# The total surgery obstruction

# The total surgery obstruction

The 1960's Browder-Novikov-Sullivan-Wall high-dimensional surgery theory for deciding if an n-dimensional Poincare duality space X is homotopy equivalent to an n-dimensional topological manifold has two obstructions. There is a primary topological K-theory obstruction to the existence of a topological bundle structure on the Spivak spherical fibration \nu{X \subset S^{n+k}}. There is also a secondary algebraic L-theory surgery obstruction in the Wall group L_n(Z[\pi_1(X)]) of quadratic forms over the fundamental group ring Z[\pi_1(X)], which depends on the resolution of the primary obstruction. In 1979 (in Princeton) I united these two obstructions in a single "total surgery obstruction" s(X) \in S_n(X). This homotopy invariant lives in an even more generalized Witt group S_n(X) defined for any space X, such that s(X)=0 if (and for n>4 only if) X is homotopy equivalent to a manifold. The object of the talk is to describe the construction of s(X) from the stable homotopy class \rho \in \pi_{n+k}(T(\nu_X)) of the Pontrjagin-Thom map \rho:S^{n+k} \to T(\nu_X) to the Thom space. The description involves the algebraic surgery theory analogue of the homotopy groups of a Thom space, for any spherical fibration.