Cone and space homeomorphism question

Question

I am wondering:

Under what conditions is a space homeomorphic to its cone? Moreover, if $M$ is a compact manifold, is it always homeomorphic to its cone? What about if for manifolds of different dimensions? That is, does there exists a dimension where $M^n$ is homeomorphic to its cone?

Answer 1

There are certainly spaces where $CX \approx X$, but manifolds do never have this property.

Let $M$ be manifold of dimension $n$. Then $CM$ contains an open subspace homeomorphic to $M \times (0,1)$ (simply remove the tip and the base $X \times \{0\}$ from $CX$). If $CM \approx M$, then $M$ would contain an open subspace homeomorphic to $M \times (0,1)$ i.e. a manifold of dimension $n+1$. This is impossible.

A positive example is the Hilbert cube $Q$ (the countable infinite product of closed unit intervals). We have $CQ \approx Q$; see here.