Vipul: Created page with "==Definition== ===For a continuous map between spaces with finitely generated homology=== Suppose $X$ and $Y$ are topological spaces, and each of them..."

2011-04-02T15:30:41Z

Created page with "==Definition== ===For a continuous map between spaces with finitely generated homology=== Suppose <math>X</math> and <math>Y</math> are topological spaces, and each of them..."

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==Definition==

===For a continuous map between spaces with finitely generated homology===

Suppose <math>X</math> and <math>Y</math> are [[topological space]]s, and each of them is a [[defining ingredient::space with finitely generated homology]]. Suppose <math>f</math> is a [[continuous map]] from <math>X</math> to <math>Y</math>. The '''Lefschetz number''' or '''Lefschetz trace''' of <math>f</math>, denoted <math>\lambda(f)</math>, is defined as follows:

For each <math>i</math>, denote by <math>r_i</math> the rank of the ''free'' part of the map <math>H_i(f): H_i(X) \to H_i(Y)</math>. One way of thinking of this is that we consider the sub-map between the free part of <math>H_i(X)</math> and the free part of <math>H_i(Y)</math>, and look at the rank of the matrix used to describe this map.

Then, the Lefschetz number of <math>f</math> is:

<math>\lambda(f) = \sum_{i=0}^n (-1)^i r_i</math>

==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 <math>0</math>.
* The Lefschetz number of a map ''from'' a [[contractible space]] to any space is <math>1</math>.
* The Lefschetz number of a map from any space ''to'' a contractible space is <math>1</math>.
* [[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.

Lefschetz number - Revision history

Vipul: Created page with "==Definition== ===For a continuous map between spaces with finitely generated homology=== Suppose X and Y are topological spaces, and each of them..."

Vipul: Created page with "==Definition== ===For a continuous map between spaces with finitely generated homology=== Suppose $X$ and $Y$ are topological spaces, and each of them..."