C*-algebras are thought be be non-commutative topological spaces because of Gelfand's theorem that any commutative C*-algebra are isomorphic to C(X) for some locally compact Hausdorff space X. I've been working on C*-algebras and I saw there are many topological methods used in the study of C*-algebras. I am curious about the other direction. Could anyone give me some examples that topology problems can be solved using C*-algebra technics?
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Topology is way simpler than noncommutative geometry so you should not expect an easy application of noncommutative geometry in topology. I can give a reference which is one of the very first applications of noncommutative geometry in topology.
Alain Connes and Henry Moscovici solved the NoviKov conjecture for hyperbolic groups in the following paper. Their method uses a number of advanced techniques of noncommutative geometry beyond basics of $C^*$-algebras.
Connes, A.,Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29(3), 345–388 (1990)
All papers on the Baum-Connes conjecture (which generalizes the Novikov conjecture) can also be considered as other applications of noncommutative geometry and $C^*$-algebras in topology.