This short monograph offers a new vista of geometric topology from the standpoint of operator algebras. The common topic are functors from topological and geometric spaces to a category of operator algebras; such functors help to solve open problems in topology which are out of reach otherwise. Chapter 1 deals with new invariants of three-dimensional manifolds obtained from operator algebras. Chapter 2 links homology invariants of three-dimensional manifolds with the K-theory of Cuntz-Krieger algebras. Chapter 3 settles the Harvey conjecture for the mapping class group using ideas and methods of the noncommutative geometry.