S_{J-p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}^{\binom{J-p}{d-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{Z}^{\binom{J}{d}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\mathbb{Z}^{\binom{J-p}{d}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$

with (A.1) on the vertical arrows, $\alpha(1_{L})=1_{L+p}$,
$\beta(1_{L})=1_{L}$ for all $L\subseteq J-p$ and $\beta(1_{L})=0$ otherwise
($p\in L$). The bottom row is split exact and thus the five lemma concludes
the induction step. ∎

######  Proposition A.6.

Take any partition $I\sqcup J=\\{1,\ldots,m\\}$ of all $m$ indices. Then the
iterated Leray coboundaries induce canonical splittings

(A.2) $H_{d-\lvert J\rvert}(W\cap S^{I+J})\oplus\bigoplus_{L\subset J,\lvert
L\rvert=d}H_{0}(W\cap S^{I+L}\setminus S_{J-L})\xrightarrow{\cong}H_{d}(W\cap
S^{I}\setminus S_{J}).$

###### Proof.

The claim is trivial for $J=\varnothing$, and generalizes to $\lvert
J\rvert>0$ by induction using the residue sequence. The same reasoning from
the proof of A.5 applies, except for the reduction to $I=\varnothing$ (which
no longer holds). In particular, we still have $\varpi=0$, since its domain
$H_{\bullet}(W\cap S^{I}\setminus S_{J-p})$ involves less hypersurfaces and is
thus generated by coboundaries (A.1) of points. ∎

This decomposition implies the following two corollaries, which describe
$H_{n-\lvert I\rvert}(W\cap S^{I}\setminus S_{J})$; c.f. the _marginal cases_
in [51, Footnote 10 on page 93].

######  Corollary A.7.

For every bipartition $I\sqcup J=\\{1,\ldots,m\\}$ of a quadratic pinch
($m\leq n$), the iterated Leray coboundary gives an isomorphism

$\delta_{J}\colon H_{n-m}(W\cap S^{1,\ldots,m})\xrightarrow{\cong}H_{n-\lvert
I\rvert}(W\cap S^{I}\setminus S_{J}).$

###### Proof.

The classes $\delta_{L}1_{L}$ from points $1_{L}\in H_{0}(W\cap
S^{I+L}\setminus S_{J-L})\cong\mathbb{Z}$ in (A.2) contribute only to degrees
$\lvert L\rvert<\lvert J\rvert=m-\lvert I\rvert\leq n-\lvert I\rvert$. ∎

######  Corollary A.8.

For a linear pinch ($m=n+1$) and $I\sqcup J=\\{1,\ldots,n+1\\}$, the $\lvert
J\rvert$ iterated Leray coboundaries $\delta_{J-p}\colon\mathbb{Z}\cong
H_{0}(S^{I+J-p})\rightarrow H_{n-\lvert I\rvert}(W\cap S^{I}\setminus S_{J})$
provide a canonical isomorphism

$\mathbb{Z}^{\lvert J\rvert}\cong\bigoplus_{p\in
J}H_{0}(S^{I+J-p})\xrightarrow[\sum\limits_{p\in
J}\delta_{J-p}]{\cong}H_{n-\lvert I\rvert}(W\cap S^{I}\setminus S_{J}).$

###### Proof.

The set $S^{I+J}=\varnothing$ is empty (A.2), thus the first contribution in
(A.2) is absent and only $L\subset J$ with $\lvert L\rvert=n-\lvert
I\rvert=\lvert J\rvert-1$ remain. These $n$-fold intersections $W\cap
S^{I+J-p}\setminus S_{p}=S^{I+J-p}$ are points: precisely those corners of the
simplex which lie opposite to the facets $S_{p}$ with $p\in J$. ∎

###  A.1. Duality

Since $W\cap S^{I}$ is contractible for $\lvert I\rvert<m$ (A.1), Poincaré-
Lefschetz duality implies that

$H_{\bullet}(\overline{W}\cap S^{I},\partial W)\cong H^{2n-2\lvert
I\rvert-\bullet}(W\cap S^{I})\cong\mathbb{Z}$

is concentrated in degree $\bullet=2n-2\lvert I\rvert$, spanned by the
fundamental class of the manifold $\overline{W}\cap S^{I}$ with boundary
$\partial W\cap S^{I}$. For a quadratic pinch with its retraction to the
vanishing sphere, the duality shows that

$\displaystyle H_{\bullet}(\overline{W}\cap S^{1,\ldots,m},\partial W)$
$\displaystyle\cong H^{2n-2m-\bullet}(W\cap S^{1,\ldots,m})$
$\displaystyle\cong H^{2n-2m-\bullet}(\mathbb{S}^{n-m})\cong
H_{\bullet-n+m}(\mathbb{S}^{n-m})$

is concentrated in degrees $n-m$ and $2n-2m$.

From this starting point, all proofs above formally dualize upon swapping
boundaries $\partial$ and coboundaries $\delta$, swapping $J$ with $K$,
replacing the pairs $(W\cap S^{I},\varnothing)$ with $(\overline{W}\cap
S^{I},\partial W)$, and shifting degrees $\bullet$ to $2n-2\lvert
I\rvert-\bullet$. In particular, one finds:

  1. (1)

For all strict subsets $I\sqcup J\sqcup K\subset\\{1,\ldots,m\\}$ with
$J\neq\varnothing$,

$H_{\bullet}(\overline{W}\cap S^{I}\setminus S_{J},\partial W\cup S_{K})=0.$

  2. (2)

For all $I\sqcup J\sqcup K=\\{1,\ldots,m\\}$ with $J\neq\varnothing$, the
groups are concentrated in degree $n-\lvert I\rvert$ and iterated
(co)boundaries yield isomorphisms

$\delta_{I\sqcup K}\partial_{K}\colon H_{n-\lvert I\rvert}(\overline{W}\cap
S^{I}\setminus S_{J},\partial W\cup
S_{K})\xrightarrow{\cong}H_{n}(\overline{W}\setminus S_{1,\ldots,m},\partial
W)\cong\mathbb{Z}.$

  3. (3)

For $I\sqcup K=\\{1,\ldots,m\\}$ in a quadratic pinch ($m\leq n$), the
boundaries are isomorphisms in the lowest degree $n-\lvert I\rvert$:

$\partial_{K}\colon H_{n-\lvert I\rvert}(\overline{W}\cap S^{I},\partial W\cup
S_{K})\xrightarrow{\cong}H_{n-m}(\overline{W}\cap S^{1,\ldots,m},\partial W).$

###### Proof.

The first claim above is dual to A.3. That proof via long exact sequences used
only that $H_{0}(W\cap S^{I+p})\rightarrow H_{0}(W\cap S^{I})$ is an
isomorphism. So dually, one only has to check that the intersection
$\varpi\colon H_{\bullet}(\overline{W}\cap S^{I},\partial W)\rightarrow
H_{\bullet-2}(\overline{W}\cap S^{I+p},\partial W)$ with $S_{p}$ is an
isomorphism. This is clear, since these groups $\cong\mathbb{Z}$ are generated
by the respective fundamental classes.

The second claim is the dual of A.4; the isomorphisms follow just from the
existence of the exact sequences. All that is left is to identify the groups
at a single slot, like $I=\\{1,\ldots,m-1\\}$, $J=\\{m\\}$ and
$K=\varnothing$. For a linear pinch, $\overline{W}\cap S^{1,\ldots,n}\setminus
S_{n+1}=\\{0\\}$ is a point with homology $\mathbb{Z}$. For a quadratic pinch,
consider the residue sequence for $S_{m}$. The intersection

$\varpi_{m}\colon\mathbb{Z}\cong H_{\bullet+2}(\overline{W}\cap
S^{1,\ldots,m-1},\partial W)\hookrightarrow H_{\bullet}(\overline{W}\cap
S^{1,\ldots,m},\partial W)$

is injective, sending the generator (fundamental class) to the (non-zero)
fundamental class $[\overline{W}\cap S^{1,\ldots,m}]$. Hence the coboundary

$\delta_{m}\colon H_{\bullet}(\overline{W}\cap S^{1,\ldots,m},\partial
W)\twoheadrightarrow H_{\bullet+1}(\overline{W}\cap S^{1,\ldots,m-1}\setminus
S_{m},\partial W)$

is surjective and identifies the image with the quotient of
$H_{\bullet-n+m}(\mathbb{S}^{n-m})$ by $\mathbb{Z}[\overline{W}\cap
S^{1,\ldots,m}]$, which cancels degree $\bullet=2n-2m$ and leaves only
$H_{0}(\mathbb{S}^{n-m})\cong\mathbb{Z}$ in degree $\bullet=n-m$ for $m<n$.
The case $m=n$ where $\mathbb{S}^{0}=\\{P_{+},P_{-}\\}$ consists of two points
is slightly different: Here we reduce $\mathbb{Z}^{2}\cong
H_{0}(\mathbb{S}^{0})$ modulo the sum $\varpi_{m}[\overline{W}\cap
S^{1,\ldots,m-1}]=[P_{+}]+[P_{-}]$, leaving $\mathbb{Z}$.

