The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows: the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the lefschetz number $L(f)$.

First it is clear, that the intersection number at a fixpoint is independent of the orientation attached to a neighborhood of the fixpoint. So the intersection number makes sense.

There is the weaker lefschetz fixpoint theorem for finite simplicial complexes, which also applies in this case, where you only get the existence of fixpoints.

The question is if the above version with the intersection number holds for non-orientable manifolds too.

  • $\begingroup$ I added the question. $\endgroup$ – Raphael Reinauer Jan 21 '13 at 20:57
  • $\begingroup$ Yes. You can reduce to the orientable case. There is a bundle of $1$-disks over $M$ that is orientable. Use $f$ to make a map from this to itself. $\endgroup$ – Tom Goodwillie Jan 22 '13 at 1:43

I agree with Tom, but I would use the 2-to-1 oriented cover of a non-oriented manifold (described for example in Allen Hatcher's book 'Algebraic Topology', p.234) to reduce the problem to the orientable case. I believe you can lift your map between the oriented covers. (I am not very familiar with bundles, so I apologize if my answer is identical to Tom's.)

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  • $\begingroup$ A beginner's question about the customs of mathoverflow: Should I have added this answer as a comment to the question (as Tom Goodwillie did) or was it okay to add an 'actual' answer? $\endgroup$ – Rami Luisto Jan 22 '13 at 22:55
  • $\begingroup$ If it answers the question, or makes substantial progress to an answer, or is too long to be a comment, then yes it's appropriate. Sometimes, a sufficient answer is really trivial and people will just leave it as a comment, but is a matter of taste. $\endgroup$ – Chris Gerig Jan 22 '13 at 23:01
  • $\begingroup$ Thanks, Chris. Good to know even though it will apparently take some time before I amass the required 50 reputation to actually be able to leave comments to questions. $\endgroup$ – Rami Luisto Jan 22 '13 at 23:04

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