Surgery obstruction groups

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1 Introduction

The surgery obstruction groups L_n(\bZ\pi, w) of C.T.C. Wall [Wall1999], [Wall1976] contain the obstructions to doing surgery on a degree 1 normal map (f,b)\colon M \to X to obtain a homotopy equivalence. In this setting, X is an n-dimensional Poincaré complex \pi = \pi_1(X, x_0) is the fundamental group of X, and w is the orientation character of X, which is determined by the first Stiefel-Whitney class of X, w_1(X).

A homotopy equivalence f\colon M \to X has a Whitehead torsion \tau(f) \in Wh(\pi) in the Whitehead group of \pi. The surgery obstruction groups for surgery up to a homotopy equivalence with torsion in a prescribed subgroup U \subseteq Wh(\pi) are denoted L^U_n(\bZ, w). The most important cases for geometric applications are U = \{ 0 \}, denoted L^s_n(\bZ, w), or U = Wh(\pi), denoted L^h_n(\bZ, w). The main problems are the following:

  1. to develop methods for the computation of the surgery obstruction groups,
  2. to define invariants of degree 1 normal maps which detect the surgery obstruction.

The surgery obstruction groups L_n(R\pi, w) depend on a coefficient ring R, a discrete group \pi and an orientation character w\colon \pi \to \{\pm 1\}. In general the surgery obstruction groups are abelian groups. For finite groups \pi the L-groups are finitely-generated and the only torsion is 2-primary.

A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-4

On the classification of hermitian forms: VI Group Rings, Wall (1976), pp. 1-2

2 References

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