Surgery obstruction groups

Revision as of 16:02, 8 June 2010

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 Poincar\'e complex $\pi = \pi_1(X, x_0)$ is the fundamental group of $X$, and $w = w_1(X)$ is the first Stiefel-Whitney class. The 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 $-primary. [[Media:Filename.pdf|A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-3]] [[Media:wall_intro.pdf|On the classification of hermitian forms: VI, Wall (1976)]] </wikitex> == References == {{#RefList:}} [[Category:Theory]]L_n(\bZ\pi, w)$ 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 Poincar\'e complex $\pi = \pi_1(X, x_0)$ is the fundamental group of $X$, and $w = w_1(X)$ is the first Stiefel-Whitney class.

The 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

[[Media:wall_intro.pdf|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