From Algebraic to Geometry Surgery

Theorem 1. Let X be a Poincare pair of dimension n ≥ 5, ζ a stable PL bundle on X, and f : M → X a degree one normal map, where M is a PL manifold. Let σ f ∈ Ω∞Lvq(X, ζX) be the relative signature of f , and suppose we are given a path p from σ f to the base point of Ω ∞Lvq(X, ζX). (We can identify such a path with a Lagrangian in the Poincare object representing σ f , which is well-defined up to bordism). Then there exists a ∆-family of degree one normal maps F : B → X × ∆, where B is a bordism from M = F−1(X × {0}) to a PL manifold N = F−1(X × {1}) such that F induces a homotopy equivalence f ′ : N → X. Moreover, we can arrange that F determines a path from σ f to σ vq f ′ = 0 which is homotopic to p.