The third claim is a very similar dualization, of A.7. ∎

######  Remark A.9.

Modulo torsion, these claims follow immediately from a generalization of
Poincaré-Lefschetz duality to transverse arrangements, spelled out in C.2.

###  A.2. Variation of a quadric

The well-known classical Picard-Lefschetz formula (e.g. [34, §(5.3.3)] or [35,
§(6.3.3)]) computes the variation of a quadratic pinch in the case
$J=K=\varnothing$, i.e., for the complete intersection of all
$I=\\{1,\ldots,m\\}$. We can ignore the coordinates $x_{1}=\ldots=x_{m-1}=0$,
so

$S^{I}_{t}\cong
Q_{t}=\\{x_{1}^{2}+\ldots+x_{r+1}^{2}=t\\}\subset\mathbb{C}^{r+1}$

is a family of complex quadrics $Q_{t}$ with dimension $r=n-m$. They are
localized $W\cap S^{I}_{t}\cong\mathbb{B}\cap Q_{t}$ in the ball
$\mathbb{B}=\\{\lvert x_{1}\rvert^{2}+\ldots+\lvert
x_{r+1}\rvert^{2}<1\\}\subset\mathbb{C}^{r+1}$. For $0<\lvert t\rvert<1$, the
embedded vanishing sphere

(A.3) $\mathbb{S}^{r}\hookrightarrow\mathbb{B}\cap Q_{t},\quad u\mapsto
x=u\cdot\sqrt{t}$

is a deformation retract $H_{\bullet}(\mathbb{B}\cap Q_{t})\cong
H_{\bullet}(\mathbb{S}^{r})$, and its fundamental class
$\nu=\widetilde{\nu}=\pm[\mathbb{S}^{r}]\in H_{r}(\mathbb{B}\cap Q_{t})$
provides the vanishing cycle. In polar coordinates $t=\rho\cdot
e^{2\mathrm{i}\pi\tau}$, the embedding (A.3) extends (for $\rho>0$) to a
diffeomorphism

(A.4)
$\begin{split}T\mathbb{S}^{r}=\\{(u,v)\in\mathbb{S}^{r}\times\mathbb{R}^{r+1}\colon
u\cdot v=0\\}\xrightarrow{\quad\cong\quad}Q_{t}\subset\mathbb{C}^{r+1},\\\
(u,v)\mapsto e^{\mathrm{i}\pi\tau}\cdot\Big{(}u\sqrt{\rho+\lVert
v\rVert^{2}\cdot\tfrac{1-\rho}{2}}+\mathrm{i}v\sqrt{\tfrac{1-\rho}{2}}\Big{)}\end{split}$

of the quadric with the tangent bundle of the vanishing sphere. Restricted to
the disk bundle $\overline{V}=\\{\lVert v\rVert\leq 1\\}\subset
T\mathbb{S}^{r}$ and its boundary, the sphere bundle $\partial V=\\{\lVert
v\rVert=1\\}$, we obtain local trivializations of the pairs

(A.5) $(\overline{\mathbb{B}}\cap Q_{t},\partial\mathbb{B}\cap
Q_{t})\cong(\overline{V},\partial V).$

The corresponding isotopy $g_{\tau}\colon Q_{\rho}\cong Q_{t},x\mapsto
e^{\mathrm{i}\pi\tau}\cdot x$ over the path $\gamma(\tau)=\rho\cdot
e^{2\mathrm{i}\pi\tau}$ ends up in the antipode $g_{1}(x)=-x$, which is not
the identity on $\partial\mathbb{B}$. Hence we compose it with the
diffeomorphisms $\Phi_{\tau}\colon T\mathbb{S}^{r}\longrightarrow
T\mathbb{S}^{r}$ defined by

$\Phi_{\tau}(u,v)=\big{(}u\cos\theta+(v/\lVert v\rVert)\sin\theta,\;-u\,\lVert
v\rVert\sin\theta+v\cos\theta\big{)}\big{|}_{\theta=\pi\tau\cdot\lVert
v\rVert}$

which rotate in the plane spanned by $u$ and $v$. By continuity in $\tau$, the
map $\Phi_{\tau}$ is homotopic to the identity $\Phi_{0}=\operatorname{id}$,
and therefore $g_{\tau}$ is homotopic to
$g^{\prime}_{\tau}=g_{\tau}\circ\Phi_{\tau}$. The angle $\theta$ of rotation
decreases from $\pi\tau$ on the boundary $\partial V$, down to $0$ at the zero
section $\mathbb{S}^{r}=\\{v=0\\}$. The resulting modified isotopy (fig. 25)

$g_{1}^{\prime}(u,v)=-\big{(}u\cos\theta+(v/\lVert
v\rVert)\sin\theta,\;-u\lVert
v\rVert\sin\theta+v\cos\theta\big{)}\big{|}_{\theta=\pi\cdot\lVert v\rVert}$

restricts to the identity map on $\partial V=\\{\lVert v\rVert=1\\}$.

Figure 25. Illustration of the map $v\mapsto g^{\prime}_{1}(u,v)$ for $r=2$.

In fact, $g_{\tau}^{\prime}$ trivializes the sphere bundle globally: The
isomorphism $g_{\tau}^{\prime}|_{\partial V}\colon\partial V\rightarrow
Q_{t}\cap\partial\mathbb{B}$ is given by

$\displaystyle(u,v)$ $\displaystyle\mapsto
e^{\mathrm{i}\theta}\Big{(}[u\cos\theta+v\sin\theta]\sqrt{\tfrac{1+\rho}{2}}+\mathrm{i}[v\cos\theta-u\sin\theta]\sqrt{\tfrac{1-\rho}{2}}\Big{)}\Big{|}_{\theta=\pi\tau}$
$\displaystyle=\left(\frac{u+\mathrm{i}v}{\sqrt{2}}\frac{R}{2}+\frac{u-\mathrm{i}v}{\sqrt{2}}\frac{t}{R}\right)_{R=\sqrt{1+\rho}+\sqrt{1-\rho}}$

and therefore well-defined and smooth also over $t=0$, where it identifies
$\partial V\cong Q_{0}\cap\partial\mathbb{B}$ as
$(u,v)\mapsto(u+iv)/\sqrt{2}$. Hence, we can use $g_{1}^{\prime}$ to compute
the localized variation. With the identification (A.5), the domain of

$\operatorname{var}=(g_{1}^{\prime})_{*}-\operatorname{id}\colon\quad
H_{r}(\overline{V},\partial V)\rightarrow H_{r}(V)$

is $H_{r}(\overline{V},\partial V)\cong H_{0}(\mathbb{S}^{r})$ by the Thom
isomorphism of the bundle $\mu\colon\overline{V}\rightarrow\mathbb{S}^{r}$,
$(u,v)\mapsto u$. So for $r\geq 1$, $H_{r}(\overline{V},\partial
V)\cong\mathbb{Z}$ is generated by the class of a fibre $\mu^{-1}(p)$. Over
the point $p=(0,\ldots,0,1)\in\mathbb{S}^{r}$, this class
$h=j_{*}[\overline{\mathbb{D}}]$ is represented by the $r$-dimensional disk
$\overline{\mathbb{D}}=\\{\lVert w\rVert\leq 1\\}\subseteq\mathbb{R}^{r}$,
embedded as

