Surgery obstruction groups
(→Introduction) |
(→Introduction) |
||
Line 4: | Line 4: | ||
<wikitex>; | <wikitex>; | ||
\newcommand{\bZ}{\mathbf Z} | \newcommand{\bZ}{\mathbf Z} | ||
− | The surgery obstruction groups $L_n(\bZ\pi, w)$ of C.T.C. Wall \cite{wall-book}, \cite{wall-VI} 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 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. | + | The surgery obstruction groups $L_n(\bZ\pi, w)$ of C.T.C. Wall \cite{wall-book}, \cite{wall-VI} 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. | ||
[[Media:Filename.pdf|A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-3]] | [[Media:Filename.pdf|A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-3]] |
Revision as of 11:11, 8 June 2010
The user responsible for this page is Hambleton. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
\newcommand{\bZ}{\mathbf Z} The surgery obstruction groups of C.T.C. Wall [wall-book], [wall-VI] contain the obstructions to doing surgery on a degree 1 normal map to obtain a homotopy equivalence. In this setting, is an -dimensional Poincar\'e complex is the fundamental group of , and is the first Stiefel-Whitney class.
The groups depend on a coefficient ring , a discrete group and an orientation character . In general the surgery obstruction groups are abelian groups. For finite groups the -groups are finitely-generated and the only torsion is -primary.
A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-3