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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 $M\times M$

  1. Show that $\int_\Delta \eta_\Gamma = L(f)$
  2. Show that if $f$ has no fixed points, then $L(f)$ is zero.
  3. 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

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1 Answer 1

up vote 3 down vote accepted

I think, I've got it now. One can use the fact that $η_\Delta$ is just the Thom class of the normal bundle of $\Delta$ and by transversality, this implies that the normal bundle of $\Delta$ is the tangent bundle of $\Gamma$ at the intersection points and vice versa. So integration boils down to integration along the fiber.

This gives that the integrals around the intesection points have values $\pm 1$, depending on $sign(\det(Df_p - I))$. (Using the idea already mentioned after the question)

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