$j\colon(\overline{\mathbb{D}},\partial\mathbb{D})\hookrightarrow(\overline{V},\partial
V),\quad w\mapsto(p,(w,0))\in T_{p}\mathbb{S}^{r}.$

Because the retraction $\mu$ induces an isomorphism $\mu_{*}\colon
H_{r}(V)\cong H_{r}(\mathbb{S}^{r})\cong\mathbb{Z}$, we can compute the
variation by pushing forward along $\mu$:

$\mu_{*}(\operatorname{var}h)=\mu_{*}(g_{1}^{\prime})_{*}j_{*}[\overline{\mathbb{D}}]-\mu_{*}j_{*}[\overline{\mathbb{D}}].$

Since $\mu j\colon\overline{\mathbb{D}}\rightarrow\mathbb{S}^{r}$ is the
constant map $w\mapsto p$, the second term is zero in
$H_{r}(\mathbb{S}^{r})\cong H_{r}(\mathbb{S}^{r},\\{p\\})$. The remaining term
is the push-forward under

$\mu
g_{1}^{\prime}j\colon(\overline{\mathbb{D}},\partial\mathbb{D})\rightarrow(\mathbb{S}^{r},\\{p\\}),\quad
w\mapsto\left(-\frac{w}{\lVert w\rVert}\sin(\pi\lVert
w\rVert),\;-\cos(\pi\lVert w\rVert)\right).$

This map is bijective on the interior
$\mathbb{D}\cong\mathbb{S}^{r}\setminus\\{p\\}$ and realizes a homeomorphism
$\overline{\mathbb{D}}/\partial\mathbb{D}\cong\mathbb{S}^{r}$. Therefore, we
proved that

(A.6) $\operatorname{var}\colon H_{r}(\overline{V},\partial
V)\cong\mathbb{Z}\longrightarrow H_{r}(V)\cong\mathbb{Z},\quad h\mapsto\pm\nu$

is an isomorphism. To fix the sign, we endow $\mathbb{D}\subset\mathbb{R}^{r}$
with the orientation $\mathrm{d}w_{1}\wedge\ldots\wedge\mathrm{d}w_{r}$. Under
the differential of $u=\mu g_{1}^{\prime}j(w)$ at $w=0$, this becomes
$(-1)^{r}\mathrm{d}u_{1}\wedge\ldots\wedge\mathrm{d}u_{r}$ at
$T_{-p}\mathbb{S}^{r}$. In terms of the standard orientation424242The standard
orientation is induced by the ambient volume form
$\mathrm{d}u_{1}\wedge\ldots\wedge\mathrm{d}u_{r+1}$ of $\mathbb{R}^{r+1}$, by
contraction with the outward normal vector. At $p$, the outward normal is
$\partial/\partial u_{r+1}$, hence the standard orientation of
$T_{p}\mathbb{S}^{r}$ is
$(-1)^{r}\mathrm{d}u_{1}\wedge\ldots\wedge\mathrm{d}u_{r}$. At $-p$, the
outward normal is $-\partial/\partial u_{r+1}$ and thus the standard
orientation of $T_{-p}\mathbb{S}^{r}$ is
$(-1)^{r+1}\mathrm{d}u_{1}\wedge\ldots\wedge\mathrm{d}u_{r}$. on the vanishing
sphere $\nu=[\mathbb{S}^{r}]$, this gives $\operatorname{var}h=-\nu$ in (A.6).
The disk $j(\overline{\mathbb{D}})$ and the vanishing sphere
$\mathbb{S}^{r}=\\{v=0\\}$ intersect in precisely one point, namely $(p,0)\in
T_{p}\mathbb{S}^{r}$. According to appendix C, the intersection number

$\langle{\nu}\,|\,{h}\rangle=\langle{[\mathbb{S}^{r}]}\,|\,{[j(\overline{\mathbb{D}})]}\rangle=\epsilon_{p}=\pm
1$

is determined by comparing the orientation of the complex manifold $Q_{t}$
with the orientation of $T_{(p,0)}j(\overline{\mathbb{D}})\oplus
T_{(p,0)}\mathbb{S}^{r}$. With our choices above, the latter is

($\sharp$)
$\mathrm{d}v_{1}\wedge\ldots\wedge\mathrm{d}v_{r}\wedge(-1)^{r}\mathrm{d}u_{1}\wedge\ldots\wedge\mathrm{d}u_{r}.$

In terms of real and imaginary parts $x_{k}=\Re x_{k}+\mathrm{i}\Im x_{k}$,
the orientation of $T_{p\sqrt{t}}Q_{t}$ induced from the complex structure is
$\mathrm{d}\Re x_{1}\wedge\mathrm{d}\Im x_{1}\wedge\ldots\wedge\mathrm{d}\Re
x_{r}\wedge\mathrm{d}\Im x_{r}$. In the parametrization (A.4), this is equal
to

($\natural$)
$\mathrm{d}u_{1}\wedge\mathrm{d}v_{1}\wedge\ldots\wedge\mathrm{d}u_{r}\wedge\mathrm{d}v_{r}\cdot\big{(}\sqrt{\lvert
t\rvert(1-\lvert t\rvert)/2}\big{)}^{r}.$

The orientations ($\sharp$ ‣ A.2) and ($\natural$ ‣ A.2) differ by the sign
$\epsilon_{p}=(-1)^{r}(-1)^{r(r+1)/2}$. We therefore arrive at the Picard-
Lefschetz formula

$\operatorname{var}h=-\nu=-(-1)^{r}(-1)^{r(r+1)/2}\langle{\nu}\,|\,{h}\rangle\nu=(-1)^{(r+1)(r+2)/2}\langle{\nu}\,|\,{h}\rangle\nu.$

######  Remark A.10.

If $r=0$, then $V=\overline{V}=\mathbb{S}^{0}=\\{p,-p\\}$ consists of two
points and $\partial V=\varnothing$. In this case, $\operatorname{var}\colon
H_{0}(\mathbb{S}^{0})\rightarrow H_{0}(\mathbb{S}^{0})$ is a map from
$\mathbb{Z}^{2}$ to $\mathbb{Z}^{2}$. The variations
$\operatorname{var}[p]=-\nu$ and $\operatorname{var}[-p]=\nu=[p]-[-p]$ from
3.5 are indeed compatible with $\operatorname{var}h=(-1)^{1\cdot
2/2}\langle{\nu}\,|\,{h}\rangle\nu$.

##  Appendix B Partial boundaries and relative residues

###  B.1. Partial boundaries

Given any two topological subspaces $B,S\subseteq X$, we can consider the
triple $B\subseteq B\cup S\subseteq X$ and its exact sequence

$\cdots\rightarrow H_{\bullet}(X,B)\rightarrow H_{\bullet}(X,B\cup
S)\xrightarrow{\partial}H_{\bullet-1}(B\cup S,B)\rightarrow
H_{\bullet-1}(X,B)\rightarrow\cdots$

The inclusion of pairs $(S,B\cap S)\rightarrow(B\cup S,B)$ induces a morphism

(B.1) $H_{\bullet}(S,B\cap S)\longrightarrow H_{\bullet}(B\cup S,B).$

######  Definition B.1.

A finite collection $(S_{i})_{i\in I}$ of smooth closed submanifolds
$S_{i}\subset X$ is _transverse_ if, for every subset $J\subseteq I$ and at
every point $p\in S^{J}$ of the intersection $S^{J}=\bigcap_{i\in J}S_{i}$,
the canonical projections $N_{p}S^{J}\longrightarrow N_{p}S_{i}$ assemble into
isomorphisms $NS^{J}\cong\bigoplus_{i\in J}NS_{i}|_{S^{J}}$ of the normal
bundles.

By the implicit function theorem, a transverse collection admits local
coordinates $(x,(y_{i})_{i\in J})\colon
U\longrightarrow\mathbb{R}^{d}\times\prod_{i\in J}\mathbb{R}^{r_{i}}$ near any
point $p\in S^{J}$ such that $S_{i}\cap U=\\{y_{i}=0\\}$ for all $i\in J$,
where $r_{i}=\operatorname{codim}S_{i}$ and $d=\dim S^{J}$. If each $S_{i}$ is
a complex hypersurface ($r_{i}=2$) in a complex manifold, a transverse
collection is also called _simple normal crossings divisor_.

