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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 $C^0(X)$)?.

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    $\begingroup$ Homotopy groups are defined in terms of maps and homotopies all of which only involve compact Hausdorff spaces, so you can straightforwardly rewrite all the definitions in terms of C*-algebras by dualizing all of the maps. Unfortunately this notion of homotopy group only sees the abelianization of a C*-algebra. But it does recover the ordinary homotopy groups. $\endgroup$ Apr 15, 2017 at 6:27
  • $\begingroup$ A standard invariant for C*-algebras is (are) homotopy groups of the unitary group (or of their matrix rings). The first ($\pi_1$) one is closely related to K$_0$ (of the C*-algebra). $\endgroup$ Apr 15, 2017 at 18:13

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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.

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