# What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? Otherwise, is there any counterexample? Moreover, what conditions should be added to homotopy to get homeomorphism?

We assume additionally both X and Y are orientable.

• Nitpick: Maps are homotopic. Spaces are homotopy equivalent.
– Neal
Commented Jan 18, 2013 at 12:27

Let $X$ be the letter

$$\ \ \ \ \ \mathsf{X}\ \ \ \ \$$ and $Y$ be the letter

$$\ \ \ \ \ \mathsf{Y}\ \ \ \ \$$

Then $X$ and $Y$ are homotopy-equivalent, but they are not homeomorphic.

Sketch proof: let $f:X\to Y$ map three of the prongs of the $\mathsf{X}$ on to the $\mathsf{Y}$ in the obvious way, and let it map the fourth prong to the point at the centre. Let $g:Y\to X$ map the $\mathsf{Y}$ into those three prongs of the $\mathsf{X}$. Then $f$ and $g$ are both continuous, and $f$ is a surjection but is not injective, while $g$ is an injection but is not surjective. Now the compositions $f\circ g$ and $g\circ f$ are both easily seen to be homotopic to the identities on $X$ and $Y$, so $X$ and $Y$ are homotopy-equivalent.

In other words, observe that $\mathsf Y$ is a deformation retract of $\mathsf X$. Alternatively, observe that $\mathsf X$ and $\mathsf Y$ both retract on to the point at the centre.

On the other hand, $X$ and $Y$ are not homeomorphic. For example, removing the point at the centre of the $\mathsf{X}$ yields a space with four connected components, while removing any point from the $\mathsf{Y}$ yields at most three connected components.

• What would be the homotopies for $f \circ g$ and $g \circ f$ and their respective identities? And how are you defining $g$ with respect to the center point? Commented Apr 13, 2017 at 19:41
• @OliverG Maybe it's easier just to define $f$ to be the map that maps the whole of $\mathsf{X}$ to the point at the centre of $\mathsf{Y}$ and to let $g$ be the map that maps the whole of $\mathsf{Y}$ to the point at the centre of $\mathsf{X}$. Then the compositions $g\circ f$ and $f\circ G$ each map the whole letter to the point at the centre, and this map is homeomorphic to the identity on the letters (i.e., $\mathsf{X}$ and $\mathsf{Y}$ are contractible). Commented Dec 6, 2017 at 11:48
• Bonus points for using the letter X for the geometric shape X and the letter Y for the geometric shape Y... genius! Commented May 28, 2022 at 19:58

When you say $X$ and $Y$ are homotopic, I assume you mean that they are homotopy equivalent. Anyways, homotopy equivalence is weaker than homeomorphic.

Counterexample to your claim: the 2-dimensional cylinder and a Möbius strip are both 2-dimensional manifolds and homotopy equivalent, but not homeomorphic.

Unfortunately I'm not an expert on the subject so I'm not sure what are the weakest assumptions to add to homotopy to get a homeomorphism.

• The counterexample is right. While what about if we assume additionally both X and Y are orientable? Commented Jan 18, 2013 at 12:28
• @liufu: technically orientiability has no meaning for arbitrary spaces. If you restrict to spaces like manifolds where such ideas are well-defined then the answer is still no. Sigur's lens space example is one. Commented Sep 11, 2013 at 22:46

This is the content of certain rigidity theorems.

You should check out the Mostow rigidity theorem. It implies, that given two smooth closed manifolds which are homotopy equivalent and both hyperbolic (constant sectional curvature = -1) then they are diffeomorphic (see http://en.wikipedia.org/wiki/Mostow_rigidity_theorem ).

Note that by the Cartan-Hadamard theorem it follows that a hyperbolic manifold is what is called aspherical, i.e. its universal cover is contractible.

There is a very beautiful conjecture due to Borel (the Borel Conjecture) that can be phrased as follows.

Let $f: M \to N$ be a homotopy equivalence of closed aspherical manifolds. Then $f$ is homotopic to a homeomorphism, and in particular $M$ and $N$ are homeomorphic.

Note that this conjecture assumes less about the manifolds (every hyperbolic manifold is aspherical, but not every aspherical manifold is hyperbolic) but you also get a weaker conclusion (the manifolds are homeomorphic not diffeomorphic or isometric).

• For hyperbolic manifolds (of finite volume), Mostow rigidity is seemingly even stronger. Namely, any two such manifolds with isomorphic fundamental group are isometric. (Since hyperbolic manifolds are $K(\pi, n)s$, this isn't really stronger, but it certainly seems stronger on first notice). Commented Jan 18, 2013 at 16:56
• Yes, but I did not want to mention it in this way in order to stay close to the question :) Commented Jan 18, 2013 at 19:45

Look for 3-dimensional lens spaces of type $L(p,q)$, quotient of $S^3$ by a free orthogonal action of the cyclic group $\mathbb{Z}_p$. More precisely, look for the spaces $L(5,1), L(5,2)$ and $L(7,1), L(7,2)$.

I you don't necessarily want compact manifolds, in dimension $$3$$ and up there are manifolds which are contractible (i.e., homotopy equivalent to a point), but not homeomorphic to$$\mathbb{R}^n$$. See, for example, the Whitehead manifold.

I you insist on compact simply connected manifolds, I don't know of too many examples. For example, in Kamerich's thesis "Transitive transformation groups on products of two spheres", he proves

The homogeneous space $$(Sp(24)\times Sp(2))/(Sp(23)\times \Delta Sp(1) \times Sp(1))$$ given by the embedding $$(A,p,q)\mapsto \big(\operatorname{diag}(A,p), \operatorname{diag}(p,q)\big)$$ where $$Sp(n)$$ denotes $$n\times n$$ quaternionic unitary matrices is homotopy equivalent but not homeomorphic to $$S^{95}\times S^4$$. Further, this is best result possible in the sense that replacing $$24$$ and $$23$$ with smaller numbers always gives examples which are not homotopy equivalent to any product of spheres.

Onishchik's book "Topology of Transitive Transformation Groups" pg. 275 contains more details.

• Nice book, thanks. Commented Jan 18, 2013 at 14:03
• I just received a downvote on this question (2 years after posting it). If anyone would like to suggest an improvement, I'd love to hear it.... Commented May 19, 2015 at 16:48
• This also just happened with my answer below... strange. Commented Oct 11, 2018 at 19:11
• @JasonDeVito: I couldn't find the book you quoted its title by googling. Is there such book? Commented Jul 2, 2020 at 19:26
• @C.F.G: Sorry, it's "Topoloy of Transitive Transformation Groups". It's by Arkadi L. Onishchick. So I got the title wrong and mispelled the author's name. Editing the post now... Commented Jul 3, 2020 at 0:27

Simplest counter-example is $X = B_n(0)$ (a ball of dimension $n$ around $0$) and $Y = \{0\}$ with the contraction $H(t,x) = (1-t)x$, i.e. $H(0,x) = x$, $H(1,x) = 0$. You will not find a homeomorphism though, since the sets don't even have the same cardinality.

• This example is not valid here, since the OP asked for examples with the same dimension. Commented Jan 18, 2013 at 12:22
• that and the map isn't written correctly. Pick $H(t,x)=(1-t)x$ Commented May 18, 2015 at 17:27