The total surgery obstruction

The total surgery obstruction is an invariant introduced by the speaker in 1978, uniting the 2 stages of the Browder-Novikov-Sullivan-Wall obstruction theory for the existence of a topological manifold in the homotopy type of a space X with n-dimensional Poincare duality for n>4. There is also a version for deciding if a homotopy equivalence of manifolds is homotopic to a homeomorphism. The talk will report on some recent advances relating to the invariant. One of these involves a joint project with Michael Crabb on the applications to surgery theory of a geometric Hopf invariant of a stable map.