Homotopy type and the fixed simplex property

JONATHAN ARIEL BARMAK

Barranquilla

Congreso; V Congreso latinoamericano de matemáticos; 2016

It can be proved that every compact polyhedron homotopy equivalent to a sphere admits a fixed point free map. However, for dimension n greater than 1, there exist triangulations of the n-sphere with the fixed simplex property, that is, each simplicial endomorphism has a fixed point. We will prove that given any compact, connected polyhedron, there exists a simplicial complex with the same homotopy type and the fixed simplex property. As a consequence we obtain that there are finite topological spaces with the fixed point property and arbitrary weak homotopy type.