This is the mirror of previous post.

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in N\times N | x_1\neq x_2, f(x_1)=f(x_2)\rbrace/(x_1,x_2)\sim(x_2,x_1)$.

Second, for each $x=[(x_1,x_2)]\in S_2[f]$, we can think an equivariant intersection number $I[x]=a(x)w(x)\in \lbrace \pm g|g\in\pi_1(M)\rbrace\subset \mathbb{Z}[\pi_1(M)]$ as follows:

$a(x)\in \pi_1(M)$ satisfies $a(x)\circ\tilde{f}(x_1)=\tilde{f}(x_2)$, here we are regarding $a(x)\colon \tilde{M}\to \tilde{M}$.

$w(x)$ is defined to be $1$ or $-1$ if the isomorphism between oriented vector spaces $(d(a(x)\tilde{f})_{x_1},d\tilde{f}_{x_2})\colon T_{x_1}N\oplus T_{x_2}N\to T_{y}\tilde{M}$, where $y=a(x)\circ\tilde{f}(x_1)=\tilde{f}(x_2)$.

Eventually, we define the Wall's self intersection $\mu(f)=\sum_{x\in S_2[g]}I[x]\in \mathbb{Z}[\pi_1(M)]/\lbrace\sum a_g(g-(-1)^ng^{-1})\rbrace$ as in Wall's Surgery on compact manifolds Section 5 or Andrew Ranicki's book page 259.

Here, the the equivalence relation $\sum a_g(g-(-1)^ng^{-1})$ is added to eliminate the ambiguity in the choice of representative element in $S_2[f]$, namely $[(x_1,x_2)]=[(x_2,x_1)]\in S_2[f]$.

Now, For $t\in [0,1]$, let $f_t\colon N^n\to M^{2n}$ be an immersion (or regular homotopy by definition) and the regular homotopy $\tilde{f_t}\colon N\to \tilde{M}$ is also given.

I want to prove that in this situation $\mu(f_0)=\mu(f_1)$. i.e. the self-intersection number is an invariant under the regular homotopy.

Here is the argument given in Ranicki's book page 259.

Let $f\colon N\times[0,1]\to M\times [0,1]$ be the trace of $f_t$. i.e. $f(x,t)=(f_t(x),t)$.

Then, $f\times f^{-1}(\Delta(M\times[0,1])-\Delta(N\times[0,1])$ is 1-dimensional manifolds.

i.e. The unordered double point set of $f_1$ is obtained from the unordered double point set of $f_0$ by a sequence of "birth" and "death" of cancelling pair consisting two points in unordered double point set. Hence, such cancelling pair contributes to the self intersection number by $\underline{a-(-1)^nw(a)a=0}\in Q_{(-1)^n}(\mathbb{Z}[\pi_1(M)])$. Hence, $\mu(f_0)=\mu(f_1)$.

I want to prove rigorously that the "birth" or "death" of each cancelling pair contributes to self-intersection number as much as the underlined part. I think that I should compare the orientation of tangent space of $N$ at some double points and tangent space of $M$ at the image of them. But I can't fill in the details.

Are there anybody who can help me?