Equivariant intersection number of π-trivial immersions
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| − | == Alternative | + | == Alternative Description: Lifts == |
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| − | Let $ | + | As in the [[Intersection_number_of_immersions|non-equivariant case]] the equivariant intersection form has a geometric interpretation. Let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|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 [[Π-trivial_map|$\pi$-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 $$\widetilde{f}_2(x_2) = g(x)\widetilde{f}_1(x_1)\in \widetilde{M}.$$ 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 $$\widetilde{x} = (x_1,x_2) \in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$$ and there is defined an isomorphism of oriented $(n_1+n_2)$-dimensional vector spaces $$d\widetilde{f}(x) = (d(g(x)\widetilde{f}_1),d\widetilde{f}_2): \tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2) \to \tau_{\widetilde{M}}(\widetilde{f}_1(x_1))$$ 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 $$I(x) = \epsilon(x)g(x)\in{\pm\pi}\subset \Z[\pi]$$ with | |
| + | $$\epsilon(x) = \left\{ \begin{array}{cc} +1, & \mathrm{if}\; d\widetilde{f}(x) \;\mathrm{preserves}\; \mathrm{orientations} \\ -1, & \mathrm{otherwise}.\end{array}\right.$$ | ||
| − | The ''equivariant | + | The ''geometric equivariant intersection number'' of $f_1$ and $f_2$ is then defined to be $$\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2):= \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z[\pi].$$ |
| − | $$ | + | |
| + | The effect on the equivariant index of a change of order in the double point is given by $$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)$$ as $g(x_2,x_1) = g(x_1,x_2)^{-1}$ and the orientation of $\tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2)$ disagrees withe the orientation of $\tau_{N_2}(x_2)\oplus \tau_{N_1}(x_1)$ if and only if $n_1$ and $n_2$ are both odd. Consequently $$\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_2,f_1) = (-1)^{n_1n_2}\overline{\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2)}.$$ | ||
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| + | Observe that $\epsilon(x)$ agrees with the non-equivariant index $I(\widetilde{x})\in \Z$ of the transverse double point $\widetilde{x}=(x_1,x_2)\in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$ from which it follows that $$\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].$$ | ||
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| + | == Alternative Description: Paths == | ||
| + | <wikitex> | ||
| + | Let $b_i\in N_i$, $b\in M$ and $\widetilde{b}\in\widetilde{M}$ such that $p(\widetilde{b})=b$ be basepoints. Then there is a bijective correspondence $$ \{ \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}) $$ as explained in the page on [[Π-trivial_map|$\pi$-trivial maps]]. Thus there are two equivalent conventions we can use for the data of a $\pi$-trivial map: either a choice of lift or a choice of path from $b$ to $f_i(b_i)$ modulo $\pi_1(\widetilde{M})$. Both conventions have equivalent definitions for the equivariant intersection number of transversely intersecting $\pi$-trivial immersions. | ||
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Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$-trivial immersions with prescribed equivalence classes of paths $[w_i:I\to M]$ such that $w_i(0)=b$ and $w_i(1)=f_i(b_i)$ for $i=1,2$. At a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ define $g(x)\in\pi$ as the loop $$g(x):= w_1 * f_1(u_1) * f_0(u_0)^-*w_0^-$$ where $u_i:I\to N_i$ is any path from $b_i$ to $x_i$. This loop is well-defined since a different choice of representative of $w_i$ or a different path $u_i$ results in a loop that differs from the other by an element of $\pi_1(\widetilde{M})$ or $(f_i)_*(\pi_1(N_i))$ which is trivial in $\pi$. | Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$-trivial immersions with prescribed equivalence classes of paths $[w_i:I\to M]$ such that $w_i(0)=b$ and $w_i(1)=f_i(b_i)$ for $i=1,2$. At a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ define $g(x)\in\pi$ as the loop $$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 $x_i$. This loop 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 $\pi$. | ||
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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. | 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 | The ''geometric intersection number'' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is | ||
Revision as of 14:13, 30 May 2014
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This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
This is a work in progress! Initial blurb.
Let 
be an oriented cover of a connected manifold 
and let 
, 
be 
-trivial immersions of manifolds in 
. The equivariant intersection number ![\lambda([N_1],[N_2])\in\Z[\pi]](/images/math/b/c/d/bcdfc6dc4b56e4b2b7b24ca843038bd8.png)
counts with elements of ![\Z[\pi]](/images/math/1/e/c/1ecba37f7392a1697efda3ee93c740e2.png)
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 
-dimensional manifold and in turn its signature - a very important invariant used in the classification of manifolds and the primary surgery obstruction.