######  Proposition B.2.

If $S=S_{1}$ and $B=\bigcup_{i>1}S_{i}$ form a transverse collection of smooth
closed submanifolds $S_{i}\subset X$, then (B.1) is an isomorphism.

###### Proof.

Pick a tubular neighbourhood $u\colon N\hookrightarrow X$ of $S$, that is, a
smooth embedding of the normal bundle $N=TX|_{S}/TS$ such that
$u\iota=\operatorname{id}_{S}$ for the zero section $\iota\colon
S\hookrightarrow N$. Let $U=u(N)$, then excise $B\setminus U$ to get

$H_{\bullet}(B\cup S,B)\cong H_{\bullet}(S\cup(B\cap U),B\cap U).$

Contraction $F_{\lambda}(v)=(1-\lambda)v$ of the fibres $N_{\pi(v)}$ shows
that the projection $\pi=F_{1}\colon U\cong N\longrightarrow S$ is a
deformation retract. Using the transversality assumption we may construct $u$
in such a way that $F_{\lambda}(U\cap B)\subseteq U\cap B$ for all $\lambda$,
see [56, Lemma 4.3] or [27, Corollary 4.1]. Then $F_{\lambda}$ restricts to
homotopies on $B\cap U$ and $S\cup(B\cap U)$, which proves that

$\pi|_{B\cap U}\colon B\cap U\longrightarrow B\cap
S\quad\text{and}\quad\pi|_{S\cup(B\cap U)}\colon S\cup(B\cap U)\longrightarrow
S$

are deformation retracts and induce isomorphisms $H_{\bullet}(B\cap U)\cong
H_{\bullet}(B\cap S)$ and $H_{\bullet}(S\cup(B\cap U))\cong H_{\bullet}(S)$.
To conclude, apply the five lemma to the long exact sequences of the pairs
$(S\cup(B\cap U),B\cap U)$ and $(S,B\cap S)$. ∎

In this proof, we exploited the fact that one may choose tubular
neighbourhoods $u_{i}\colon TX|_{S_{i}}/TS_{i}\hookrightarrow X$ for all
$S_{i}$ in a mutually compatible way, such that
$u_{i}^{-1}(S_{j})=TS_{j}|_{S_{i}\cap S_{j}}/TS_{i}|_{S_{i}\cap S_{j}}$. We
will repeatedly use such compatible systems of tubes in this appendix. They
are constructed in the references given in the proof above, see also [41, §7],
and illustrated in fig. 26.

######  Definition B.3.

In the situation of B.2, the _boundary in $S$_ is the map $\partial_{S}\colon
H_{\bullet}(X,B\cup S)\longrightarrow H_{\bullet-1}(S,B\cap S)$ obtained from
the connecting morphism $\partial$ of the triple $B\subseteq B\cup S\subseteq
X$ under the isomorphism (B.1):

$\textstyle{H_{\bullet}(X,B\cup
S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\partial_{S}}$$\textstyle{H_{\bullet-1}(B\cup
S,B)}$$\textstyle{H_{\bullet-1}(S,B\cap
S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$

Now the exact sequence of the triple $B\subseteq B\cup S\subseteq X$ takes the
form

$\cdots\rightarrow H_{\bullet}(X,B)\rightarrow H_{\bullet}(X,B\cup
S)\xrightarrow{\partial_{S}}H_{\bullet-1}(S,B\cap S)\rightarrow
H_{\bullet-1}(X,B)\rightarrow\cdots$

###  B.2. Relative residues

If we replace $S$ by its complement $S^{\mathsf{c}}=X\setminus S$ and consider
instead the triple $B\subseteq B\cup S^{\mathsf{c}}\subseteq X$, we get the
sequence

$\cdots\rightarrow H_{\bullet}(X,B)\rightarrow H_{\bullet}(X,B\cup
S^{\mathsf{c}})\xrightarrow{\partial}H_{\bullet-1}(B\cup
S^{\mathsf{c}},B)\rightarrow H_{\bullet-1}(X,B)\rightarrow\cdots$

The inclusion of pairs $(S^{\mathsf{c}},B\setminus S)\rightarrow(B\cup
S^{\mathsf{c}},B)$ induces a morphism

(B.2) $H_{\bullet}(X\setminus S,B\setminus S)\longrightarrow H_{\bullet}(B\cup
S^{\mathsf{c}},B).$

######  Proposition B.4.

If $S=S_{1}$ and $B=\bigcup_{i\neq 1}S_{i}$ form a transverse collection of
smooth closed submanifolds $S_{i}\subset X$, then (B.2) is an isomorphism.

###### Proof.

The zero section $S_{i}\hookrightarrow U_{i}$ of any tubular neighbourhood
$NS_{i}\cong U_{i}\subseteq X$ induces an isomorphism $H_{\bullet}(S_{i})\cong
H_{\bullet}(U_{i})$. Following [56, 27] we may assume the compatibility that
every intersection $U^{J}=\bigcap_{i\in J}U_{i}$ is a tubular neighbourhood of
$S^{J}$ (fig. 26), so that $H_{\bullet}(S^{J})\cong H_{\bullet}(U^{J})$.

Figure 26. Tubular neighbourhoods of transverse submanifolds.

Then also the inclusion of $B$ into $U\mathrel{\mathop{:}}=\bigcup_{i\neq
1}U_{i}$ induces an isomorphism $H_{\bullet}(B)\cong H_{\bullet}(U),$ which
follows from the Mayer-Vietoris sequence and the five lemma via induction over
the number $\lvert I\rvert-1$ of components of $B$. Analogously, the
inclusions $B\setminus S\hookrightarrow U\setminus S$ and $B\hookrightarrow
B\cup(U\setminus S)$ induce isomorphisms

$H_{\bullet}(B\setminus S)\cong H_{\bullet}(U\setminus S)\quad\text{and}\quad
H_{\bullet}(B)\cong H_{\bullet}(B\cup(U\setminus S)),$

because compatibility with $S_{1}$ ensures that for $1\notin J$, the
projections $U^{J}\rightarrow S^{J}$ map $U^{J}\cap S$ to $S^{J}\cap S$ and
restrict to deformation retracts $U^{J}\setminus S\rightarrow S^{J}\setminus
S$ and $U^{J}\cap(B\cup S^{\mathsf{c}})\rightarrow S^{J}$. These thickenings
of $B\setminus S$ and $B$ induce $H_{\bullet}(S^{\mathsf{c}},B\setminus
S)\cong H_{\bullet}(S^{\mathsf{c}},U\setminus S)$ and $H_{\bullet}(B\cup
S^{\mathsf{c}},B)\cong H_{\bullet}(B\cup S^{\mathsf{c}},B\cup(U\setminus S))$,
and these isomorphisms identify (B.2) with the canonical map

$H_{\bullet}(S^{\mathsf{c}},U\setminus S)\rightarrow H_{\bullet}(B\cup
S^{\mathsf{c}},B\cup(U\setminus S)).$

The interior of $B\cup(U\setminus S)$ in $B\cup S^{\mathsf{c}}$ contains
$B\cap S$, thus the map above is an isomorphism by excision. ∎

To describe the group $H_{\bullet}(X,B\cup S^{\mathsf{c}})$, let
$r=\operatorname{codim}S$ and consider a subbundle $\mu\colon V\rightarrow S$
of a tubular neighbourhood $NS\cong U\rightarrow S$ such that the fibres of
$V\subset U$ are isomorphic to closed discs
$\overline{\mathbb{B}}_{r}=\\{x\in\mathbb{R}^{r}\colon\lVert x\rVert\leq
1\\}$. Then we can excise the complement of $V$ to localize near $S$,

($\ast$) $H_{\bullet}(X,B\cup S^{\mathsf{c}})\cong H_{\bullet}(V,(B\cap
V)\cup(V\setminus S)).$

For a transverse collection $S\cup B$, recall that we can choose $U\rightarrow
S$ such that it preserves $B$. Then $\mu^{-1}(B\cap S)=V\cap B=V|_{B\cap S}$
is itself a disc bundle, namely the restriction of $V$ to $B\cap S$.

