Fixed point property of spaces having same homotopy type

Question

Suppose X and Y have same homotopy type.X as a topological space has fixed point property.Can we conclude anything about fixed point property of Y?

Answer 1

No. For example, $X$ could be a point. Then you're asking whether a contractible space $Y$ has the fixed point property. This isn't true in general (although the Brouwer fixed point theorem is a weaker result along these lines): for example, $Y = \mathbb{R}$ doesn't have the fixed point property.

More generally, if $X$ is any space, then $Y = X \times \mathbb{R}$ is a homotopy equivalent space which doesn't have the fixed point property. This suggests that at the very least we should take $X$ and $Y$ to both be compact. Unfortunately, this is also not enough.

More can be said with more restrictions. For example, the Lefschetz fixed point theorem can be used to show that a compact triangulable space which is homotopy equivalent to $\mathbb{CP}^{2n}$ has the fixed point property (in fact it suffices that it have the same cohomology ring).