Equivariant intersection number of π-trivial immersions

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1 Introduction

This is a work in progress! Initial blurb.

Let $(\widetilde{M},\pi,w)$${{Stub}} == Introduction == <wikitex>; This is a work in progress! Initial blurb. Let $(\widetilde{M},\pi,w)$ be an oriented cover of a connected manifold $M^{n_1+n_2}$ and let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$-trivial immersions of manifolds in $M$. The equivariant intersection number $\lambda([N_1],[N_2])\in\Z[\pi]$ counts with elements of $\Z[\pi]$ the number of intersection points that the two immersions have. The intersection number is an obstruction to perturbing the immersions into being disjoint, which when zero can often be achieved using the Whitney trick. The intersection number is also used in defining the intersection form of a k$-dimensional manifold and in turn its signature - a very important invariant used in the classification of manifolds and the primary surgery obstruction. </wikitex> == Definition == <wikitex>; The ''homology intersection pairing'' of $M$ with respect to an oriented cover $(\widetilde{M},\pi,w)$ $$\begin{array}{rcl}\lambda: H_n(\widetilde{M})\times H_{m-n}(\widetilde{M}) &\to& \Z[\pi]\ (a,b) &\mapsto& \lambda(a,b)\end{array}$$ is the sesquilinear pairing defined by $$\lambda(a,b) = a^*(b)\in \Z[\pi]$$ with $a^*\in H^{m-n}(\widetilde{M})$ the Poincar\'{e} dual of $a$ with respect to Universal Poincar\'{e} duality, such that $$\lambda(b,a) = (-1)^{n(m-n)}\overline{\lambda(a,b)}\in \Z[\pi].$$ The ''algebraic intersection number'' of $\pi$-trivial maps $f_1:N_1^{n_1} \to M^{n_1+n_2}$, $f_2:N_2^{n_2} \to M^{n_1+n_2}$ with prescribed lifts $\widetilde{f}_1:N_1\to \widetilde{M}$, $\widetilde{f}_2:N_2\to\widetilde{M}$ is the homology intersection of the homology classes $(\widetilde{f}_1)_*[N_1]\in H_{n_1}(\widetilde{M})$, $(\widetilde{f}_2)_*[N_2]\in H_{n_2}(\widetilde{M})$: $$\lambda^{\mathrm{alg}}(N_1,N_2) = \lambda((\widetilde{f}_1)_*[N_1],(\widetilde{f}_2)_*[N_2])\in \Z[\pi].$$ </wikitex> == Alternative Description == <wikitex> Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$-trivial immersions with prescribed lifts to the oriented cover $\widetilde{f}_1: N_1 \to \widetilde{M}$, $\widetilde{f}_2:N_2 \to \widetilde{M}$. At a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ let $g(x)\in \pi$ be the unique covering translation $\widetilde{M}\to \widetilde{M}$ such that $$\widetilde{f}_2(x_2) = g(x)\widetilde{f}_1(x_1)\in \widetilde{M}.$$ Note that the lifted immersions $g(x)\widetilde{f}_1:N_1\looparrowright\widetilde{M}$, $\widetilde{f}_2:N_2\looparrowright\widetilde{M}$ have a transverse double point $$\widetilde{x} = (x_1,x_2) \in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$$ and there is defined an isomorphism of oriented $m$-dimensional vector spaces $$d\widetilde{f}(x) = (d(g(x)\widetilde{f}_1),d\widetilde{f}_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_{\widetilde{M}}(\widetilde{f}_1(x_1)).$$ The ''equivariant index'' $I(x)\in\Z[\pi]$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is $$I(x) = w(x)g(x)\in{\pm\pi}\subset \Z[\pi]$$ with $$w(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; d\widetilde{f}(x) \;\mathrm{preserves}\; \mathrm{orientations} \ -1, & \mathrm{otherwise}.\end{array}\right.$$ Instead of thinking in terms of lifts the equivariant index can also be defined in terms of paths. Let $b$ be a basepoint of $M$, $b_i$ a basepoint of $N_i$ for $i=1,2$ and let $\widetilde{b}\in\widetilde{M}$ be some choice of lift. For a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$, an isotopy class of paths from $b_i$ to $x_i$ corresponds to a lift $\widetilde{f}_i$ as follows. The effect on the equivariant index of a change of order in the double point is given by $$I(x_2,x_1) = (-1)^{n_1n_2}\overline{I(x_1,x_2)}\in\Z[\pi]$$ with $\Z[\pi]\to \Z[\pi];a \mapsto \overline{a}$ the $w$-twisted involution $$a=\sum_{g\in\pi}n_gg \mapsto \sum_{g\in\pi}n_gw(g)g^{-1}\,(n_g\in\Z).$$ The ''geometric intersection number'' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is $$\lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z[\pi].$$ </wikitex> == Equivalence of definitions == <wikitex>; The algebraic and geometric intersection numbers agree. See REFERENCE </wikitex> == Examples == <wikitex>; ... </wikitex> == References == {{#RefList:}} [[Category:Definitions]](\widetilde{M},\pi,w)$ be an oriented cover of a connected manifold $M^{n_1+n_2}$$M^{n_1+n_2}$ and let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$$f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$$f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$$\pi$-trivial immersions of manifolds in $M$$M$. The equivariant intersection number $\lambda([N_1],[N_2])\in\Z[\pi]$$\lambda([N_1],[N_2])\in\Z[\pi]$ counts with elements of $\Z[\pi]$$\Z[\pi]$ the number of intersection points that the two immersions have. The intersection number is an obstruction to perturbing the immersions into being disjoint, which when zero can often be achieved using the Whitney trick.

