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December 11th, 2009, 01:32 AM   #1
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path-connected

If you have a path connected space & a path connected subset, does the inclusion map between them induce an injective map between their fundamental groups?

Obviously I realise that for the induced group homomorphism we have injectivity iff ker = 0 and that the indentity is that class of all path homotopic to a constant path. But I just have no idea where to go from here
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December 11th, 2009, 03:55 PM   #2
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Re: path-connected

No. Take S^1 in R^2. Any homomorphism to the fundamental group of R^2 must be trivial, and the fundamental group of S^1 is Z.

Edit: Or perhaps an example that ties in a little better to other theory would be taking D^2 instead of R^2, with S^1 being the boundary. You may be interested in looking at some results about the way S^1 and D^2 interact. I would assume these would be in whatever book(s) you're working out of.
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