I'm trying to get a feel for why different operations on spaces are useful. I realize this question is very long if someone wants to give a response to all the cases. With ''operations on spaces'' I mean:
product, wedge product, cone, suspension, smash product, loop space etc.
My question is essentially what is their relation to the following.
- Homology & cohomology including cup product structure.
- Homotopy groups
Some of these have answers I already know of. Like the product commutes with homotopy and the Kunneth formula takes care of products of homology and cohomology. However, I don't know what, if anything, can be said about the resulting cup product structure if it's known for each individual space. The cone is also trivial, because it's contractible, but other things like the smash product I'm completely clueless about.