######  Definition B.5.

Suppose that the disc bundle $V\rightarrow S$ is oriented. Then the pullback
bundle $\sigma^{\ast}V\cong\overline{\mathbb{B}}_{r}\times\Delta$ over a
singular simplex $\sigma\colon\Delta\rightarrow S$ inherits an orientation,
and hence descends to a well-defined map on homology:434343The maps (B.3) and
(B.4) only depend on the orientation of the normal bundle $NS$ (equivalently
the orientation of $V$), but not on the choice of tubular neighbourhoods,
since tubular neighbourhoods are unique up to isotopy.

(B.3) $\mu_{S}^{\ast}\colon H_{\bullet}(S,B\cap S)\longrightarrow
H_{\bullet+r}(X,B\cup S^{\mathsf{c}}).$

Similarly, the pullback of the sphere bundle $\partial V\rightarrow S$ induces
a map

(B.4) $\delta_{S}\colon H_{\bullet}(S,B\cap S)\longrightarrow
H_{\bullet+r-1}(S^{\mathsf{c}},B\setminus S)$

called the _Leray coboundary_ , such that
$\partial(\mu_{S}^{\ast}[\sigma])=\delta_{S}[\sigma]+(-1)^{r}\mu_{S}^{\ast}(\partial[\sigma])$.

######  Proposition B.6.

If $S=S_{1}$ and $B=\bigcup_{i\neq 1}S_{i}$ form a transverse collection of
smooth closed submanifolds $S_{i}\subset X$, and $S$ and $X$ are oriented,
then the fibration-by-discs map (B.3) is an isomorphism.

###### Proof.

The orientations of $S$ and $X$ induce an orientation of the disc bundle $V$,
and the complement $V^{\prime}\mathrel{\mathop{:}}=V\setminus S$ of the zero
section deformation retracts onto the sphere bundle $\partial V$. So if
$B=\varnothing$, then the excision ($\ast$ ‣ B.2) reduces the claim to the
Thom isomorphism $H_{\bullet}(S)\rightarrow
H_{\bullet+r}(V,V^{\prime})$.444444 For this homological version of the Thom
isomorphism, see e.g. [58, Theorem 18.1.2] or [44, Corollary §10.7].

For non-empty $B$, let $V_{B}\mathrel{\mathop{:}}=V\cap B=V|_{B\cap S}$ denote
the restriction of $V$ to $B\cap S$ and consider the commutative diagram of
the long exact sequences of the triple $V^{\prime}\subseteq V^{\prime}\cup
V_{B}\subseteq V$ and the pair $B\cap S\subseteq S$:

$\textstyle{\cdots\rightarrow
H_{\bullet}(V,V^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\bullet}(V,V^{\prime}\cup
V_{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\bullet-1}(V^{\prime}\cup
V_{B},V^{\prime})\rightarrow\cdots}$$\textstyle{\cdots\rightarrow
H_{\bullet-r}(S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H_{\bullet-r}(S,B\cap
S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{S}^{\ast}}$$\textstyle{H_{\bullet-1-r}(B\cap
S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\rightarrow\cdots}$$\scriptstyle{\nu}$

The vertical maps are induced by the pullbacks of the disc bundle
$V\rightarrow S$ and its restriction $V_{B}\rightarrow B\cap S$. The claim
follows from the five lemma because $\nu$ is an isomorphism. Indeed, by
construction $\nu$ factors into

$H_{\bullet}(B\cap S)\rightarrow
H_{\bullet+r}(V_{B},V_{B}^{\prime})\rightarrow H_{\bullet+r}(V_{B}\cup
V^{\prime},V^{\prime})$

where $V_{B}^{\prime}\mathrel{\mathop{:}}=V_{B}\setminus S=V_{B}\cap
V^{\prime}$ and the first map is the Thom isomorphism for the bundle $V_{B}$.
The second map is an isomorphism because $V^{\prime}$ and $V_{B}$ constitute
an excisive couple, by the thickening principle from the proof of B.4: replace
$V_{B}$ by an open neighbourhood $V\cap U\cap(V^{\prime}\cup V_{B})$ inside
$V^{\prime}\cup V_{B}$ with the same homology as $V_{B}$, obtained from a
union $U$ of compatible tubular neighbourhoods of the components of $B$. ∎

Combining this result with B.4, we can write the long exact sequence of the
triple $B\subseteq B\cup S^{\mathsf{c}}\subseteq X$ for a transverse
collection of oriented smooth closed submanifolds as the _relative residue
sequence_

$\cdots\rightarrow
H_{\bullet}(X,B)\xrightarrow{\varpi_{S}}H_{\bullet-r}(S,B\cap
S)\xrightarrow{\delta_{S}}H_{\bullet-1}(X\setminus S,B\setminus S)\rightarrow
H_{\bullet-1}(X,B)\rightarrow\cdots$

where the map $\varpi_{S}$ can be interpreted as “transverse intersection with
$S$”. It is defined by the commutative diagram

$\textstyle{H_{\bullet}(X,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi_{S}}$$\textstyle{H_{\bullet}(X,B\cup
S^{\mathsf{c}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{H_{\bullet-1}(B\cup
S^{\mathsf{c}},B)}$$\textstyle{H_{\bullet-r}(S,B\cap
S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\ast}_{S}}$$\scriptstyle{\cong}$$\scriptstyle{\delta_{S}}$$\textstyle{H_{\bullet-1}(X\setminus
S,B\setminus
S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eqref{eq:(X-S,B-S)->(X-S+B,B)}}$

###  B.3. Functoriality and commutativity

Let $r_{i}$ denote the codimension of a submanifold $S_{i}\subset X$ in a
transverse family. Then the maps
$(\partial_{S_{i}},\delta_{S_{i}},\varpi_{S_{i}})$ have homological degrees
$(-1,r_{i}-1,-r_{i})$, and for $i\neq j$ they are graded commutative with the
same degrees:

(B.5) $\displaystyle\partial_{S_{i}}\partial_{S_{j}}$
$\displaystyle=-\partial_{S_{j}}\partial_{S_{i}}$
$\displaystyle\delta_{S_{i}}\delta_{S_{j}}$
$\displaystyle=(-1)^{(r_{i}-1)(r_{j}-1)}\delta_{S_{j}}\delta_{S_{i}}$
$\displaystyle\partial_{S_{i}}\delta_{S_{j}}$
$\displaystyle=(-1)^{r_{j}-1}\delta_{S_{j}}\partial_{S_{i}}$
$\displaystyle\delta_{S_{i}}\varpi_{S_{j}}$
$\displaystyle=(-1)^{(r_{i}-1)r_{j}}\varpi_{S_{j}}\delta_{S_{i}}$
$\displaystyle\partial_{S_{i}}\varpi_{S_{j}}$
$\displaystyle=(-1)^{r_{j}}\varpi_{S_{j}}\partial_{S_{i}}$
$\displaystyle\varpi_{S_{i}}\varpi_{S_{j}}$
$\displaystyle=(-1)^{r_{i}r_{j}}\varpi_{S_{j}}\varpi_{S_{i}}$

The first identity amounts to $\partial^{2}=0$454545Consider a single simplex
$\Delta$. Its orientation induces orientations on its facets, which in turn
induce orientations on their respective facets. These respective orientations
must be opposite, in order to ensure $\partial^{2}=0$. and the other signs
compare the induced orientations on the disc and sphere bundles, see also [39,
Proposition 39]. Furthermore, suppose that a smooth map $F\colon
X\longrightarrow X^{\prime}$ restricts to submersions $S_{i}\longrightarrow
S_{i}^{\prime}\mathrel{\mathop{:}}=F(S_{i})$ onto a transverse family
$(S_{i}^{\prime})_{i\in I}$ in $X^{\prime}$. Then the maps commute with
pushforward:

(B.6) $F_{\ast}\partial_{S_{i}}=\partial_{S_{i}^{\prime}}F_{\ast},\quad
F_{\ast}\delta_{S_{i}}=\delta_{S_{i}^{\prime}}F_{\ast},\quad
F_{\ast}\varpi_{S_{i}}=\varpi_{S_{i}^{\prime}}F_{\ast}.$

##  Appendix C Relative intersection numbers

Consider a smooth compact oriented manifold $\overline{M}$ of dimension $n$,
with boundary $\partial M=\overline{M}\setminus M$ (possibly empty) and
interior $M$. The cap product with the fundamental class $[M]\in
H_{n}(\overline{M},\partial M)$ defines a homomorphism

$\Psi\colon H^{d}(M)\longrightarrow H_{n-d}(\overline{M},\partial
M),\quad\alpha\mapsto[M]\frown\alpha$

from singular cohomology, to homology in complementary degree. Poincaré-
Lefschetz duality states that this is an isomorphism [62, Theorem 6.25, p.
171]. The duality $H_{d}(M)\times H^{d}(M)\rightarrow\mathbb{Z}$ can thus be
viewed as a bilinear map

$\langle{\cdot}\,|\,{\cdot}\rangle\colon H_{d}(M)\times
H_{n-d}(\overline{M},\partial
M)\longrightarrow\mathbb{Z},\quad\langle{x}\,|\,{y}\rangle=(\Psi^{-1}(y))(x).$

These _intersection numbers_ can be computed as follows: If $x=[Q]$ and
$y=[R]$ are fundamental classes of compact oriented submanifolds $Q$ and $R$
($\partial Q=\varnothing$ and $\partial R\subseteq\partial M$) that intersect
transversally, then $Q\cap R\subseteq M$ is also a compact oriented
submanifold—in fact, it is a finite set of points. Then

$\langle{[Q]}\,|\,{[R]}\rangle=\sum_{p\in Q\cap R}\epsilon_{p},$

with signs $\epsilon_{p}=\pm 1$ determined by the orientations. We follow the
convention in [51], namely that $\epsilon=+1$ precisely when the orientation
of $M$ at $p$ agrees with the orientation of first $R$, then $Q$, under the
identification

($\flat$) $T_{p}M=T_{p}R\oplus T_{p}Q.$

When $x$ and $y$ are not presented by transverse submanifolds, one can perturb
the singular simplices slightly to obtain homologous chains with transverse
simplices, and apply the same recipe; cf. [54, §73] and [17, §VII.4].

If $\partial M=\varnothing$, we can swap $x$ and $y$. Then from ($\flat$ ‣ C)
it follows that

(C.1) $\langle{x}\,|\,{y}\rangle=(-1)^{d(n-d)}\cdot\langle{y}\,|\,{x}\rangle.$

######  Remark C.1.

The sign convention is more intuitive in terms of the normal bundles
$NR=TM|_{R}/TR$ and $NQ=TM|_{Q}/TQ$. From ($\flat$ ‣ C) we see $N_{p}R\cong
T_{p}Q$ and $N_{p}Q\cong T_{p}R$, hence $\epsilon_{p}=+1$ if the orientations
on the normal bundles agree, under the canonical isomorphism

($\natural$) $N_{p}(Q\cap R)\cong N_{p}Q\oplus N_{p}R$

with the normal bundles of _first $Q$, then $R$_ (the same order as in
“$\langle{[Q]}\,|\,{[R]}\rangle$”). Note that ($\natural$ ‣ C.1) holds for
transverse intersections of arbitrary dimensions $i=\dim Q$ and $j=\dim R$,
whereas ($\flat$ ‣ C) only applies for $i+j=n$. The orientation of $Q\cap R$
induced by ($\natural$ ‣ C.1) generalizes the intersection numbers to the
_intersection product_ $[Q]\bullet[R]=[Q\cap R]\in H_{i+j-n}(M)$, see [17,
§VIII.13].

###  C.1. Transverse arrangements

Inside a smooth compact oriented manifold $\overline{M}$ as above, consider a
transverse arrangement $D=A\cup B=\bigcup_{i}D_{i}$ of smooth oriented closed
submanifolds $D_{i}$. If $\partial M\neq\varnothing$, we also require that $D$
is transverse to $\partial M$. Then the intersection numbers define a bilinear
pairing

(C.2) $\langle{\cdot}\,|\,{\cdot}\rangle\colon H_{d}(M\setminus B,A)\times
H_{n-d}(\overline{M}\setminus A,\partial M\cup B)\longrightarrow\mathbb{Z}.$

For transverse submanifolds $Q\subseteq M\setminus B$,
$R\subseteq\overline{M}\setminus A$ with $\partial Q\subseteq A$ and $\partial
R\subseteq B\cup\partial M$, their intersection is a finite set of points
inside $M\setminus D$, and $\langle{[Q]}\,|\,{[R]}\rangle$ can be computed as
above. In general, the intersection pairing is constructed in [17, §VII.4]. In
fact, we can define (C.2) as the composition of:

  * •

the Künneth morphism for the product $Y=(M\setminus
B)\times(\overline{M}\setminus A)$, relative to
$D^{\prime}=A\times(\overline{M}\setminus A)\cup(M\setminus B)\times(\partial
M\cup B)$:

$H_{d}(M\setminus B,A)\otimes H_{n-d}(\overline{M}\setminus A,\partial M\cup
B)\longrightarrow H_{n}(Y,D^{\prime})$

  * •

intersection with the diagonal $\Delta=\\{(p,p)\colon p\in M\\}\cap Y\cong
M\setminus D$:

$H_{n}(Y,D^{\prime})\xrightarrow{\varpi_{\Delta}}H_{0}(\Delta,D^{\prime}\cap\Delta)\cong
H_{0}(M\setminus D)\cong\mathbb{Z}.$

Here note that $D^{\prime}\cap\Delta=\varnothing$, and the map
$\varpi_{\Delta}$ from the Gysin/residue sequence is constructed in appendix
B: work relative to $D^{\prime}\cup(Y\setminus\Delta)$, excise all but a
tubular neighbourhood of $\Delta$, and apply the Thom isomorphism.

###  C.2. Duality

Consider a submanifold $S\in\mathfrak{B}$ of real codimension $r$. Let
$B^{\prime}\subseteq B$ denote the union of the remaining $B_{i}\neq S$. Then
the intersection pairings on $M$ and $S$ are compatible in the following
sense: If $x\in H_{d}(S\setminus B^{\prime},A)$ and $y\in
H_{n-d-r+1}(\overline{M}\setminus A,\partial M\cup B)$, then

(C.3)
$\langle{\delta_{S}\,x}\,|\,{y}\rangle=(-1)^{1+(r-1)(n-d)}\cdot\langle{x}\,|\,{\partial_{S}\,y}\rangle.$

This is a simple generalization of [51, §III.2.3]. Equivalently, swapping the
roles of $x$ and $y$ as in (C.1) when $S\in\mathfrak{A}$, we have for $x\in
H_{d}(M\setminus B,A)$ and $y\in H_{n-d-r+1}(\overline{S}\setminus
A^{\prime},\partial S\cup B)$ the identity

$\langle{\partial_{S}\,x}\,|\,{y}\rangle=(-1)^{1+d+n}\cdot\langle{x}\,|\,{\delta_{S}\,y}\rangle.$

Hence in the case of a complex hypersurface $S\subset M$ ($r=2$ and $n$ even),
we obtain for homology classes $x$ of degree $d$ that

(C.4) $\displaystyle\langle{\delta_{S}\,x}\,|\,{y}\rangle$
$\displaystyle=(-1)^{1+d}\cdot\langle{x}\,|\,{\partial_{S}\,y}\rangle\quad\text{and}$
$\displaystyle\langle{\partial_{S}\,x}\,|\,{y}\rangle$
$\displaystyle=(-1)^{1+d}\cdot\langle{x}\,|\,{\delta_{S}\,y}\rangle.$

For an Abelian group $G$, let $G^{\vee}=\operatorname{Hom}(G,\mathbb{Z})$
denote its dual. The pairing (C.2) defines a map $H_{d}(M\setminus
B,A)\rightarrow H_{n-d}(\overline{M}\setminus A,\partial M\cup B)^{\vee}$.
Extend to rational coefficients, then the codomain is canonically isomorphic
to singular cohomology, by the universal coefficient theorem. This defines a
linear map

(C.5) $H_{d}(M\setminus B,A;\mathbb{Q})\longrightarrow
H^{n-d}(\overline{M}\setminus A,\partial M\cup B;\mathbb{Q}).$

######  Theorem C.2.

The intersection pairing (C.2) is perfect modulo torsion. In other words,
(C.5) is an isomorphism.

