# Lefschetz number of a transformation of the sphere.

In differential topology the Lefschetz number of an automorphism of a compact manifold is the oriented intersection number of the graph of that automorphism with the diagonal.

I would like a proof or a significant hint to establish that on the sphere, this is 1 plus (or minus in odd dimensions) the degree (winding number) of that map.

• I think what you need is the proof of the fact that oriented intersection of diagonal and graph is $= \sum_q (-1)^q \text{trace } H^q(f)$ as pointed out below. You can find the proof of this computation on page 421 of the book principles of algebraic geometry - griffiths & harris. Commented Dec 1, 2015 at 17:28
• Many thanks to you both. I will study that text with the question quoted by Gareth as a reading guideline. Commented Dec 1, 2015 at 17:31

## 1 Answer

This looks like essentially the same question as Exercise concerning the Lefschetz fixed point number for the special case of a sphere.

• I guess it is, but I will have to study. The cohomology of the sphere is trivial except in the lowest and highest dimension, so my right hand side is the algebraic definition of the Lefschetz number. I will try to understand from that question how the two definitions are equivalent. Upvote for the link already. Commented Dec 1, 2015 at 17:24