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Analogue of homotopy groups: mapping from tori instead of spheres

  1. Apr 17, 2013 #1
    So the question is simple: we have the homotopic groups of a topological space X, [itex]\pi_n(X)[/itex], which are based on maps [itex]S^n \to X[/itex]. What I am wondering is: are there analogous groups [itex]\tau_n(X)[/itex] which is based on [itex]T^n \to X[/itex], i.e. we map tori onto our space?

    I know little about homotopy theory, so I basically googled the question and got an interesting hit: http://mathoverflow.net/questions/37792/a-possible-generalization-of-the-homotopy-groups
    If you check the upvoted answer, it says that you can define it, but [itex]\tau_n(X)[/itex] won't have a group structure, the thing is: I don't know enough homotopy theory to understand the reason for this ("counital", "cohomology", "cup-product structure").

    I have no problem accepting this statement on faith, but I tried fiddling around with a specific case and there I think we have a group structure, although I wouldn't be able to prove it: look at X being [itex]\mathbb R^3 \backslash \{0\}[/itex], and look at [itex]T^2 \to \mathbb R^3 \backslash \{0\}[/itex], then I would think that [itex]\tau_n(\mathbb R^3 \backslash \{0\}) = \mathbb Z[/itex], and I think the group operation works out.

    But so my question is twofold (an answer on any is fine!):
    - Is [itex]\tau_n(\mathbb R^3 \ \{0\}) = \mathbb Z[/itex]
    - Is it (im)possible to have a "sensible" [itex]\tau_n(X)[/itex]?
     
  2. jcsd
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