/deficiency in a given energy range. It is
also worth noting that these effects are local, and do not affect the global
parameters of the continuum. In fact, the residuals are flat outside the
ranges that are affected by omitting a given energy. However, these residuals
could be confused with emission/absorption features in the ranges of interest.

###  4.2 Uncertainty on the PSF

Figure 10: The reconstructed simulated spectra (normalised by the effective
area) assuming a flux level of 200 mCrab and an exposure of 10 ks, for the
different event grades, using the modified PSF as explained in Sect. 4.2. The
bottom panels show the residuals for each event grade. All the spectra are
rebinned for clarity reasons.

Figure 11: Top: The reconstructed simulated spectra of GRO J1655$-$40 for
$t_{\rm exp}=10$ ks at a flux level of 1 Crab. The dotted and dashed lines
represent the blackbody and the power-law components, respectively. The
spectra are rebinned for clarity reasons. Bottom: the $\log\xi$ vs. $\log
N_{\rm H}$ confidence contours at the 1, 2, and $3\sigma$ levels. The black
cross shows the input values used for the simulations. The inset shows the
typical precision (at $1\sigma$) that can be achieved using current
facilities. We plot the $3\sigma$ contours obtained with the X-IFU, for
comparison.

We investigate in this section the effects on the reconstructed ARFs
introduced by the variation in the effective areas from each PSF modules in
the mirror aperture. In fact, the modules are arranged in 15 rows (annular
rings) in the aperture. The model assumes that the modules within each row are
all identical and have exactly the same effective area and PSF as a function
of energy. The resulting model PSFs are the ones that have been used in the
previous sections. However, in reality the effective area and PSF of the
modules will vary because of manufacturing tolerances and imperfections in the
reflecting surface coatings. Thus, it is expected that the real effective area
and PSF will show variations at the level of the modules. These variations
will modulate the brightness of the individual blobs in the defocused PSF. To
simulate these effects, we randomize the value of the PSF at each of the
module. We consider a grid of $10\arcsec\times 10\arcsec$ that is equivalent
to the module size, and we randomize the value of the PSF in each of the grid
by assuming a normal distribution around the nominal PSF value with a standard
deviation of 10%. The choice of the scale size is dictated by the angular
resolution ($2\times\rm HEW$ of the in-focus PSF). In fact, the area cannot
vary over a scale size that is less than the angular resolution. The amplitude
of the variations in effective area is controlled by the error budget
tolerances used in the manufacturing process. Using a standard deviation of
10% with a normal distribution is a conservative upper limit to the expected
variations over the modules and is in line with the top level requirements on
the absolute effective area of Athena mirror. In Fig. 9, we show the nominal
(model) and modified defocused PSF at 1 keV. We also show the ratio of the two
PSFs.

Then, we reconstructed the ARFs for all grades using the modified PSF. Figure
10 shows the reconstructed spectra using the modified PSF, for the 200-mCrab
simulation with $t_{\rm exp}=10$ ks. The residuals show a systematic trend
notably below ${\sim}3$ keV in all grades. In the right-hand side of Fig. 7,
we show the best-fit results and the deviation of the C-stat from the expected
value, for $t_{\rm exp}=2$ ks and 10 ks. It is clear from this figure that
using the modified PSF, the best-fit C-stat is more than 3$\sigma$ larger than
the expected value, for $t_{\rm exp}=10$ ks, resulting in an unaccepted fit.
However, the fit is statistically accepted for the lower exposure time
simulation. Our results show that our capabilities to observe bright sources
with the X-IFU depend on our knowledge of the variations in effective area for
each individual module, especially for the deepest exposures.

##  5 Application: The case of GRO J1655–40

In this section, we show an example of the capabilities of X-IFU in studying
bright sources. We use GRO J1655$-$40 as source for our simulations. It is a
typical bright, hard state black hole with strong wind signatures and can be
regarded as a canonical source to study absorption lines caused by accretion
disk winds. We use a model, with spectral parameters taken from Miller et al.
(2008), which consists of a neutral Galactic absorption in the line-of-sight
of the source, in addition to a strong ionized absorption from a wind (with a
column density and ionisation parameter of $N_{\rm H}=4\times
10^{23}\leavevmode\nobreak\ \rm cm^{-2}$ and $\log\left(\xi/\rm
erg\leavevmode\nobreak\ cm\leavevmode\nobreak\ s^{-1}\right)=4.5$,
respectively) that acts on the sum of a disk blackbody (with a temperature
$kT_{\rm in}=1.35$ keV) and a powerlaw (with a photon index $\Gamma=3.5$). We
assume a source flux that is equivalent to 1 Crab with an exposure time of 10
ks.

