Danya Mamaev — Lefschetz Fixed-Point Theorem (Junior Geometry Seminar)

Title: Lefschetz fixed-point theorem

Speaker: Danya Mamaev

Abstract:  To any continuous self-map f of a good topological space X there corresponds its Lefschetz number L(f), which is an integer defined in terms of the maps on homology of X induced by f. The Lefschetz fixed-point theorem is a collection of statements relating L(f) with the number of fixed points of f, in the talk we will discuss in some details two of them.

First, if L(f) is non-zero, then f has a fixed point. This is a consequence of the existence of cellular (or simplicial) approximations for maps between good topological spaces.

Secondly, if X is a (compact, orientable) manifold, then L(f) is equal to the (algebraic) number of fixed points of f. As long as one figures the correct definition for the number of fixed points, this is simply a calculation extensively involving Poincaré duality.

After sketching the proofs of the above statements, I will explain one or two applications, depending on the audience’s preferences it will be a way to count either the number of zeroes of a vector field, the number of fixed points of a selp-map of a torus, or the number of points on an elliptic curve over a finite field.

I will try to keep the talk largely self-contained, yet some familiarity with point-set topology and (singular, simplicial, or cellular) homology groups is desirable.

Some snacks will be provided before and after the talk.

This talk will be broadcasted via Zoom. Subscribe to the mailing list or contact organisers to get the Zoom details.