Homology groups and fundamental group need not determine homotopy groups
It is possible to have two path-connected spaces and such that:
- , i.e., and have isomorphic Fundamental group (?)s.
- for each , i.e., and have isomorphic Homology group (?)s in each homology.
- There exists some such that the Homotopy group (?)s and are not isomorphic.
In fact, we can choose both and to be Simply connected space (?)s, i.e., they both have trivial fundamental group.
- Fundamental group of wedge sum relative to basepoints with neighborhoods that deformation retract to them is free product of fundamental groups
Also, they are both simply connected spaces ( is simply connected by Fact (1), for , see homology of complex projective space).
However, they do not have the same isomorphism class of : is nontrivial whereas is trivial. For this, note that:
- There is a retraction that sends all points in the piece to themselves and all points in the piece to the point of wedging. This retraction induces a retraction on each homotopy group, so is a retraction. In particular, since is nontrivial, is also nontrivial.
- is the trivial group, based on the homotopy of complex projective space.