Homotopy groups of noncommutative spaces In the approach to noncommutative geometry of Alain Connes any Hausdorff compact space $X$ is replaced by its algebra of complex valued continuous functions $C^0(X)$, and one regard general (that is, possibly noncommutative) $C^*$-algebras as algebras of "functions" over hypotetical noncommutative spaces.
My questions is: Is there a sensible way to define the homotopy groups of a  noncommutative space (which recovers, in particular, the classical homotopy groups of a space $X$ purely in terms of the corresponding algebra $C^0(X)$)?. 
 A: If one thinks of homotopy groups built from the $C^*$ analogue of maps $\mathbb S^1\to X$, as said above, one can develop a theory. But such theory will be empty outside of abelian $C^*$--algebras. Consider, as an example, that $X$ may have no classical points and therefore a theory of loops with fixed endpoint will be not possible.
There are some attempts to define homotopy of NC $C^*$-algebras, basically inspired by methods developed in algebraic geometry (where rather than thinking of loops you think about coverings). The first such example I could keep track of is introduced here:
 Čerin, Z. - Homotopy groups for C∗-algebras, Topology with applications, 29-45, Bolyai Soc. Math. Stud., 4, János Bolyai Math. Soc., Budapest, 1995.
and commented here: Agrigoroiaei
On the web you can also find some preprints of a different attempt, e.g. here
The fact that such attempts haven't had much followers may indicate they are (still) not quite successfull. 
If such a generalization exists it should be quite nontrivial. The quantum torus, having no classical points and a classical analogue with nontrivial homotopy, should be the first test case to be analyzed.
