The Geometric Surgery Obstruction Group and Surgery Obstruction

This chapter is the equivalent to Chapter 9 inWall’s book [414].We give a geometric approach to the L-groups and the surgery obstruction based on bordism theory in Section 13.4. We prove that this geometric surgery obstruction is a normal bordism with cylindrical ends invariant and vanishes in high dimensions if and only if we can transform a normal map relative boundary to a (simple) homotopy equivalence by surgery on the interior, see Theorem 13.45. This illuminating geometric approach will only require the notion of surgery kernels, but not the notions of quadratic forms and algebraic L-groups.