Fixed-point property

Definition
A topological space is said to have the fixed-point property if every continuous map (not necessarily a self-homeomorphism) from the topological space to itself has a fixed point.

Stronger properties

 * acyclic compact polyhedron (nonempty)
 * rationally acyclic compact polyhedron (nonempty)

Fixed-point theorems

 * Lefschetz fixed-point theorem
 * Brouwer fixed-point theorem

In general, we combine the Lefschetz fixed-point theorem with the structure of the cohomology ring of the space to determine whether or not it has the fixed-point property. For instance, we can show that complex projective space in even dimensions has the fixed-point property, by combining the Lefschetz fixed-point theorem with the fact that the trace on the $$(2k)^{th}$$ homology is $$d^k$$ where $$d$$ is the trace on the second homology.

Products
The product of any topological space that does not satisfy the fixed-point property with any nonempty topological space gives a space that does not satisfy the fixed-point property.

Metaproperties
Every retract of a space with the fixed-point property also has the fixed-point property.