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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.

  1. Homology & cohomology including cup product structure.
  2. 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.

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    difficult, is that the homotopy excision theorem is pretty weak and the result depends on how connected the initial spaces are (a space is *n-connected* if $\pi_{\leq n-1}=0$). However, taking loops shifts all homotopy groups down by 1, so that's nice. Smash product certainly gives you a map $\pi_n(X)\times \pi_m(Y)\rightarrow \pi_{n+m}(X\wedge Y)$, but I don't know what there is to say about it. This should be closely related to the question of excision, though, because of how the smash product is defined.2011-04-29

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Specifically for suspension:

In K-theory you have Bott periodicity, relating $\mathrm K(X)$ to $\mathrm K(S^2X)$, where $S^2X$ denotes the twice suspended space.

Similarly, for the action of taking the loop space, you also have Bott Periodicity - see for instance this Wikipedia page.

Quite a lot of information about the actions you ask about, in the setting of K-theory, can be found in Hatcher's (unfinished) book "Vector Bundles and K-Theory", available from his webpage.

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    @Aaron: Thanks for the clarifications!2011-04-29