Equivariant intersection number of π-trivial immersions

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Contents

[edit] 1 Introduction

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 4k-dimensional manifold and in turn its signature - a very important invariant used in the classification of manifolds and the primary surgery obstruction.

[edit] 2 Definition

The homology intersection pairing of M with respect to an oriented cover (\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}
is the sesquilinear pairing defined by
\displaystyle \lambda(a,b) = a^*(b)\in \Z[\pi]
with a^*\in H^{m-n}(\widetilde{M}) the Poincaré dual of a with respect to Universal Poincaré duality, such that
\displaystyle \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}):
\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2) = \lambda((\widetilde{f}_1)_*[N_1],(\widetilde{f}_2)_*[N_2])\in \Z[\pi].


[edit] 3 Alternative Description: Lifts


As in the non-equivariant case the equivariant intersection form has a geometric interpretation. Let (\widetilde{M},\pi,w) be an oriented cover of a connected manifold M^{n_1+n_2} with w-twisted fundamental class [\widetilde{M}]\in H_{n_1+n_2}(M;\Z^w) corresponding to the lift \widetilde{b}\in\widetilde{M} of the basepoint b\in M. 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 transverse \pi-trivial immersions of oriented manifolds with prescribed lifts \widetilde{f}_1:N_1^{n_1} \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) there is a unique covering translation 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}.
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
\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)-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))
where \tau_{N_1}(x_1), \tau_{N_2}(x_2) and \tau_{\widetilde{M}}(\widetilde{f}_1(x_1)) all inherit orientations from the given orientations of N_1, N_2 and \widetilde{M}. The equivariant index 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) is defined to be
\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.
The geometric equivariant intersection number of f_1 and 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].
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]
with \Z[\pi]\to \Z[\pi];a \mapsto \overline{a} the 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)
as 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) agrees with the orientation of \tau_{N_2}(x_2)\oplus \tau_{N_1}(x_1) if and only if n_1 and 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)}.
Observe that \epsilon(x) agrees with the non-equivariant index 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) 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].


[edit] 4 Alternative Description: Paths


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

Thus there are two equivalent conventions for the data of a \pi-trivial map f_i/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_OalaI0: either a choice of lift of f_i or the homotopy class of a choice of path from b to f_i(b_i) modulo \pi_1(\widetilde{M}). In the previous section the geometric equivariant intersection number of f_1 and 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_2:N_2^{n_2} \looparrowright M^{n_1+n_2} be \pi-trivial immersions with prescribed equivalence classes of paths [w_i:I\to M] such that w_i(0)=b and w_i(1)=f_i(b_i) for i=1,2. At a transverse double point x=(x_1,x_2)\in S_2(f_1,f_2) define 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^-]
where u_i:I\to N_i is any path from b_i to x_i, i=1,2, w_0^- denotes the path w_0 in reverse and * denotes concatenation of paths. This is well-defined since a different choice of representative of w_i or a different path u_i results in a loop that differs from the other by an element of \pi_1(\widetilde{M}) or (f_i)_*(\pi_1(N_i)) which is trivial in \pi.


Definition of \epsilon(x):

Equivalence: 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 geometric intersection number of transverse immersions f_i:N_i^{n_i} \looparrowright M^{n_1+n_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].

[edit] 5 Equivalence of definitions

The algebraic and geometric intersection numbers agree. See REFERENCE


[edit] 6 Examples

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[edit] 7 References

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