I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing Homology/Cohomology (oh, let's compute the homology of the Klein Bottle !).
I'm looking for interesting "toys spaces" for training myself on compute algebraic invariant (homology groups, cohomology, verify general theorems etc...) and why not for motivation for further theory. For example, there are singular varieties which does not verifiy Poincare Duality, in my sense it's a good motivation for introduce a more general homology (here intersection homology). It's of course an "algebraic-topology" question, but if there is link with other areas in mathematics algebraic geometry or complex geometry I'm very interested also.
Thanks in advance!