TITLE: Does there exist any non-contractible manifold with fixed point property?
QUESTION [30 upvotes]: Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the quasi circle on $\mathbb R^2$ is an example. Yet I've not written down the (dirty) proof. But in this case all its homotpy groups are trivial. So if I assume my space as a manifold, then (QUESTION:) does this fixed point property force it to become a contractible manifold? I read somewhere that there exists a contractible compact manifold which does not satisfy this fixed point property. So does there exist any non-contractible manifold (compact) where this property follows? Or otherwise can anyone please provide an outline of how to prove that such a manifold is contractible?

REPLY [5 votes]: Another nice example is $\mathbb RP^{2n}$. Here is a simple proof...
If $f:\mathbb RP^{2n} \to \mathbb RP^{2n}$ is a continuous function then look at the lift of the map $\bar{f} :S^{2n}\to S^{2n}$. A fixed point of $f$ is equivalent to a point $x\in S^{2n}$ s.t. $\bar{f}(x)=x \ or \ -x$. If such a point doesn't exist, then we can have a homotopy between Identitity and Antipodal map via $H(x,t)=cos(t)x+sin(t)\bar{f}(x)$.[This is well defined map from sphere to sphere]. But for even dimensional sphere antipodal map has degree $-1$. That's a contradiction.