Gene asked: <<

Can we say that two compact n-dimensional manifolds M1 and M2 are homotopy equivalent if for all k, 1<=k<=n the homotopy groups pi[k](M1) and pi[k](M2) are isomorphic?

Dylan answered: << No, that's not true. What is true is that if there is a map between M1 and M2 which induces the identity on all homotopy groups (not just in this range) then M1 and M2 are homotopy equivalent.

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A simple example of spaces that have the same homotopy groups but are not homotopy equivalent is P^k x S^m and P^m x S^k, where say m > k > 1. (Note to Dylan: The condition you mention -- usually called "weak homotopy equivalence" -- is the same as homotopy equivalence for "nice" spaces like manifolds, and more generally simplicial or CW complexes, but there are stranger spaces that can be weakly homotopy equivalent without being homotopy equivalent.) --Dan --Dan