###### Proof.

When $A=B=\varnothing$, this is just a restatement of the Poincaré-Lefschetz
theorem. With this as a base case, the proof follows by induction over the
number of elements of the arrangement
$\mathfrak{D}=\mathfrak{A}\sqcup\mathfrak{B}$:

Consider an element $S\in\mathfrak{B}$, so that $B=S\cup B^{\prime}$ where
$B^{\prime}$ is the union of all $B_{i}$ in $\mathfrak{B}\setminus\\{S\\}$.
The residue and boundary sequences from appendix B fit into the diagram

$\scriptstyle{\varpi_{S}}$$\textstyle{H_{d+1-r}(S\setminus
B^{\prime},A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\delta_{S}}$$\textstyle{H_{n-d-1}(\overline{S}\setminus
A,\partial S\cup
B^{\prime})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{S}^{\vee}}$$\textstyle{H_{d}(M\setminus
B,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n-d}(\overline{M}\setminus
A,\partial M\cup
B)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{d}(M\setminus
B^{\prime},A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\varpi_{S}}$$\textstyle{H_{n-d}(\overline{M}\setminus
A,\partial M\cup
B^{\prime})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

built out of the residue sequence in the left column, the dual of the boundary
sequence in the right column, and the horizontal maps induced by the
intersection pairings. We can add signs such that this diagram commutes: one
of the squares is (C.3), and similarly one checks that the other squares also
commute up to sign. We may assume that $\alpha\otimes\mathbb{Q}$ and
$\beta\otimes\mathbb{Q}$ are isomorphisms, because they arise from
intersection pairings of smaller arrangements $\mathfrak{D}\setminus\\{S\\}$.
The induction step then follows from the five-lemma.

Analogously, we can add an element $S\in\mathfrak{A}$, by considering the
diagram built from the boundary sequence (left column) and the dual of the
residue sequence (right column). ∎

###  C.3. de Rham cohomology

Under the identification (C.5), the intersection pairing can be viewed as a
perfect pairing in singular cohomology,

$H^{d}(M\setminus B,A;\mathbb{Q})\times H^{n-d}(\overline{M}\setminus
A,\partial M\cup B;\mathbb{Q})\longrightarrow\mathbb{Q}.$

Consider now the special case when $M$ is a compact complex manifold without
boundary, and $D\subset M$ is a transverse union of smooth complex
hypersurfaces. Let $H_{\mathsf{dR}}^{\bullet}$ denote the cohomology of smooth
complex differential forms. By the de Rham theorem
$H_{\mathsf{dR}}^{\bullet}\cong H^{\bullet}\otimes\mathbb{C}$, we obtain a
perfect pairing

(C.6) $H_{\mathsf{dR}}^{d}(M\setminus B,A)\times
H_{\mathsf{dR}}^{n-d}(M\setminus A,B)\longrightarrow\mathbb{C}.$

This incarnation of C.2 is derived in [11, §3] from Verdier duality.
Furthermore, by [11, Proposition 3.10] the pairing (C.6) equals the integral

$([\nu],[\omega])\longmapsto\int_{M}\nu\wedge\omega.$

Here the representative forms $\nu$ ($\omega$) have to be chosen such that
they have at most simple poles on $B$ ($A$). This is always possible [39, pp.
121–122], and ensures that the integral converges. When $M$ and $D$ are
algebraic over $\mathbb{Q}$, then the pairing (C.6) also respects the
$\mathbb{Q}$-mixed Hodge structures on algebraic de Rham cohomology [11].

##  Appendix D Fundamental group and small loops

Consider a closed analytic subvariety $L\subsetneq T$ of a connected complex
manifold $T$. Let $\mathfrak{L}$ denote the subset of irreducible components
of $L$ which have complex codimension one. Their union forms a subset
$L^{\prime}=\bigcup_{\ell\in\mathfrak{L}}\ell\subseteq L$.

######  Proposition D.1.

Fix a point $t\in T\setminus L$ and take all fundamental groups in the sequel
relative to $t$ as the base point. Then:

  1. (1)

The inclusion $T\setminus L\hookrightarrow T\setminus L^{\prime}$ induces an
isomorphism

$\pi_{1}(T\setminus L)\xrightarrow{\;\cong\;}\pi_{1}(T\setminus L^{\prime}).$

  2. (2)

The inclusion $T\setminus L\hookrightarrow T$ induces a surjection

$\iota\colon\pi_{1}(T\setminus
L)\relbar\joinrel\twoheadrightarrow\pi_{1}(T).\qquad\quad$

  3. (3)

The kernel of $\iota$ is generated by simple loops (3.1).

######  Corollary D.2.

The fundamental group of $T\setminus L$ depends only on the codimension one
part $L^{\prime}$ of $L$. When $T$ is simply connected, then
$\pi_{1}(T\setminus L)$ is generated by simple loops.

D.1 follows from the existence of stratifications by smooth subvarieties
(section 2) and transverse deformations. A smooth map $f\colon X\rightarrow Y$
of manifolds is called _transverse_ to a closed submanifold $D\subset Y$, if
at every point $x\in X$ with $f(x)\in D$, the tangent space of $Y$ is spanned
by the tangent space of $D$ and the image of the differential of $f$:

$T_{f(x)}Y=T_{f(x)}D+\mathrm{d}f_{x}\left(T_{x}X\right).$

Under this condition, $f^{-1}(D)$ is a closed submanifold of $X$, with
codimension $\dim Y-\dim D$. In particular, if $\dim D+\dim X<\dim Y$, then
transversality implies that the image of $f$ does not intersect $D$, that is
$f(X)\subseteq Y\setminus D$.

######  Theorem D.3 (Transversality theorem [33, §IV.Corollary (2.5)]).

Given a smooth closed submanifold $D\subset Y$ and a smooth map $f\colon
X\rightarrow Y$, there exists a smooth map that is homotopic to $f$ and
transverse to $D$.

######  Corollary D.4 ([22, Théorème 2.3]).

For a closed smooth submanifold $D\subset Y$ of real codimension $r\geq 2$,
the morphism $\pi_{1}(Y\setminus D)\rightarrow\pi_{1}(Y)$, induced by the
inclusion $Y\setminus D\hookrightarrow Y$, is surjective. Whenever $r\geq 3$,
it is also injective.

###### Proof.

Every loop $\gamma\colon[0,1]\rightarrow Y$ is homotopic to a loop
$\gamma^{\prime}$ that is transverse to $D$. Because $\dim D+\dim\,[0,1]=\dim
Y-r+1$ and $r\geq 2$, $\gamma^{\prime}$ does not intersect $D$, proving
surjectivity. For injectivity, suppose $[\gamma]\in\pi_{1}(Y\setminus D)$ is
in the kernel, i.e. there is a contracting homotopy
$F\colon[0,1]^{2}\rightarrow Y$ of $\gamma$ in $Y$. Choose $F$ transverse to
$D$. Then since $\dim([0,1]^{2})=2$ and $r\geq 3$, the homotopy does not
intersect $D$. Thus $\gamma$ is contractible already in $Y\setminus D$. ∎

###### Proof of D.1.

The subvariety $L\setminus L^{\prime}$ of $Y=T\setminus L^{\prime}$ has
complex codimension at least two (definition of $L^{\prime}$). Consider a
stratification $L\setminus L^{\prime}=C_{n-2}\supseteq\cdots\supseteq C_{0}$
by closed analytic subvarieties $C_{k}$, so that each difference
$D_{k}=C_{k}\setminus C_{k-1}$ is a smooth closed complex submanifold of
$Y\setminus C_{k-1}$, with real codimension $2n-2k\geq 4$. By D.4, inclusions
induce isomorphisms

($\ast$) $\iota_{k}\colon\pi_{1}(Y\setminus
C_{k})\longrightarrow\pi_{1}(Y\setminus C_{k-1})$

for each $k\leq n-2$. Their composition $\iota_{0}\cdots\iota_{n-2}$ yields
the isomorphism (1) between the fundamental groups of $Y\setminus
C_{n-2}=T\setminus L$ and $Y=T\setminus L^{\prime}$.

The claim (2) is proved analogously, by considering a stratification of
$L=C_{n-1}\supseteq\cdots\supseteq C_{0}$ inside $Y=T$. As above, the maps
($\ast$ ‣ D) are isomorphisms for $k\leq n-2$. At $k=n-1$, we still have
surjectivity (real codimension two).

