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Let $X$ be a path connected topological space. Is it always true that any two homeomorphisms $f$ and $g$ from $X$ to itself are homotopic? If not, is there a minimal condition on $X$ which guarantees that this will be the case for all possible $f$ and $g$?

I know that this is true for a contractible space (indeed, and two maps on such a space are homotopic). However, if it is not true in general, I want to determine a necessary and sufficient condition for this property to hold.

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up vote 4 down vote accepted

It's not true in general. For example, consider a torus $S^1\times S^1$. The two homeomorphisms to be considered are the identity map and the map which swaps the factors.

The key point is that homotopic maps induce the same map on fundamental group, but these two maps don't. The fundamental group of a torus is $\mathbb{Z}\times\mathbb{Z}$ and the first map will act as the identity while the second will swap the two $\mathbb{Z}$ factors.

I'll think a bit about necessary and sufficient conditions, but I think this question may be quite hard.

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For anyone who sees this later, it turns out that finding examples of noncontractible manifolds where every homeomorphism is homotopic to the identity is hard. For example, among all closed surfaces, only $\mathbb{R}P^2$ has this property (it has the stronger property that the homotopy can be chosen so that each times gives a homeomorphism). – Jason DeVito Oct 24 '11 at 18:18

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