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Splitting along a submanifold pair

Published online by Cambridge University Press:  04 March 2008

Rolando Jimenez ,
Yu. V. Muranov  and
Dušan Repovš
Rolando Jimenez
Affiliation:
[email protected] de Matematicas, UNAM, Avenida Universidad S/N, Col. Lomas de Chamilpa, 62210 Cuernavaca, Morelos, Mexico
Yu. V. Muranov
Affiliation:
[email protected] Tecnologica de la Mixteca, k.m. 2.5 Carretera Huajuapan-Acatlima, 69000 Huajuapan de Leon, Oaxaca, Mexico
Dušan Repovš
Affiliation:
[email protected] for Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia
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Abstract

The paper introduces a group LSP of obstructions to splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the LSP-groups with various surgery obstruction groups for a manifold triple and structure sets arising from a manifold triple. The natural map from the surgery obstruction group of the ambient manifold to the LSP-group provides an invariant when elements of the Wall group are not realized by normal maps of closed manifolds. Some LSP-groups are computed precisely.

Keywords

Splitting along submanifoldsurgery of manifoldssurgery and splitting obstruction groupssurgery exact sequenceBrowder-Livesay invariantstructure sets
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Special Issue 2: In Memory of Yurii Petrovich Solovyev October 8, 1944 – September 11, 2003 , October 2008 , pp. 385 - 404
Copyright
Copyright © ISOPP 2008

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References

1.Bak, A., Muranov, Yu.V., Splitting along submanifolds, and -spectra. Sovrem. Mat. Prilozh. No. 1, Topol., Anal. Smezh. Vopr. (in Russian) (2003), 3–18; English transl. in J. Math. Sci. (N. Y.) 123 (2004), 4169–4184Google Scholar
2.Bak, A., Muranov, Yu.V., Normal invariants of manifold pairs and assembly maps. Mat. Sbornik (in Russian) 197 (2006), 3–24; English transl. in Sbornik Math 197:6, 791–811Google Scholar
3.Browder, W., Livesay, G.R., Fixed point free involutions on homotopy spheres. Bull. Amer. Math. Soc. 73 (1967), 242–245CrossRefGoogle Scholar
4.Browder, W., Quinn, F., A surgery theory for G-manifolds and stratified spaces. in Manifolds, 1 Univ. of Tokyo Press (1975), 27–36Google Scholar
5.Cappell, S.E., Shaneson, J.L., Pseudo-free actions. I. Lecture Notes in Math. 763 (1979), 395–447CrossRefGoogle Scholar
6.Hambleton, I., Projective surgery obstructions on closed manifolds. Lecture Notes in Math. 967 (1982), 101–131CrossRefGoogle Scholar
7.Hambleton, I., Pedersen, E., Topological equivalences of linear representations for cyclic groups. Preprint MPI (1997)Google Scholar
8.Hambleton, I., Ranicki, A., and Taylor, L., Round L-theory. J. Pure Appl. Algebra 47 (1987), 131–154CrossRefGoogle Scholar
9.Hambleton, I., Milgram, J., Taylor, L., and Williams, B., Surgery with finite fundamental group. Proc. London Mat. Soc. 56 (1988), 349–379CrossRefGoogle Scholar
10.Hambleton, I., Kharshiladze, A. F.A spectral sequence in surgery theory. Mat. Sbornik (in Russian) 183 (1992), 3–14; English transl. in Russian Acad. Sci. Sb. Math. 77 (1994); 1–9Google Scholar
11.Kharshiladze, A.F., Surgery on manifolds with finite fundamental groups. Uspehi Mat. Nauk (in Russian) 42 (1987), 55–85; English transl. in Russian Math. Surveys 42 (1987), 65–103Google Scholar
12.de Medrano, S. LopezInvolutions on manifolds. Springer-Verlag: Berlin-Heidelberg-New York. 1971CrossRefGoogle Scholar
13.Muranov, Yu.V., Splitting problem. Trudy MIRAN (in Russian) 212 (1996), 123–146; English transl. in Proc. of the Steklov Inst. of Math. 212 (1996), 115–137Google Scholar
14.Muranov, Yu. V., Repovš, D., Spaggiari, F., Surgery on triples of manifolds. Mat. Sbornik (in Russian) 194 (2003), 1251–1271; English transl. in Sbornik Math. 194 (2003), 1251–1271CrossRefGoogle Scholar
15.Muranov, Yu. V., Jimenez, Rolando, Transfer maps for triples of manifolds. Matem. Zametki (in Russian) 79 (2006), 420–433; English transl. in Math. Notes 79 (2006), 387–398Google Scholar
16.Muranov, Yu. V., Repovš, D., Jimenez, Rolando, Surgery spectral sequence and manifolds with filtration. Trudy MMO (in Russian) 67 (2006), 294–325Google Scholar
17.Ranicki, A.A., The total surgery obstruction. Lecture Notes in Math. 763 (1979), 275–316CrossRefGoogle Scholar
18.Ranicki, A.A., Exact Sequences in the Algebraic Theory of Surgery. Math. Notes 26, Princeton Univ. Press, Princeton, N. J. 1981Google Scholar
19.Ranicki, A.A., The L-theory of twisted quadratic extensions. Canad. J. Math. 39 (1987), 245–364CrossRefGoogle Scholar
20.Switzer, R., Algebraic Topology–Homotopy and Homology. Grund. Math. Wiss. 212, Springer–Verlag: Berlin–Heidelberg–New York. 1975Google Scholar
21.Wall, C.T.C., Surgery on Compact Manifolds. Academic Press:London – New York. 1970; Second Edition, Ranicki, A. A. Editor, Amer. Math. Soc., Providence, R.I. 1999Google Scholar
22.Weinberger, S., The Topological Classification of Stratified Spaces. The University of Chicago Press: Chicago and London. 1994Google Scholar