Homotopy groups need not determine homology groups
- For every positive integer , there is an isomorphism of groups between the Homotopy group (?)s.
- There exists a positive integer such that the Homology group (?)s and are not isomorphic.
In particular, because Weak homotopy-equivalent topological spaces (?) have isomorphic homology groups, this gives an example of spaces that have isomorphic homotopy groups but are not weak homotopy-equivalent.
Consider the spaces and . We note that:
- Both the spaces have universal cover (product of 3-sphere and 2-sphere) which is a double cover, so they both have fundamental group .
- By fact (1), the covering maps and both induce isomorphisms on . Thus, we get and and we thus get .
Thus, we see that all the homotopy groups match.
However, the homology groups do not. One simple way of seeing this is to note that is orientable, and therefore has , whereas is non-orientable, so .
The full homology descriptions are below: