# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Weinberger, The Topological Classification of Stratified Spaces (1994) · Zbl 0826.57001 [2] Hambleton, Mat. Sbornik (in Russian) 183 pp 3– (1992) [3] Switzer, Grund. Math. Wiss. 212 (1975) [4] DOI: 10.1112/plms/s3-56.2.349 · Zbl 0665.57026 · doi:10.1112/plms/s3-56.2.349 [5] DOI: 10.1016/0022-4049(87)90057-0 · Zbl 0638.18003 · doi:10.1016/0022-4049(87)90057-0 [6] DOI: 10.1007/BFb0061900 · doi:10.1007/BFb0061900 [7] DOI: 10.1007/BFb0088095 · doi:10.1007/BFb0088095 [8] Browder, A surgery theory for G-manifolds and stratified spaces pp 27– (1975) [9] DOI: 10.1090/S0002-9904-1967-11700-2 · Zbl 0156.21903 · doi:10.1090/S0002-9904-1967-11700-2 [10] Bak, Mat. Sbornik (in Russian) 197 pp 3– (2006) · doi:10.4213/sm1568 [11] Bak, Sovrem. Mat. Prilozh. No. 1, Topol., Anal. Smezh. Vopr. (in Russian) none pp 3– (2003) [12] Ranicki, Canad. J. Math. 39 pp 245– (1987) · Zbl 0635.57017 · doi:10.4153/CJM-1987-017-x [13] Ranicki, Exact Sequences in the Algebraic Theory of Surgery (1981) · Zbl 0471.57012 [14] DOI: 10.1007/BFb0088091 · doi:10.1007/BFb0088091 [15] Muranov, Trudy MMO (in Russian) 67 pp 294– (2006) [16] Muranov, Matem. Zametki (in Russian) 79 pp 420– (2006) · doi:10.4213/mzm2711 [17] DOI: 10.1070/SM2003v194n08ABEH000764 · Zbl 1067.57032 · doi:10.1070/SM2003v194n08ABEH000764 [18] Muranov, Trudy MIRAN (in Russian) 212 pp 123– (1996) [19] de Medrano, Involutions on manifolds (1971) · doi:10.1007/978-3-642-65012-3 [20] Kharshiladze, Uspehi Mat. Nauk (in Russian) 42 pp 55– (1987) [21] Wall, Surgery on Compact Manifolds (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.