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Surgery with finite fundamental group. (English) Zbl 0665.57026

In the paper the authors introduce a new technique for evaluating the (weakly simple) surgery obstruction for degree 1 normal maps of closed manifolds. The results are applied to oriented manifolds with finite fundamental group. The main result is the following theorem.
Theorem A. Let (X,\(\partial X)\) be an n-dimensional Poincaré pair and \((h_ 0,\hat h_ 0)\) a degree 1 normal map with torsion of \(h_{| \partial}\) in \(U\subseteq WH({\mathbb{Z}}[\pi_ 1X])\). If \(\tilde Y\subseteq U\) and \(\pi_ 1X\) is finite, then for any f: X/\(\partial X\to G/TOP\), \[ \sigma^ U_ 0(f) =^{Def}\lambda^ U((h_0,\hat h_0)*f) - \lambda^ U(h_0,\hat h_0)\text{ in }L^ U_ n({\mathbb{Z}}[\pi_ 1X]) \] is equal to \[ \begin{aligned} \text{Index}(f) + \kappa^ U_2 \{c_* (ARF_2(f))\} \qquad&\text{for }n\equiv 0\pmod4, \tag{a}\\ \kappa^ U_3 \{c_* (ARF_3(f))\} \qquad&\text{for }n\equiv 1\pmod4, \tag{b}\\ ARF(f) + \kappa^ U_4 \left\{c_*\left( \sum_{r\geq 0} s_ r\left(ARF_{2^{r+2}}(f)\right)\right)\right\} \qquad&\text{for }n \equiv 2\pmod4, \tag{c}\\ \kappa^ U_1 \left\{ c_* (ARF_1(f))\right\} \qquad&\text{for }n \equiv 3\pmod4. \tag{d} \end{aligned} \] Here U is a conjugation invariant subgroup \(U\subseteq Wh({\mathbb{Z}}[\pi_ 1X])\), \((h_ 0,\hat h_ 0)\) is a chosen base point in the set of degree 1 normal maps (h: (M,\(\partial M)\to (X,\partial X)\); \(\hat h:\) \(\nu_ M\to \xi)\), and \((h_ 0,\hat h_ 0)*f\) denotes the degree 1 normal map induced by the action of an element f in [X/\(\partial X,G/TOP]\) on the base point. Furthermore, ARF(f) and Index(f) denote the change in the ordinary Arf invariant and index of the surgery problem given by f (considered as element of \(L_*({\mathbb{Z}}))\). The homomorphisms \(\kappa^ U_ j: H_ j(\pi,{\mathbb{Z}}/2)\to L^ U_{j+2}({\mathbb{Z}}\pi)_{(2)}\) are defined in Section 1 of the paper, and ARF is defined by a Rourke- Sullivan type formula. For other definitions used in Theorem A the reader is referred to the paper.
Reviewer: K.H.Dovermann

MSC:

57R67 Surgery obstructions, Wall groups
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R19 Algebraic topology on manifolds and differential topology
57R65 Surgery and handlebodies
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