Surgery obstruction groups
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== Introduction == | == Introduction == | ||
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− | 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 | + | 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 [[Wikipedia:Normal_invariant|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 [[Wikipedia:Stiefel-Whitney_class|Stiefel-Whitney]] class of $X$, $w_1(X)$. |
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+ | A homotopy equivalence $f\colon M \to X$ has a [[Wikipedia:Whitehead torsion|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: | ||
# to develop methods for the computation of the surgery obstruction groups, | # to develop methods for the computation of the surgery obstruction groups, | ||
# to define invariants of degree 1 normal maps which detect the surgery obstruction. | # to define invariants of degree 1 normal maps which detect the surgery obstruction. | ||
− | The surgery | + | 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. |
− | [[Media: | + | [[Media:Hambleton&Taylor-pp-1-4.pdf|A Guide to the Calculation of Surgery Obstruction Groups, Hambleton & Taylor (2000), pp. 1-4]] |
[[Media:wall_intro.pdf|On the classification of hermitian forms: VI Group Rings, Wall (1976), pp. 1-2]] | [[Media:wall_intro.pdf|On the classification of hermitian forms: VI Group Rings, Wall (1976), pp. 1-2]] | ||
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Latest revision as of 12:04, 28 January 2013
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1 Introduction
The surgery obstruction groups of C.T.C. Wall [Wall1999], [Wall1976] contain the obstructions to doing surgery on a degree 1 normal map to obtain a homotopy equivalence. In this setting, is an -dimensional Poincaré complex is the fundamental group of , and is the orientation character of , which is determined by the first Stiefel-Whitney class of , .
A homotopy equivalence has a Whitehead torsion in the Whitehead group of . The surgery obstruction groups for surgery up to a homotopy equivalence with torsion in a prescribed subgroup are denoted . The most important cases for geometric applications are , denoted , or , denoted . 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 obstruction 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. 1-4
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