Geodesic
Splitting a simple homotopy equivalence along a submanifold with filtration
Sbornik. Mathematics, Tome 199 (2008) no. 6, pp. 787-809 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A simple homotopy equivalence $f\colon M^n\to X^n$ of manifolds splits along a submanifold $Y\subset X$ if it is homotopic to a map that is a simple homotopy equivalence on the transversal preimage of the submanifold and on the complement of this preimage. The problem of splitting along a submanifold with filtration is a natural generalization of this problem. In this paper we define groups $\mathit{LSF}_*$ of obstructions to splitting along a submanifold with filtration and describe their properties. We apply the results obtained to the problem of the realization of surgery and splitting obstructions by maps of closed manifolds and consider several examples. Bibliography: 36 titles. 
@article{SM_2008_199_6_a0,
     author = {A. Bak and Yu. V. Muranov},
     title = {Splitting a~simple homotopy equivalence along a~submanifold with filtration},
     journal = {Sbornik. Mathematics},
     pages = {787--809},
     year = {2008},
     volume = {199},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_6_a0/}
}
TY  - JOUR
AU  - A. Bak
AU  - Yu. V. Muranov
TI  - Splitting a simple homotopy equivalence along a submanifold with filtration
JO  - Sbornik. Mathematics
PY  - 2008
SP  - 787
EP  - 809
VL  - 199
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2008_199_6_a0/
LA  - en
ID  - SM_2008_199_6_a0
ER  - 
%0 Journal Article
%A A. Bak
%A Yu. V. Muranov
%T Splitting a simple homotopy equivalence along a submanifold with filtration
%J Sbornik. Mathematics
%D 2008
%P 787-809
%V 199
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2008_199_6_a0/
%G en
%F SM_2008_199_6_a0
A. Bak; Yu. V. Muranov. Splitting a simple homotopy equivalence along a submanifold with filtration. Sbornik. Mathematics, Tome 199 (2008) no. 6, pp. 787-809. http://geodesic.mathdoc.fr/item/SM_2008_199_6_a0/

[1] S. P. Novikov, “Algebraic construction and properties of Hermitian analogs of $K$-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. I, II”, Math. USSR-Izv., 4:2 (1970), 257–292 | DOI | MR | Zbl | Zbl | Zbl

[2] C. T. C. Wall, Surgery on compact manifolds, London Math. Soc. Monogr. Ser., 1, Academic Press, London–New York, 1970 | MR | Zbl

[3] W. Browder, Surgery on simply-connected manifolds, Ergeb. Math. Grenzgeb., 65, Springer-Verlag, Berlin–Heidelberg–New York, 1972 | MR | Zbl

[4] A. Ranicki, “The total surgery obstruction”, Algebraic Topology, Aarhus 1978 (Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., 763, Springer-Verlag, Berlin–Heidelberg, 1979, 275–316 | DOI | MR | Zbl

[5] A. Ranicki, Exact sequences in the algebraic theory of surgery, Math. Notes, 26, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo, 1981 | MR | Zbl

[6] W. Browder, F. Quinn, “A surgery theory for $G$-manifolds and stratified sets”, Manifolds – Tokyo 1973, Proc. Internat. Conf. Manifolds (Tokyo, 1973), Univ. Tokyo Press, Tokyo, 1975, 27–36 | MR | Zbl

[7] Sh. Weinberger, The topological classification of stratified spaces, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, IL, 1994 | MR | Zbl

[8] Yu. V. Muranov, D. Repovs, R. Jimenez, “A spectral sequence in surgery theory and manifolds with filtrations”, Trans. Moscow Math. Soc., 2006, 261–288 | DOI | MR | Zbl

[9] Yu. V. Muranov, D. Repovs, F. Spaggiari, “Surgery on triples of manifolds”, Sb. Math., 194:8 (2003), 1251–1271 | DOI | MR | Zbl

[10] A. Bak, Yu. V. Muranov, “Normal invariants of manifold pairs and assembly maps”, Sb. Math., 197:6 (2006), 791–811 | DOI

[11] R. Jimenez, Yu. V. Muranov, D. Repovš, “Splitting along a submanifold pair”, $K$-Theory (to appear)

[12] W. Browder, G. R. Livesay, “Fixed point free involutions on homotopy spheres”, Bull. Amer. Math. Soc., 73 (1967), 242–245 | DOI | MR | Zbl

[13] S. López de Medrano, Involutions on manifolds, Ergeb. Math. Grenzgeb., 59, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[14] S. E. Cappell, J. L. Shaneson, “Pseudo-free actions. I”, Algebraic topology, Aarhus 1978 (Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., 763, Springer-Verlag, Berlin–Heidelberg, 1979, 395–447 | DOI | MR | Zbl

