Lefschetz number

For a continuous map between spaces with finitely generated homology
Suppose $$X$$ and $$Y$$ are topological spaces, and each of them is a defining ingredient::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.