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If we define $X\subset\ell^2$ to be the unit sphere, i.e. $$X=\left\{(x_i)_{i=1}^\infty\ \Bigg|\ \sum_{i=1}^\infty x_i^2 = 1\right\}$$ then since $X$ is path connected and not compact, and since $X\setminus\{x\}$ is contractible for all $x\in X$, then a simple calculation shows that $\tilde{H}_n(X)=0$ for all $n$.

Now, I wish to show that $X$ is not contractible. Assuming we have a contraction to a point $x$, we can use a change of orthonormal basis to obtain a contraction to the point $e_1=(1,0,0,\ldots)$, i.e. a continuous map $h:X\times I\to X$ such that $h(\bullet,0)=\mathrm{id}_X$ and $h(\bullet,1)\equiv e_1$. However, I'm struggling to get a contradiction here, since I know of no stronger invariant than homology/homotopy groups, and by compactness arguments, these are all trivial.

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  • $\begingroup$ Unit sphere, not unit ball. And this is contractible. Bad luck. $\endgroup$
    – user98602
    Commented Dec 27, 2016 at 4:13
  • $\begingroup$ Huh. What's the contraction? $\endgroup$ Commented Dec 27, 2016 at 4:14
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    $\begingroup$ mathoverflow.net/a/199/40804 $\endgroup$
    – user98602
    Commented Dec 27, 2016 at 4:14
  • $\begingroup$ Ah, I see. So when can we have noncontractible spaces with zero homology groups? $\endgroup$ Commented Dec 27, 2016 at 4:17
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    $\begingroup$ The Hilbert sphere is even diffeomorphic to the Hilbert space. $\endgroup$ Commented Dec 27, 2016 at 4:19

1 Answer 1

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There are two different observations here. One is that the fundamental group can be nonzero but the homology groups all zero. (For instance, one may take the Poincare homology sphere and delete a point.) When the fundamental group is also zero, we go into the realm of Whitehead's theorem: a CW complex whose homotopy groups are all zero is contractible. By combining this with the Hurewicz theorem, we see that if the fundamental group and all homology groups are zero, a CW complex (or a space with the homotopy type thereof) is contractible. But some spaces are not, and e.g. the "Warsaw circle" (closed-up topologists since curve) has all homotopy groups, and hence homology groups, zero, but is not contractible. (It actually has nontrivial "shape", where you probe the shape by maps out instead of maps in.)

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