ON EXOTIC HOMOTOPY EQUIVALENCES OF 3-MANIFOLDS

Abstract

This chapter provides an overview on exotic homotopy equivalences of 3-manifolds. A homotopy equivalence between 3-manifolds will be called exotic if it cannot be deformed into a homeomorphism. It is well-known that such homotopy equivalences exist between 3-manifolds with non-empty boundaries. A theorem is stated in the chapter that can be considered as a classification of exotic homotopy equivalences between 3-manifolds. This chapter presents the study of exotic homotopy equivalences between 3-manifolds that is with those homotopy equivalences that cannot be deformed into a homeomorphism. In such a case, 3-manifold is always compact, orientable, and irreducible. As Waldhausen has shown that there are no exotic homotopy equivalences between closed 3-manifolds that are sufficiently large, in the chapter only 3-manifolds with non-empty boundary is considered.