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I have heard that some issues in group theory prevent classifying all manifolds upto homotopy using the fundamental group invariant. Does anyone know what are those issues?

Thanks, K.

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    Your question is not quite interesting because the fundamental group focuses on curves - it does not really reflect on 2-cells, 3-cells, etc, as shown in the case where S^2 and a point are both simply connected. A more interesting question would be whether two non-homotopy equivalent spaces can have identical homotopy groups. See http://mathoverflow.net/questions/3540/are-there-two-non-homotopy-equivalent-spaces-with-equal-homotopy-groups2010-08-09
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    I think that the OP is alluding to the insolvability of the word problem, and the impact that this has on the feasibility of giving an algorithmic classification of higher dimensional manifolds. See my answer below for more details.2010-08-09
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    I also read this in the way Soarer does, maybe the OP could clarify. However Matt E makes a good and interesting point.2010-08-10

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