The intersection number is also used in defining the intersection form of a $4k$$4k$-dimensional manifold and in turn its signature - a very important invariant used in the classification of manifolds and the primary surgery obstruction.

2 Definition

The homology intersection pairing of $M$$M$ with respect to an oriented cover $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$

$\displaystyle \begin{array}{rcl}\lambda: H_n(\widetilde{M})\times H_{m-n}(\widetilde{M}) &\to& \Z[\pi]\\ (a,b) &\mapsto& \lambda(a,b)\end{array}$ $$\displaystyle \begin{array}{rcl}\lambda: H_n(\widetilde{M})\times H_{m-n}(\widetilde{M}) &\to& \Z[\pi]\\ (a,b) &\mapsto& \lambda(a,b)\end{array}$$

is the sesquilinear pairing defined by

$\displaystyle \lambda(a,b) = a^*(b)\in \Z[\pi]$ $$\displaystyle \lambda(a,b) = a^*(b)\in \Z[\pi]$$

with $a^*\in H^{m-n}(\widetilde{M})$$a^*\in H^{m-n}(\widetilde{M})$ the Poincar\'{e} dual of $a$$a$ with respect to Universal Poincar\'{e} duality, such that

$\displaystyle \lambda(b,a) = (-1)^{n(m-n)}\overline{\lambda(a,b)}\in \Z[\pi].$ $$\displaystyle \lambda(b,a) = (-1)^{n(m-n)}\overline{\lambda(a,b)}\in \Z[\pi].$$

The algebraic intersection number of $\pi$$\pi$-trivial maps $f_1:N_1^{n_1} \to M^{n_1+n_2}$$f_1:N_1^{n_1} \to M^{n_1+n_2}$, $f_2:N_2^{n_2} \to M^{n_1+n_2}$$f_2:N_2^{n_2} \to M^{n_1+n_2}$ with prescribed lifts $\widetilde{f}_1:N_1\to \widetilde{M}$$\widetilde{f}_1:N_1\to \widetilde{M}$, $\widetilde{f}_2:N_2\to\widetilde{M}$$\widetilde{f}_2:N_2\to\widetilde{M}$ is the homology intersection of the homology classes $(\widetilde{f}_1)_*[N_1]\in H_{n_1}(\widetilde{M})$$(\widetilde{f}_1)_*[N_1]\in H_{n_1}(\widetilde{M})$, $(\widetilde{f}_2)_*[N_2]\in H_{n_2}(\widetilde{M})$$(\widetilde{f}_2)_*[N_2]\in H_{n_2}(\widetilde{M})$:

$\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2) = \lambda((\widetilde{f}_1)_*[N_1],(\widetilde{f}_2)_*[N_2])\in \Z[\pi].$ $$\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2) = \lambda((\widetilde{f}_1)_*[N_1],(\widetilde{f}_2)_*[N_2])\in \Z[\pi].$$

