Spaces homeomorphic to the Hilbert cube are extremely common, e.g., all compact, convex, infinite-dimensional subsets of locally convex metrizable
topological vector spaces. A consequence of a celebrated theorem of PA Smith is that the fixed point set F of an involution of the Hilbert cube must be acyclic in Czech homology with Z/2Z coefficients. Using finite-dimensional work of Lowell Jones, Jan van Mill and I have the

Theorem: If an Absolute Neighborhood Retract X is acyclic in Czech homology with Z/2Z coefficients, then X is the fixed point set of an involution of the Hilbert cube.