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Given an $n$-manifold $M$ (say), we can talk about its Euler characteristic $\chi(M)$.

  1. This can be generalized to the Euler number of any $n$-dimensional bundle ${\mathcal V}$. Or indeed, the Euler class of any $k$-dimensional oriented bundle.

  2. Or, given a map $f:M\to M$, we can talk about its Lefschetz number, where $\chi(M)$ is the Lefschetz number of the identity.

  3. Or, we can compute $M$'s Betti numbers, and see $\chi(M)$ as the alternating sum.

I know how to combine 2 & 3, replacing the Betti number by $Tr(f|_{H^i(M)})$.

Are 1 & 2 or 1 & 3 combinable? Or 1, 2, and 3?

I don't have an application in mind, just the usual mathematician's hankering to take LCMs wherever possible.

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    $\begingroup$ Given a map $f : M \to M$ you can post-compose it with the $0$-section of a vector bundle, this gives a map $M \to E$ where $E \to B$ is a vector bundle. There is a primary obstruction to homotoping this map to be disjoint from the $0$-section (like in Milnor and Stasheff). You could interpret that as a combination of 1, 2 and 3, in that it has all three cases as special cases. If the bundle was orientable, the primary obstruction would be an $m-k$ dimensional integral homology class in $M$, where the bundle fibers have dimension $k$. $\endgroup$
    – Ryan Budney
    Commented Aug 22, 2011 at 7:45
  • $\begingroup$ How does it include 3? What is the analogue of a Betti number, that I would alternating-sum to get this obstruction class? $\endgroup$
    – Allen Knutson
    Commented Aug 22, 2011 at 19:16
  • $\begingroup$ In my comment above, $B=M$, sorry about that confusion. I took point (3) as a type of "local summation formula" computation of $\chi M$. When you perturb $M \to E$ to be transverse to the $0$-section, the intersection will have various connected components, and your obstruction is the sum of those (with appropriate orientations). (3) can be interpreted as a version of that. $\endgroup$
    – Ryan Budney
    Commented Aug 23, 2011 at 0:11

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