Surgery obstruction groups

Revision as of 00:02, 12 April 2012

1 Introduction

The surgery obstruction groups ${{Authors|Hambleton}} {{Stub}} == Introduction == <wikitex>; The surgery obstruction groups $L_n(\bZ\pi, w)$ of C.T.C. Wall \cite{Wall1999}, \cite{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 $w = w_1(X)$ is the first Stiefel-Whitney class. A homotopy equivalence $f\colon M \to X$ has a [[Wikipedia:Whitehead torsion|Whitehead torsion]] $\tau(f) \in Wh(\pi)$. 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: # to develop methods for the computation of the surgery obstruction groups, # to define invariants of degree 1 normal maps which detect the surgery obstruction. The surgery obstuctions 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 $-primary. [[Media:Filename.pdf|A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-3]] [[Media:wall_intro.pdf|On the classification of hermitian forms: VI Group Rings, Wall (1976), pp. 1-2]] </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Surgery]]L_n(\bZ\pi, w)$ 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 $w = w_1(X)$ is the first Stiefel-Whitney class. A homotopy equivalence $f\colon M \to X$ has a Whitehead torsion $\tau(f) \in Wh(\pi)$. 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 obstuctions 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. 1-3

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

2 References

• [Wall1976] C. T. C. Wall, Classification of Hermitian Forms. VI. Group rings, Ann. of Math. (2) 103 (1976), no.1, 1–80. MR0432737 (55 #5720) Zbl 0328.18006
• [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003