The Lefschetz fixed point theorem tells us that if an endomorphism on a space has Lefschetz number (the alternating sum of the traces of its induced maps on homology) not equal to zero, then it must have a fixed point. Groups famously have lots of topological automorphisms with no fixed points (left-multiply by any nontrivial element g). One can also prove that such a function is homotopic to the identity whenever the group is path connected (just take any path from 1 to g and appropriate translations thereof performed simultaneously).

Hence, since homotopic maps induce the same maps on homology, the Lefschetz number of the identity map must be zero, and the Lefschetz number of the identity map is precisely the Euler characteristic!

This is really cool. In particular, let’s think about 2-manifolds. The only compact triangulable 2-manifolds of Euler characteristic 0 are the torus and the Klein bottle. We know the torus has a natural group structure. What Lefschetz has told us is in fact, no other orientable surfaces have a group structure! In particular, we recover the reason elliptic curves have street cred. An algebraic curve (over the complex numbers) has a topological group structure iff it is an elliptic curve.

This also raises the question of whether the Klein bottle also has a nice group structure. Given it’s sort of a twisted torus, maybe we can build one somehow…

Advertisements