Given the isomorphisms $\pi_{1}(T\setminus C_{n-2})\cong\pi_{1}(T\setminus
C_{n-3})\cong\cdots\cong\pi_{1}(T)$, we set $Y=T\setminus C_{n-2}$ and
identify the map $\iota$ in claim (3) with the morphism

$\pi_{1}(T\setminus L)=\pi_{1}(Y\setminus
D)\longrightarrow\pi_{1}(Y)\cong\pi_{1}(T)$

induced by the inclusion of the complement $T\setminus L=Y\setminus D$ of the
smooth submanifold $D=L\setminus C_{n-2}\subset Y$, which has real codimension
two. We apply the proof of [51, Proposition §V.1.3]: For
$[\gamma]\in\pi_{1}(Y\setminus D,t)$ in the kernel of $\iota$, there is a
homotopy $F\colon\square\rightarrow Y$ on the square $\square=[0,1]^{2}$ with
$F(\tau,1)=F(0,\tau)=F(1,\tau)=t$ constant and $\gamma(\tau)=F(\tau,0)$.
Choose $F$ transverse to $D$, so $P=\\{p_{1},\ldots,p_{k}\\}=F^{-1}(D)$ is a
submanifold—with dimension zero (a finite set of points). In the group
$\pi_{1}(\square\setminus P)$, the boundary of the square is a product of
simple loops $\gamma_{i}$ (fig. 27), hence, applying $F$,

$[\gamma]=\eta_{1}\star\cdots\star\eta_{k}$

is a product of simple loops $\eta_{i}=[F\circ\gamma_{i}]\in\pi_{1}(Y\setminus
D,t)$ around $t_{i}=F(p_{i})\in D$. Consider a holomorphic chart
$(z_{1},\ldots,z_{d})$ of $Y$, centered at $t_{i}$, with locally
$D=\\{z_{1}=0\\}$. By transversality of $F$, the map $q\mapsto z_{1}(F(q))$ is
a local diffeomorphism and embeds a sufficiently small disk
$\mathbb{D}=\\{q\in\square\colon\lVert q-p\rVert<\epsilon\\}$. As $\gamma_{i}$
is conjugate to the circle $\partial\mathbb{D}$, the loop $\eta_{i}$ is
conjugate to the closed curve $F(\partial\mathbb{D})=\partial(F(\mathbb{D}))$.
The latter winds once around $t_{i}=\\{z=0\\}=F(\mathbb{D})\cap L$, and it is
homotopic to $\\{\lvert
z_{1}\rvert=\epsilon\\}\times\\{z_{2}=\cdots=z_{d}=0\\}\subset T\setminus L$.
Thus indeed, $\eta_{i}$ (or its inverse, depending on the orientation)
represents the class of a simple loop in the sense of 3.1. ∎

Figure 27. Decomposition of a homotopy into simple loops.

######  Remark D.5.

Instead of transversality, one can prove D.1 and D.4 also with the Seifert–van
Kampen theorem. Consider the open cover $Y=Y\setminus D\cup N$ obtained from a
tubular neighbourhood $N\supset D$. By induction over the connected
components, we may assume that $D$ is connected. Then we can write the
amalgamated product as

(D.1) $\pi_{1}(Y)\cong\pi_{1}(Y\setminus D)*_{\pi_{1}(S)}\pi_{1}(D),$

exploiting the homotopy equivalences $N\simeq D$ and $N\setminus D\simeq S$,
where $S\subset N$ denotes a sphere bundle with fibre $\mathbb{S}^{r-1}$.
Since the fibre is connected ($r\geq 2$), the long exact homotopy sequence for
the bundle $S$ ends in

$\cdots\rightarrow\pi_{1}(\mathbb{S}^{r-1})\rightarrow\pi_{1}(S)\xrightarrow{\alpha}\pi_{1}(D)\rightarrow
0.$

The surjectivity of $\alpha$ renders the factor $\pi_{1}(D)$ in (D.1)
superfluous, proving that $\iota\colon\pi_{1}(Y\setminus
D)\rightarrow\pi_{1}(Y)$ surjects. In consequence, the amalgamated product
degenerates to a quotient

$\pi_{1}(Y)\cong\pi_{1}(Y\setminus D)/\langle G\rangle,$

where $\ker\iota=\langle G\rangle$ denotes the normal closure of the leftover
relations, where $G$ denotes the image in $\pi_{1}(Y\setminus D)$ of the
kernel of $\alpha$. For $r\geq 3$, the groups $\pi_{1}(\mathbb{S}^{r-1})=0$
and thus $\ker\alpha$ are trivial, so that $\iota$ is injective.

Finally, if $r=2$, $\pi_{1}(\mathbb{S}^{1})\cong\mathbb{Z}$ shows that $G$ is
generated by a single small loop $\gamma$ around $D$. Thus the entire kernel
$\langle G\rangle$ of $\iota$ is generated by the conjugates of $\gamma$. This
generation by a single element and its conjugates is explained also in [32,
Lemme 1.2].

######  Remark D.6.

Any two simple loops $\gamma$, $\gamma^{\prime}$ around the _same_ component
$\ell\in\mathfrak{L}$ are conjugate. Indeed, by definition,

$\gamma=\eta^{-1}\star\partial\mathbb{D}\star\eta\quad\text{and}\quad\gamma^{\prime}={\eta^{\prime}}^{-1}\star\partial\mathbb{D}^{\prime}\star\eta^{\prime}$

are conjugate to the the boundary circles
$\partial\mathbb{D}\cong\mathbb{S}^{1}$ of small holomorphic discs
$\mathbb{D}$, $\mathbb{D}^{\prime}$ embedded transverse to the smooth part
$S=\ell\setminus C_{n-2}$. We can shrink them to fit into a tubular
neighbourhood $N\supset S$, so that $\partial\mathbb{D}\simeq\pi^{-1}(t_{c})$
and $\partial\mathbb{D}^{\prime}\simeq\pi^{-1}(t_{c}^{\prime})$ are homotopic
to fibres of the corresponding circle bundle $\pi\colon\partial N\rightarrow
S$. Since $S$ is connected, there is a path $\rho\colon[0,1]\rightarrow S$
that connects $t_{c}$ and $t_{c}^{\prime}$. The pullback
$\rho^{-1}\pi\cong[0,1]\times\mathbb{S}^{1}$ is a cylinder and supplies a
homotopy $\partial\mathbb{D}\simeq\partial\mathbb{D}^{\prime}$; as paths with
basepoints, these circles are conjugate.

The closure $S\mapsto\overline{S}$ gives a bijection between the connected
components $S\subseteq D=L\setminus C_{n-2}$ (strata of complex codimension
one), and the Landau singularities $\mathfrak{L}$ (the irreducible components
of $L^{\prime}$).

For every $\ell\in\mathfrak{L}$, choose an arbitrary simple loop
$\gamma_{\ell}$ around $\ell$. Then by the previous remark, their conjugates
provide the set

$H=\\{\eta^{-1}\star\gamma_{\ell}\star\eta\colon\ell\in\mathfrak{L}\
\text{and}\ \eta\in\pi_{1}(T\setminus L)\\}.$

of _all_ simple loops. Part (3) of D.1 states that the kernel of $\iota$ is
the subgroup generated by $H$.

######  Remark D.7.

The subgroup generated by the simple loops
$H^{\prime}=\\{\gamma_{\ell}\colon\ell\in\mathfrak{L}\\}$ alone, without
including conjugates, can be strictly smaller. For example, [52, Theorem 2]
constructs, among several others, a smooth embedding

$L=\ell\sqcup\ell^{\prime}\quad\hookrightarrow\quad Y=\mathbb{S}^{6}$

of two disjoint spheres $\ell\cong\ell^{\prime}\cong\mathbb{S}^{4}$ with the
following property: For every choice of simple loops $\gamma_{\ell}$ and
$\gamma_{\ell^{\prime}}$, the group generated by
$H^{\prime}=\\{\gamma_{\ell},\gamma_{\ell^{\prime}}\\}$ is strictly smaller
than the kernel of $\pi_{1}(Y\setminus L)\rightarrow\pi_{1}(Y)$.

Thus the monodromy representation is not necessarily completely determined by
the variation along only one simple loop per Landau component.