[15] I. Hambleton, “Projective surgery obstructions on closed manifolds”, Algebraic $K$-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math., 967, Springer-Verlag, Berlin–New York, 1982, 101–131 | DOI | MR | Zbl

[16] A. F. Kharshiladze, “Smooth and piecewise-linear structures on products of projective spaces”, Math. USSR-Izv., 22:2 (1984), 339–355 | DOI | MR | Zbl | Zbl

[17] Yu.,V. Muranov, “The splitting problem”, Proc. Steklov Inst. Math., 212 (1996), 115–137 | MR | Zbl | Zbl

[18] A. F. Kharshiladze, “Surgery on manifolds with finite fundamental groups”, Russian Math. Surveys, 42:4 (1987), 65–103 | DOI | MR | Zbl

[19] I. Hambleton, R. J. Milgram, L. Taylor, B. Williams, “Surgery with finite fundamental group”, Proc. London Math. Soc. (3), 56:2 (1988), 349–379 | DOI | MR | Zbl

[20] H. K. Mukerjee, “Classification of homotopy Dold manifolds”, New York J. Math., 9 (2003), 271–293 | MR | Zbl

[21] H. K. Mukerjee, “Classification of homotopy real Milnor manifolds”, Topology Appl., 139:1–3 (2004), 151–184 | DOI | MR | Zbl

[22] A. Cavicchioli, Yu. V. Muranov, F. Spaggiari, “Mixed structures on a manifold with boundary”, Glasg. Math. J., 48:1 (2006), 125–143 | DOI | MR | Zbl

[23] Yu. V. Muranov, A. F. Kharshiladze, “Browder–Livesay groups of Abelian 2-groups”, Math. USSR-Sb., 70:2 (1991), 499–540 | DOI | MR | Zbl | Zbl

[24] I. Hambleton, L. Taylor, B. Williams, “An introduction to maps between surgery obstruction groups”, Algebraic topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math., 1051, 1984, 49–127 | DOI | MR | Zbl

[25] A. Ranicki, “The $L$-theory of twisted quadratic extensions”, Canad. J. Math., 39:2 (1987), 345–364 | MR | Zbl

[26] Yu. V. Muranov, R. Himenez, “Transfer maps for triples of manifolds”, Math. Notes, 79:3–4 (2006), 387–398 | DOI | MR | Zbl

[27] A. F. Kharshiladze, “Iterated Browder–Livesay invariants and the uzing problem”, Math. Notes, 41:4 (1987), 312–315 | DOI | MR | Zbl

[28] A. Cavicchioli, Yu. V. Muranov, F. Spaggiari, On the elements of the second type in surgery groups, Preprint No 111, Max-Planck-Institut für Mathematik, Bonn, 2006

[29] I. Hambleton, A. Ranicki, L. Taylor, “Round $L$-theory”, J. Pure Appl. Algebra, 47:2 (1987), 131–154 | DOI | MR | Zbl

[30] A. A. Ranicki, Algebraic $L$-theory and topological manifolds, Cambridge Tracts in Math., 102, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[31] A. Bak, Yu. V. Muranov, “Splitting along submanifolds and $L$-spectra”, J. Math. Sci. (N. Y.), 123:4 (2004), 4169–4184 | DOI | MR | Zbl

[32] I. Khèmblton, A. F. Kharshiladze, “A spectral sequence in surgery”, Russian Acad. Sci. Sb. Math., 77:1 (1994), 1–9 | DOI | MR | Zbl

[33] P. M. Akhmetiev, A. Cavicchioli, D. Repovš, “On realization of splitting obstructions in Browder–Livesay groups for closed manifold pairs”, Proc. Edinburgh Math. Soc. (2), 43:1 (2000), 15–25 | DOI | MR | Zbl

[34] Yu. V. Muranov, “Splitting obstruction groups and quadratic extensions of anti-structures”, Izv. Math., 59:6 (1995), 1207–1232 | DOI | MR | Zbl

[35] R. M. Switzer, Algebraic topology–homotopy and homology, Grundlehren Math. Wiss., 212, Springer-Verlag, Berlin–Heidelberg–New York, 1975 ; R. M. Svittser, Algebraicheskaya topologiya – gomotopii i gomologii, Nauka, M., 1985 | MR | Zbl | MR | Zbl

[36] A. K. Bousfield, D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., 304, Springer-Verlag, Berlin–Heidelberg–New York, 1972 | DOI | MR | Zbl