Surgery obstruction groups
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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.
- [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