The fixed point property in every weak homotopy type

TL;DR

This paper proves that every weak homotopy type of a connected compact CW-complex can be represented by a non-Hausdorff space with the fixed point property, extending fixed point results beyond classical settings.

Contribution

It constructs non-Hausdorff spaces with the fixed point property that are weak homotopy equivalent to any given connected compact CW-complex, showing such spaces exist beyond polyhedra.

Findings

Existence of non-Hausdorff spaces with fixed point property for any weak homotopy type

Construction of spaces with finitely many points having the fixed point property

Counterexample showing the result does not hold for polyhedra

Abstract

The Brouwer fixed point theorem states that the disk $D^n$ has the fixed point property. More generally, by the Lefschetz fixed point theorem any compact ANR with trivial rational homology has the fixed point property. In this note we prove that for any connected compact CW-complex $K$ there exists a space $X$ weak homotopy equivalent to $K$ which has the fixed point property. The result is known to be false if we require $X$ to be a polyhedron. The space $X$ we construct is a non-Hausdorff space with finitely many points.