Surgery obstruction groups
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− | The surgery obstruction groups $L_n(\bZ\pi, w)$ of C.T.C. Wall \cite{ | + | The surgery obstruction groups $L_n(\bZ\pi, w)$ of C.T.C. Wall \cite{Wall1998}, \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. | 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. |
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1 Introduction
The surgery obstruction groups of C.T.C. Wall [Wall1998], [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