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I am currently self studying algebraic topology from the book "topology and groupoids".

I don't understand classifying spaces up to homotopy type. By "I don't understand" I don't mean that I don't understand the proof of facts about homotopy types that are in my book. These facts follow easily from the definitions and I did not face any trouble in proving them. By not understanding, I mean that I don't realize why this is being done.

I know that homeomorphic spaces are of the same homotopy type. So it seems to me that homotopy type is some sort of a topological invariant. However, it is not clear to if determining whether 2 spaces are of the same homotopy type is really easier than determining if the 2 spaces are homeomorphic. Determining whether 2 spaces are both connected seems an easier problem to me than determining if they are homeomorphic.

I think I will be much more satisfied, if someone can show me an example of showing that 2 spaces are not homeomorphic by showing that they are not of the same homotopy type.

Thanks for your time.

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Most of the invariants of topological spaces algebraic topologists have managed to construct are invariant not only under homeomorphism but also under homotopy equivaence: this is true for the fundamental group, homology, cohomology, $K$-theory, and so on. This means that they fail to distinguish two spaces which are of the same homotopy type.

A classic example of this is the problem of distinguising the three dimensional lens spaces It is not too difficult (using cohomology) to see which pairs cannot be homotopy equivalent, but it is considerably more difficult to see which pairs are not homeomorphic.

The most celebrated example, though, is that of spheres: the Poincaré conjecture is that a $3$-dimensional manifold which has the homotopy type of a sphere is homeomorphic to a sphere, and this was famously solved in the affirmative by Perelman and Hamilton a few years ago.

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  • $\begingroup$ I don't know how popular the fundamental groupoid is. Do you have any idea if spaces of the same homotopy type have isomorphic fundamental groupoids ? $\endgroup$ – Amr Apr 28 '13 at 21:24
  • $\begingroup$ No they don't. The groupoid of a point is very very small, and that of $\mathbb R^n$ is very big, yet those two spaces are of the same homotopy type. It follows that the isomorphism type of the fundamental grupoid is not an homotopy invariant. The problem is that isomorphism of grupoids is no the correct notion here: if you look in any textbook which deals with the fundamental grupoid you will find what the correct, useful statement is. $\endgroup$ – Mariano Suárez-Álvarez Apr 28 '13 at 21:27
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    $\begingroup$ If you knew what that is, why did you ask the question in your first comment? $\endgroup$ – Mariano Suárez-Álvarez Apr 28 '13 at 21:29
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    $\begingroup$ «why do we want our algebraic invariant to fail to distinguish between spaces of the same homotopy type» I don't understand where you got that from. If you insist in comparing grupoids by isomorphism, then the fundamental groupoid does distinguish spaces which are of the same homotopy type (and becomes more or less useless, really...); the correct notion of homotopy equivaence of grupoids is the correct one because it allows us to do useful things —it also happens to not distinguish homotopy equivalent spaces. $\endgroup$ – Mariano Suárez-Álvarez Apr 28 '13 at 21:40
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    $\begingroup$ The problem with insisting in isomorphism of grupoids is that it is not a useful notion: this can only be seen in practice. Just like isomorphism of categories, which is a pretty useless notion for the most part. $\endgroup$ – Mariano Suárez-Álvarez Apr 28 '13 at 21:54

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