3 Alternative Description

Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$$f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$$f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$$\pi$-trivial immersions with prescribed lifts to the oriented cover $\widetilde{f}_1: N_1 \to \widetilde{M}$$\widetilde{f}_1: N_1 \to \widetilde{M}$, $\widetilde{f}_2:N_2 \to \widetilde{M}$$\widetilde{f}_2:N_2 \to \widetilde{M}$. At a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ let $g(x)\in \pi$$g(x)\in \pi$ be the unique covering translation $\widetilde{M}\to \widetilde{M}$$\widetilde{M}\to \widetilde{M}$ such that

$\displaystyle \widetilde{f}_2(x_2) = g(x)\widetilde{f}_1(x_1)\in \widetilde{M}.$ $$\displaystyle \widetilde{f}_2(x_2) = g(x)\widetilde{f}_1(x_1)\in \widetilde{M}.$$

Note that the lifted immersions $g(x)\widetilde{f}_1:N_1\looparrowright\widetilde{M}$$g(x)\widetilde{f}_1:N_1\looparrowright\widetilde{M}$, $\widetilde{f}_2:N_2\looparrowright\widetilde{M}$$\widetilde{f}_2:N_2\looparrowright\widetilde{M}$ have a transverse double point

$\displaystyle \widetilde{x} = (x_1,x_2) \in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$ $$\displaystyle \widetilde{x} = (x_1,x_2) \in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$$

and there is defined an isomorphism of oriented $m$$m$-dimensional vector spaces

$\displaystyle d\widetilde{f}(x) = (d(g(x)\widetilde{f}_1),d\widetilde{f}_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_{\widetilde{M}}(\widetilde{f}_1(x_1)).$ $$\displaystyle d\widetilde{f}(x) = (d(g(x)\widetilde{f}_1),d\widetilde{f}_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_{\widetilde{M}}(\widetilde{f}_1(x_1)).$$

The equivariant index $I(x)\in\Z[\pi]$$I(x)\in\Z[\pi]$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ is

$\displaystyle I(x) = w(x)g(x)\in{\pm\pi}\subset \Z[\pi]$ $$\displaystyle I(x) = w(x)g(x)\in{\pm\pi}\subset \Z[\pi]$$

with

$\displaystyle w(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; d\widetilde{f}(x) \;\mathrm{preserves}\; \mathrm{orientations} \\ -1, & \mathrm{otherwise}.\end{array}\right.$ $$\displaystyle w(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; d\widetilde{f}(x) \;\mathrm{preserves}\; \mathrm{orientations} \\ -1, & \mathrm{otherwise}.\end{array}\right.$$

Instead of thinking in terms of lifts the equivariant index can also be defined in terms of paths. Let $b$$b$ be a basepoint of $M$$M$, $b_i$$b_i$ a basepoint of $N_i$$N_i$ for $i=1,2$$i=1,2$ and let $\widetilde{b}\in\widetilde{M}$$\widetilde{b}\in\widetilde{M}$ be some choice of lift. For a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$x=(x_1,x_2)\in S_2(f_1,f_2)$, an isotopy class of paths from $b_i$$b_i$ to $x_i$$x_i$ corresponds to a lift $\widetilde{f}_i$$\widetilde{f}_i$ as follows.

The effect on the equivariant index of a change of order in the double point is given by

$\displaystyle I(x_2,x_1) = (-1)^{n_1n_2}\overline{I(x_1,x_2)}\in\Z[\pi]$ $$\displaystyle I(x_2,x_1) = (-1)^{n_1n_2}\overline{I(x_1,x_2)}\in\Z[\pi]$$

with $\Z[\pi]\to \Z[\pi];a \mapsto \overline{a}$$\Z[\pi]\to \Z[\pi];a \mapsto \overline{a}$ the $w$$w$-twisted involution

$\displaystyle a=\sum_{g\in\pi}n_gg \mapsto \sum_{g\in\pi}n_gw(g)g^{-1}\,(n_g\in\Z).$ $$\displaystyle a=\sum_{g\in\pi}n_gg \mapsto \sum_{g\in\pi}n_gw(g)g^{-1}\,(n_g\in\Z).$$

The geometric intersection number of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$$f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$$(i=1,2)$ is

$\displaystyle \lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z[\pi].$ $$\displaystyle \lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z[\pi].$$

4 Equivalence of definitions

The algebraic and geometric intersection numbers agree. See REFERENCE

5 Examples

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