# Analogue of homotopy groups: mapping from tori instead of spheres

1. Apr 17, 2013

### nonequilibrium

So the question is simple: we have the homotopic groups of a topological space X, $\pi_n(X)$, which are based on maps $S^n \to X$. What I am wondering is: are there analogous groups $\tau_n(X)$ which is based on $T^n \to X$, 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 $\tau_n(X)$ 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 $\mathbb R^3 \backslash \{0\}$, and look at $T^2 \to \mathbb R^3 \backslash \{0\}$, then I would think that $\tau_n(\mathbb R^3 \backslash \{0\}) = \mathbb Z$, and I think the group operation works out.

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