Lefschetz fixed-point theorem: Difference between revisions
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* Any [[contractible space|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]] | * Any [[contractible space|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]] | ||
* More generally, every [[acyclic space|acyclic]] compact polyhedron has the fixed-point property | * More generally, every [[acyclic space|acyclic]] compact polyhedron has the fixed-point property | ||
* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic|zero]] | * The [[Euler characteristic]] of any nontrivial compact connected Lie group is [[space with zero Euler characteristic|zero]] | ||
* Any map from a sphere to itself of degree greater than 1 must have a fixed point | |||
Revision as of 22:42, 27 October 2007
Statement
If the Lefschetz number of a map from a compact polyhedron (viz a compact space that is also a polyhedron) to itself is nonzero, then the map has a fixed point.
Corollaries
- Any contractible compact polyhedron has the fixed-point property. In particular, every disc has the fixed-point property, which is Brouwer fixed-point theorem
- More generally, every acyclic compact polyhedron has the fixed-point property
- The Euler characteristic of any nontrivial compact connected Lie group is zero
- Any map from a sphere to itself of degree greater than 1 must have a fixed point