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]$$ with $a^*\in H^{m-n}(\widetilde{M})$ the Poincaré dual of $a$ with respect to Universal Poincaré duality, such that $$\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].$$ </wikitex> == Alternative Description: Lifts == <wikitex> As in the [[Intersection_number_of_immersions|non-equivariant case]] the equivariant intersection form has a geometric interpretation. Let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|oriented cover]] of a connected manifold $M^{n_1+n_2}$ with $w$-twisted fundamental class $[\widetilde{M}]\in H_{n_1+n_2}(M;\Z^w)$ corresponding to the lift $\widetilde{b}\in\widetilde{M}$ of the basepoint $b\in M$. Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be transverse [[Π-trivial_map|$\pi$-trivial immersions]] of oriented manifolds with prescribed lifts $\widetilde{f}_1:N_1^{n_1} \looparrowright \widetilde{M}$, $\widetilde{f}_2:N_2^{n_2} \looparrowright \widetilde{M}$. At a double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ there is a unique covering translation $g(x):\widetilde{M}\to\widetilde{M}$ such that $$\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.$$ The ''geometric equivariant intersection number'' of $f_1$ and $f_2$ is then defined to be $$\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)}.$$ Observe that $\epsilon(x)$ agrees with the non-equivariant index $I(\widetilde{x})\in \Z$ of the transverse double point $\widetilde{x}=(x_1,x_2)\in S_2(g(x)\widetilde{f}_1,\widetilde{f}_2)$ from which it follows that $$\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].$$ </wikitex> == Alternative Description: Paths == <wikitex> Let $b_i\in N_i$, $b\in M$ and $\widetilde{b}\in\widetilde{M}$ such that $p(\widetilde{b})=b$ be basepoints. Then there is a bijective correspondence $$ \{ \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}) $$ as explained in the page on [[Π-trivial_map|$\pi$-trivial maps]]. Thus there are two equivalent conventions we can use for the data of a $\pi$-trivial map: either a choice of lift or a choice of path from $b$ to $f_i(b_i)$ modulo $\pi_1(\widetilde{M})$. Both conventions have equivalent definitions for the equivariant intersection number of transversely intersecting $\pi$-trivial immersions. Let $f_1:N_1^{n_1} \looparrowright M^{n_1+n_2}$, $f_2:N_2^{n_2} \looparrowright M^{n_1+n_2}$ be $\pi$-trivial immersions with prescribed equivalence classes of paths $[w_i:I\to M]$ such that $w_i(0)=b$ and $w_i(1)=f_i(b_i)$ for $i=1,2$. At a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$ define $g(x)\in\pi$ as the loop $$g(x):= w_1 * f_1(u_1) * f_0(u_0)^-*w_0^-$$ where $u_i:I\to N_i$ is any path from $b_i$ to $x_i$. This loop is well-defined since a different choice of representative of $w_i$ or a different path $u_i$ results in a loop that differs from the other by an element of $\pi_1(\widetilde{M})$ or $(f_i)_*(\pi_1(N_i))$ which is trivial in $\pi$. Definition of $\epsilon(x)$: Equivalence: Let $b$ be a basepoint of $M$, $b_i$ a basepoint of $N_i$ for $i=1,2$ and let $\widetilde{b}\in\widetilde{M}$ be some choice of lift. For a transverse double point $x=(x_1,x_2)\in S_2(f_1,f_2)$, an isotopy class of paths from $b_i$ to $x_i$ corresponds to a lift $\widetilde{f}_i$ as follows. The ''geometric intersection number'' of transverse immersions $f_i:N_i^{n_i} \looparrowright M^{n_1+n_2}$ $(i=1,2)$ is $$\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})$$a^*\in H^{m-n}(\widetilde{M})$ the Poincaré dual of $a$$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)$ disagrees withe 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 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

Let $b_i\in N_i$$b_i\in N_i$, $b\in M$$b\in M$ and $\widetilde{b}\in\widetilde{M}$$\widetilde{b}\in\widetilde{M}$ such that $p(\widetilde{b})=b$$p(\widetilde{b})=b$ be basepoints. Then there is a bijective correspondence

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

as explained in the page on $\pi$$\pi$-trivial maps. Thus there are two equivalent conventions we can use for the data of a $\pi$$\pi$-trivial map: either a choice of lift or a choice of path from $b$$b$ to $f_i(b_i)$$f_i(b_i)$ modulo $\pi_1(\widetilde{M})$$\pi_1(\widetilde{M})$. Both conventions have equivalent definitions for the equivariant intersection number of transversely intersecting $\pi$$\pi$-trivial immersions. 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$ as 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$$x_i$. This loop is well-defined since a different choice of representative of $w_i$$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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