An example concerning equivariant deformations

Abstract

We give an example of $Z_2$-space $X$ with a property that the identity map ${\rm id}_X:X\to X$ as well as its restriction to the fixed point set of the group action ${\rm id}^{Z_2}:X^{Z_2}\to X^{Z_2}$ are deformable to fixed point free maps whereas there is no fixed point free map in the equivariant homotopy class of the identity $[{\rm id}_X]_{Z_2}$.