I can't see a good approach to the third part of the following problem:
Let $f: M \to M$ be a smooth map of a compact oriented manifold into itself. Denote by $H^q(f)$ the induced map on the cohomology $H^q(M)$. The Lefschetz number of f is defined to be
$L(f) = \sum_q (-1)^q \text{trace } H^q(f)$
Let $\Gamma$ be the graph of $f$ in $M\times M$
- Show that $\int_\Delta \eta_\Gamma = L(f)$
- Show that if $f$ has no fixed points, then $L(f)$ is zero.
- At a fixed point $P$ of $f$ the derivative $Df_p$ is an endomorphism of the tangent space $T_pM$. We define the multiplicity of the fixed point $P$ to be
$ \sigma_P = \text{sign} \, \det(Df_p - I)$
Show that if the graph $\Gamma$ is transversal to the diagonal $\Delta$ in $M \times M$, then
$L(f) = \sum_P \sigma_P$
where $P$ ranges over the fixed points of $f$. Here $\eta_S$ denotes the Poincaré dual of the submanifold $S$.
I'd be glad, if someone could help me out. Best would be only a hint and not an entire solution.
Ideas:
I guess one might somehow use that a form representing the Poincaré dual of a submanifold $S$ can be chosen to have support in an arbitrarily small neighborhood of $S$ to turn the problem into a local problem of computing integrals in neighborhoods of the fixed points of $f$, noting that
$\int_\Delta \eta_\Gamma = \int_{M\times M} \eta_\Gamma\wedge \eta_\Delta$
and $\eta_\Gamma\wedge \eta_\Delta$ would then be nonzero only in a arbitrarily small neighborhood of the intersections of $\Delta$ and $\Gamma$, i.e. the set of fixed points of $f$, so that one could work in local coordinates there. However, what I would find particularly strange is how -- if we could compute local integrals as suggested above -- they would turn out to be so nicely integervalued? But I don't get far trying to push this idea further...
I also actually don't really know what $\eta_\Gamma$ looks like. I've tried figuring out some expression for it, but without luck.
As usual, many thanks for your always helpful suggestions!
Best regards,
Sam