2 Definition
with respect to an oriented cover ![\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}](/images/math/0/b/4/0b4b53e55cd8acf4980d2c5535ca1469.png)
![\displaystyle \lambda(a,b) = a^*(b)\in \Z[\pi]](/images/math/b/0/f/b0fcae216fdc9e5f0f4c0c3575d15bcd.png)
the Poincaré dual of 
with respect to Universal Poincaré duality, such that ![\displaystyle \lambda(b,a) = (-1)^{n(m-n)}\overline{\lambda(a,b)}\in \Z[\pi].](/images/math/7/a/9/7a959480e00e19774331363df44ca591.png)
-trivial maps 
, 
with prescribed lifts 
, 
is the homology intersection of the homology classes ![(\widetilde{f}_1)_*[N_1]\in H_{n_1}(\widetilde{M})](/images/math/c/4/6/c46e32560c8615e67d873ca8c7ff648d.png)
, ![(\widetilde{f}_2)_*[N_2]\in H_{n_2}(\widetilde{M})](/images/math/1/f/e/1febc73b71c95f1718e9acbbff3df2c1.png)
: ![\displaystyle \lambda^{\mathrm{alg}}(N_1,N_2) = \lambda((\widetilde{f}_1)_*[N_1],(\widetilde{f}_2)_*[N_2])\in \Z[\pi].](/images/math/0/a/5/0a5d108cbbc13596efbbc744b332d2cd.png)
3 Alternative Description: Lifts
As in the non-equivariant case the equivariant intersection form has a geometric interpretation. Let be an oriented cover of a connected manifold with 
-twisted fundamental class ![[\widetilde{M}]\in H_{n_1+n_2}(M;\Z^w)](/images/math/7/5/4/754443c9bbb2af67b082b919cbd77201.png)
corresponding to the lift 
of the basepoint 
. Let , be transverse -trivial immersions of oriented manifolds with prescribed lifts 
, 
.
there is a unique covering translation 
such that 
, 
have a transverse double point 
-dimensional vector spaces 
, 
and 
all inherit orientations from the given orientations of 
, 
and 
.
The equivariant index ![I(x)=I(x_1,x_2)\in\Z[\pi]](/images/math/5/f/b/5fb6d0c9d0e9599764435c2cba0faf4a.png)
of a transverse double point is defined to be ![\displaystyle I(x) = \epsilon(x)g(x)\in{\pm\pi}\subset \Z[\pi]](/images/math/f/2/d/f2df2770489968fbd580e4eda396a118.png)
and 
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].](/images/math/d/3/e/d3ef600c38196816eeea0e2211cec383.png)
![\displaystyle I(x_2,x_1) = (-1)^{n_1n_2}\overline{I(x_1,x_2)}\in\Z[\pi]](/images/math/a/e/d/aede0071954c04f02bad98282381d414.png)
![\Z[\pi]\to \Z[\pi];a \mapsto \overline{a}](/images/math/a/8/7/a8741f19d2532c7dba3b6941e56c3b69.png)
the -twisted involution 
and the orientation of 
disagrees withe the orientation of 
if and only if 
and 
are 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)}.](/images/math/a/8/f/a8f6f60c47afc1e8d68b70373795fe68.png)
agrees with the non-equivariant index 
of the transverse double point 
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].](/images/math/2/7/d/27dd1648e043309d5177b8662ab2eab4.png)
4 Alternative Description: Paths
, and such that 
be basepoints. Then there is a bijective correspondence 
-trivial maps. Thus there are two equivalent conventions we can use for the data of a -trivial map: either a choice of lift or a choice of path from 
to 
modulo 
. Both conventions have equivalent definitions for the equivariant intersection number of transversely intersecting -trivial immersions.
Let , be -trivial immersions with prescribed equivalence classes of paths ![[w_i:I\to M]](/images/math/6/0/2/60250747a77973d74f7dd12ff5dac28d.png)
such that 
and 
for 
. At a transverse double point define 
as the loop 
is any path from 
to 
. This loop is well-defined since a different choice of representative of 
or a different path 
results in a loop that differs from the other by an element of or 
which is trivial in .
Definition of :
Equivalence:
Let be a basepoint of , a basepoint of 
for and let be some choice of lift. For a transverse double point , an isotopy class of paths from to corresponds to a lift 
as follows.
The geometric intersection number of transverse immersions 
is
![\displaystyle \lambda^{\mathrm{geo}}(N_1,N_2) = \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z[\pi].](/images/math/1/d/1/1d191be0a6f9a0c566a9030f40a3ea2a.png)
5 Equivalence of definitions
The algebraic and geometric intersection numbers agree. See REFERENCE
6 Examples
...
