# Lefschetz Fixpoint theorem for non-orientable manifolds

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.

• I added the question. Jan 21 '13 at 20:57
• 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. Jan 22 '13 at 1:43