$\varepsilon$-retracts, $Q$-manifolds, and fixed points

A generalization of one of the Noguchi fixed point theorems is presented. We prove that there exists a compact noncollapsible acyclic $Q$-manifold with the fixed point property. A topological space with the fixed point $\sigma$-property is introduced and studied and an example of a noncompact set in $R^2$ with the fixed point property is given.