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Splitting along a submanifold pair. (English) Zbl 1166.57020
For two closed oriented \(n\)-dimensional topological manifolds \(M\) and \(X\), the authors introduce a group LSP of obstructions to splitting a simple homotopy equivalence \(f: M\to X\) along a pair of submanifolds \(Z\subset Y\) of \(X\). By definition, \(f\) splits along the pair \(Z\subset Y\) if it is concordant to an \(s\)-triangulation of the triple \(Z\subset Y\subset X\). The authors study exact sequences which relate the LSP groups with various classical surgery obstruction groups for a manifold triple \(Z\subset Y\subset X\) and structure sets arizing from \(Z\subset Y\subset X\). The natural map from the surgery obstruction group of the ambient manifold to the LSP group gives an invariant for determining when elements of the Wall group are not realized by normal maps of closed manifolds. The invariant is equivalent to the pair of I. Hambleton’s invariants discussed in the paper [Projective surgery obstructions on closed manifolds, Algebraic \(K\)-theory, Proc. Conf., Oberwolfach 1980, Part II, Lect. Notes Math. 967, 101–131 (1982; Zbl 0503.57018)]. The authors of the paper compute some of the LSP groups in details.

MSC:
57R67 Surgery obstructions, Wall groups
19J25 Surgery obstructions (\(K\)-theoretic aspects)
57R05 Triangulating
19G24 \(L\)-theory of group rings
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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