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)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(\widetilde{M},\pi,w)$ be an oriented cover of a connected manifold $M^{n_1+n_2}$$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})$$a^*\in H^{m-n}(\widetilde{M})$ the Poincaré dual of $a$$a$ with respect to Universal Poincaré 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: Lifts

As in the non-equivariant case the equivariant intersection form has a geometric interpretation. Let $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$ be an oriented cover of a connected manifold $M^{n_1+n_2}$$M^{n_1+n_2}$ with $w$$w$-twisted fundamental class $[\widetilde{M}]\in H_{n_1+n_2}(M;\Z^w)$$[\widetilde{M}]\in H_{n_1+n_2}(M;\Z^w)$ corresponding to the lift $\widetilde{b}\in\widetilde{M}$$\widetilde{b}\in\widetilde{M}$ of the basepoint $b\in M$$b\in M$. 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 transverse $\pi$$\pi$-trivial immersions of oriented manifolds with prescribed lifts $\widetilde{f}_1:N_1^{n_1} \looparrowright \widetilde{M}$$\widetilde{f}_1:N_1^{n_1} \looparrowright \widetilde{M}$, $\widetilde{f}_2:N_2^{n_2} \looparrowright \widetilde{M}$$\widetilde{f}_2:N_2^{n_2} \looparrowright \widetilde{M}$.

At a double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ there is a unique covering translation $g(x):\widetilde{M}\to\widetilde{M}$$g(x):\widetilde{M}\to\widetilde{M}$ such that

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

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 $(n_1+n_2)$$(n_1+n_2)$-dimensional vector spaces

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

where $\tau_{N_1}(x_1)$$\tau_{N_1}(x_1)$, $\tau_{N_2}(x_2)$$\tau_{N_2}(x_2)$ and $\tau_{\widetilde{M}}(\widetilde{f}_1(x_1))$$\tau_{\widetilde{M}}(\widetilde{f}_1(x_1))$ all inherit orientations from the given orientations of $N_1$$N_1$, $N_2$$N_2$ and $\widetilde{M}$$\widetilde{M}$. The equivariant index $I(x)=I(x_1,x_2)\in\Z[\pi]$$I(x)=I(x_1,x_2)\in\Z[\pi]$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ is defined to be

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

with

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

The geometric equivariant intersection number of $f_1$$f_1$ and $f_2$$f_2$ is then defined to be

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

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

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

as $g(x_2,x_1) = g(x_1,x_2)^{-1}$$g(x_2,x_1) = g(x_1,x_2)^{-1}$ and the orientation of $\tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2)$$\tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2)$ agrees with the orientation of $\tau_{N_2}(x_2)\oplus \tau_{N_1}(x_1)$$\tau_{N_2}(x_2)\oplus \tau_{N_1}(x_1)$ if and only if $n_1$$n_1$ and $n_2$$n_2$ are not both odd. Consequently

$\displaystyle \lambda_{\Z[\pi]}^{\mathrm{geo}}(f_2,f_1) = (-1)^{n_1n_2}\overline{\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2)}.$ $$\displaystyle \lambda_{\Z[\pi]}^{\mathrm{geo}}(f_2,f_1) = (-1)^{n_1n_2}\overline{\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2)}.$$

Observe that $\epsilon(x)$$\epsilon(x)$ agrees with the non-equivariant index $I(\widetilde{x})\in \Z$$I(\widetilde{x})\in \Z$ of the transverse double point $\widetilde{x}=(x_1,x_2)\in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$$\widetilde{x}=(x_1,x_2)\in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$ from which it follows that

$\displaystyle \lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2) = \sum_{g\in\pi}\lambda_{\Z}^{\mathrm{geo}}(g\widetilde{f}_1,\widetilde{f}_2)g\in \Z[\pi].$ $$\displaystyle \lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2) = \sum_{g\in\pi}\lambda_{\Z}^{\mathrm{geo}}(g\widetilde{f}_1,\widetilde{f}_2)g\in \Z[\pi].$$

4 Alternative Description: Paths

As explained in the page on $\pi$$\pi$-trivial maps given basepoints $b_1\in N_1$$b_1\in N_1$, $b_2\in N_2$$b_2\in N_2$ and $b\in M$$b\in M$, a choice of lift, $\widetilde{b}$$\widetilde{b}$, of $b$$b$ to the oriented cover $\widetilde{M}$$\widetilde{M}$ defines a bijection of sets

$\displaystyle \{ \widetilde{f_i}:N_i \to \widetilde{M} : p\circ \widetilde{f_i}=f_i\} \longleftrightarrow \{ w:I \to M : w(0)=b, w(1) = f_i(b_i)\}/\pi_1(\widetilde{M}).$ $$\displaystyle \{ \widetilde{f_i}:N_i \to \widetilde{M} : p\circ \widetilde{f_i}=f_i\} \longleftrightarrow \{ w:I \to M : w(0)=b, w(1) = f_i(b_i)\}/\pi_1(\widetilde{M}).$$

Thus there are two equivalent conventions for the data of a $\pi$$\pi$-trivial map $f_i/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_OalaI0$$f_i$: either a choice of lift of $f_i$$f_i$ or the homotopy class of a choice of path from $b$$b$ to $f_i(b_i)$$f_i(b_i)$ modulo $\pi_1(\widetilde{M})$$\pi_1(\widetilde{M})$. In the previous section the geometric equivariant intersection number of $f_1$$f_1$ and $f_2$$f_2$ was defined using lifts as data. In this section we see the equivalent approach of using paths and prove the equivalence.

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$$\pi$-trivial immersions with prescribed equivalence classes of paths $[w_i:I\to M]$$[w_i:I\to M]$ such that $w_i(0)=b$$w_i(0)=b$ and $w_i(1)=f_i(b_i)$$w_i(1)=f_i(b_i)$ for $i=1,2$$i=1,2$. At a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$$x=(x_1,x_2)\in S_2(f_1,f_2)$ define $g(x)\in\pi$$g(x)\in\pi$ to be the class of the loop

$\displaystyle g(x):= [w_1 * f_1(u_1) * f_0(u_0)^-*w_0^-]$ $$\displaystyle g(x):= [w_1 * f_1(u_1) * f_0(u_0)^-*w_0^-]$$

where $u_i:I\to N_i$$u_i:I\to N_i$ is any path from $b_i$$b_i$ to $x_i$$x_i$, $i=1,2$$i=1,2$, $w_0^-$$w_0^-$ denotes the path $w_0$$w_0$ in reverse and $*$$*$ denotes concatenation of paths. This is well-defined since a different choice of representative of $w_i$$w_i$ or a different path $u_i$$u_i$ results in a loop that differs from the other by an element of $\pi_1(\widetilde{M})$$\pi_1(\widetilde{M})$ or $(f_i)_*(\pi_1(N_i))$$(f_i)_*(\pi_1(N_i))$ which is trivial in $\pi$$\pi$.

Definition of $\epsilon(x)$$\epsilon(x)$:

Equivalence: 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 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

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

5 Equivalence of definitions

The algebraic and geometric intersection numbers agree. See REFERENCE

6 Examples

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