# Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding topological spaces are homeomorphic. This means that all topological properties are encoded in structure of the algebra. I know already how connectedness is encoded, and now I am wondering in what way fundamental group (and more generally whole "homotopy structure") is encoded. I will be glad if someone can clarify or show me relevant literature.

With regards, Blazej

• Do you also know already how path connectedness is encoded? That would be an important precursor to how the fundamental group is encoded. – Lee Mosher Mar 27 '16 at 15:53
• No, not really. – Blazej Mar 27 '16 at 16:07