I am new to fundamental group.

I was reading Munkres and found that need of fundamental group was to distinguish between non-homeomorphic topological spaces.

So my question is, does fundamental group distinguish between any two non-homeomorphic topological space?

Or there exist some spaces which are non-homeomorphic but their fundamental groups are same?

My intution says it's a successful tool to distinguish between them.

Thanks in advance.

  • $\begingroup$ Any two homeomorphic topological spaces have the same fundamental group (and indeed the same homotopy groups), but homotopy groups alone (let alone just the fundamental group) cannot disinguish any two nonhomeomorphic spaces. $\endgroup$ – florence Jul 3 '16 at 12:40

The fundamental group does not, in fact, distinguish spaces up to homeomorphism.

For a simple example of this, each of the following spaces have trivial fundamental group, yet no two are homeomorphic:

  • $\begingroup$ So, is there anything in advanced topology to distinguish between them ? $\endgroup$ – User Jul 3 '16 at 12:40
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    $\begingroup$ @Human There are many other invariants. For instance, the higher homotopy group $\pi_2$ can distinguish the $2$-sphere from the others. Then there are invariant properties like compactness, connectedness, etc. More complicated arguments are also possible: the real line becomes disconnected when any single point is removed, which distinguishes it from the plane; and the long line does not have a countable dense subset, which distinguishes it from the others. But topological spaces are "wild": there isn't a simple invariant (or set of invariants) which characterizes them up to homeomorphism. $\endgroup$ – Noah Schweber Jul 3 '16 at 12:42
  • $\begingroup$ Okie, so there exist higher homotopy groups to distinguish between the spaces that can't be distinguished using first homotopic group. $\endgroup$ – User Jul 3 '16 at 12:44
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    $\begingroup$ @Human And there are even spaces which are not homeomorphic, but agree on all their homotopy groups. $\endgroup$ – Noah Schweber Jul 3 '16 at 12:44
  • 5
    $\begingroup$ @Human Well, if we did, how would we know? :) $\endgroup$ – Noah Schweber Jul 3 '16 at 12:49

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