Lefschetz number

Definition

For a continuous map between spaces with finitely generated homology

Suppose $X$ and $Y$ are topological spaces, and each of them is a space with finitely generated homology. Suppose $f$ is a continuous map from $X$ to $Y$. The Lefschetz number or Lefschetz trace of $f$, denoted $\lambda (f)$, is defined as follows:

For each $i$, denote by $r_{i}$ the rank of the free part of the map $H_{i}(f):H_{i}(X)\to H_{i}(Y)$. One way of thinking of this is that we consider the sub-map between the free part of $H_{i}(X)$ and the free part of $H_{i}(Y)$, and look at the rank of the matrix used to describe this map.

Then, the Lefschetz number of $f$ is:

$\lambda (f)=\sum _{i=0}^{n}(-1)^{i}r_{i}$

Facts

• The Lefschetz number of the identity map from a space with finitely generated homology to itself equals the Euler characteristic of the space.
• The Lefschetz number of a map from an empty space is $0$.
• The Lefschetz number of a map from a contractible space to any space is $1$.
• The Lefschetz number of a map from any space to a contractible space is $1$.
• Lefschetz fixed-point theorem: This states that if the Lefschetz number of a map from a compact polyhedron to itself is nonzero, then the map must have a fixed point.