First, we fitted the simulated spectra using the nominal ARF (i.e., without
reconstruction), simultaneously for all grades. We tied the parameters between
all of the grades. The fit is not statistically acceptable
(C-stat$\rm/dof=22143$), showing strong energy-dependent residuals. In
addition, the best-fit parameters of the continuum components (blackbody and
power law) deviate from the input parameters as shown in Table 2. In
particular, the normalization relative to each of those components deviates
from the input value. We note that given the low quality of the fit, we do not
report the uncertainty on the best-fit parameters as they do not have any
statistical significance. Instead, after applying the reconstruction, the
best-fit becomes statistically accepted ($\mathrm{C-stat/dof}=1.02$), being
consistent with the expected C-stat within $1.2\sigma$. The reconstructed
spectra are shown in the top panel of Fig. 11. The best-fit parameters are
shown in Table 2. The best-fit parameters deviate from the input values by
less than 0.7%. In the bottom panel of Fig. 11, we show the confidence
contours demonstrating the accuracy that could be obtained to determine the
wind absorption parameters.

Table 2: Best-fit parameters obtained by fitting the simulated spectra of GRO J155–40. The uncertainties correspond to an increase in C-stat by 1. We show the input parameters for comparison. Parameter | Input | Nominal | Reconstructed
---|---|---|---
$N_{\rm H}\leavevmode\nobreak\ \rm(10^{23}\leavevmode\nobreak\ cm^{-2})$ | $4.0$ | $4.04$ | $3.994\pm 0.002$
$\log\left(\xi/{\rm erg\leavevmode\nobreak\ cm\leavevmode\nobreak\ s^{-1}}\right)$ | $4.5$ | $4.5$ | $4.499\pm 0.003$
$kT_{\rm in}$ (keV) | $1.35$ | $1.32$ | $1.352\pm 0.0003$
$\rm Norm_{\rm BB}$ | $483$ | $79$ | $479.3\pm 0.7$
$\Gamma$ | $3.5$ | $3.23$ | $3.495\pm 0.003$
$\rm Norm_{\rm pow}$ | $8.3$ | $1.73$ | $8.311\pm 0.005$
C-stat/dof |  | 22143 | 1.02
$\rm\Delta C\leavevmode\nobreak\ (\sigma)$ |  | – | 1.2

##  6 Summary and conclusions

In this paper, we presented a new technique that enables the analysis of
defocused observations in which the PSF varies as a function of energy. We
applied our method to the Athena/X-IFU observations of bright sources,
operated out of focus. We have tested the method for different flux levels,
exposure times, and spectral shapes. We note that in this work we adopt a
configuration (e.g., pixel size and event grading scheme) that differs from
the one that will be adopted for the in-flight model. However, our results
will not depend on the specific configuration as it is considered as an input
to the method which will reconstruct the ARFs and the weights of each grade
accordingly. We note that the method used to generate the weighted ARFs per
event grade, presented in this work, be included in a future version of SIXTE.

We have also demonstrated that the results of the reconstruction scheme, and
thus the performance of the instrument depends heavily on the knowledge of the
PSF and its energy dependency. In this work, we have used an analytical model
of the PSF defined at 13 energies. We tested the uncertainties that are
introduced by the calibration of the PSF as a function of energy, and the
value of the PSF itself. These tests rely on a toy model to demonstrate the
dependency of the method on the PSF knowledge. In the future, a
physical/parametric model of the PSF will be available, which will allow us to
perform end-to-end simulations in which we can fold-in all the mirror-
instrument calibration uncertainties.

It is worth noting that the simulations presented in this work assume that the
observed source is stable. However, it is well known that the bright sources
that will be observed in the defocused mode (X-ray binaries for example) can
be variable on very short timescales (on a millisecond timescale). The
variability in flux and spectral shape will have to be accounted for by the
method. Variability will alter the count rate detected by each of the X-IFU
pixels and the grade selection of the events consequently. Similarly, we do
not address the effects of dithering that can affect the alignment of the PSF
during the observation, that will need to be folded in the analysis. However,
we note that for off-axis angles of up to $\sim 2\arcmin$ the defocused PSF
will be identical to the on-axis version. In addition, the positional accuracy
of Athena will be of the order of $\sim 1\arcsec$, so this will not blur the
off-axis PSF except, of course, the centre will be displaced by the off-axis
angle and the area will drop slightly because of vignetting. However, all
these effects will be taken into account during the analysis, and will be
minor compared to the effect of event grading.

###### Acknowledgements.

ESK acknowledges financial support from the Centre National d’Etudes Spatiales
(CNES). JW and TD acknowledge partial funding from BMWi through Deutsches
Zentrum für Luft- und Raumfahrt grant 50 QR 1903. This work makes use of
AstroPy (Astropy Collaboration et al., 2018), Matplotlib (Hunter, 2007), NumPy
(Harris et al., 2020), and SciPy (Virtanen et al., 2020).

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##  Appendix A Plots

Figure 12: Residuals for the different grades, assuming that the PSF is not
known at a given energy (indicated by a vertical solid line) during the
interpolation. The bottom panel shows how the C-stat changes as a function of
the removed energy. The horizontal line in this panel corresponds to the
C-stat value obtained by fitting the spectrum using all energies to
reconstruct the ARF.