Equivariant intersection number of π-trivial immersions
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This is a work in progress! Initial blurb. | 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. | + | Let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|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 [[Π-trivial_map|$\pi$-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. | 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. | ||
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== Definition == | == Definition == | ||
<wikitex>; | <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 | + | The ''homology intersection pairing'' of $M$ with respect to an [[Oriented cover|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 Poincaré dual of $a$ with respect to Universal Poincaré 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 $(\widetilde{f}_1)_*[N_1]\in H_{n_1}(\widetilde{M})$, $(\widetilde{f}_2)_*[N_2]\in H_{n_2}(\widetilde{M})$: $$\lambda^{\mathrm{alg}}(N_1,N_2) = \lambda((\widetilde{f}_1)_*[N_1],(\widetilde{f}_2)_*[N_2])\in \Z[\pi].$$ | 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> | </wikitex> | ||
− | == Alternative Description == | + | == Alternative Description: Lifts == |
<wikitex> | <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 | + | 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'' $I(x)\in\Z[\pi]$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is $$I(x) = | + | The ''equivariant index'' $I(x)=I(x_1,x_2)\in\Z[\pi]$ of a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ is defined to be $$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 ''geometric equivariant intersection number'' of $f_1$ and $f_2$ is then defined to be $$\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2):= \sum_{x\in S_2(f_1,f_2)}{I(x)}\in \Z[\pi].$$ | |
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+ | 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 $\tau_{N_1}(x_1)\oplus \tau_{N_2}(x_2)$ 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 $$\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_2,f_1) = (-1)^{n_1n_2}\overline{\lambda_{\Z[\pi]}^{\mathrm{geo}}(f_1,f_2)}.$$ | ||
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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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+ | </wikitex> | ||
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+ | == Alternative Description: Paths == | ||
+ | <wikitex> | ||
+ | As explained in the page on [[Π-trivial_map|$\pi$-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 $$ \{ \widetilde{f_i}:N_i \to \widetilde{M} : p\circ \widetilde{f_i}=f_i\} \longleftrightarrow \{ w:I \to M : w(0)=b, w(1) = f_i(b_i)\}/\pi_1(\widetilde{M}).$$ | ||
+ | Thus there are two equivalent conventions for the data of a $\pi$-trivial map $f_i$: 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. | ||
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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 $g(x)\in\pi$ to be the class of 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 $x_i$, $i=1,2$, $w_0^-$ 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 $\pi$. | ||
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+ | Definition of $\epsilon(x)$: | ||
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+ | 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. | ||
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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 |
Latest revision as of 15:34, 16 June 2014
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 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 counts with elements of 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.
[edit] 2 Definition
[edit] 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 corresponding to the lift of the basepoint . Let , be transverse -trivial immersions of oriented manifolds with prescribed lifts , .
[edit] 4 Alternative Description: Paths
Thus there are two equivalent conventions for the data of a -trivial map : either a choice of lift of or the homotopy class of a choice of path from to modulo . In the previous section the geometric equivariant intersection number of and was defined using lifts as data. In this section we see the equivalent approach of using paths and prove the equivalence.
Let , be -trivial immersions with prescribed equivalence classes of paths such that and for . At a transverse double point define to be the class of the loop
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
[edit] 5 Equivalence of definitions
The algebraic and geometric intersection numbers agree. See REFERENCE
[edit] 6 Examples
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