# Surgery on pairs of closed manifolds

Alberto Cavicchioli; Yuri V. Muranov; Fulvia Spaggiari

Czechoslovak Mathematical Journal (2009)

- Volume: 59, Issue: 2, page 551-571
- ISSN: 0011-4642

## Access Full Article

top## Abstract

top## How to cite

topCavicchioli, Alberto, Muranov, Yuri V., and Spaggiari, Fulvia. "Surgery on pairs of closed manifolds." Czechoslovak Mathematical Journal 59.2 (2009): 551-571. <http://eudml.org/doc/37940>.

@article{Cavicchioli2009,

abstract = {To apply surgery theory to the problem of classifying pairs of closed manifolds, it is necessary to know the subgroup of the group $LP_*$ generated by those elements which are realized by normal maps to a pair of closed manifolds. This closely relates to the surgery problem for a closed manifold and to the computation of the assembly map. In this paper we completely determine such subgroups for many cases of Browder-Livesay pairs of closed manifolds. Moreover, very explicit results are obtained in the case of an elementary fundamental group. Then we generalize them, and obtain several further results about the realization of elements in the Browder-Quinn surgery obstruction groups by means of normal maps to a closed manifold filtered by closed submanifolds.},

author = {Cavicchioli, Alberto, Muranov, Yuri V., Spaggiari, Fulvia},

journal = {Czechoslovak Mathematical Journal},

keywords = {surgery on manifolds; surgery obstruction groups for a manifold pair; assembly map; splitting problem; Browder-Livesay groups; Browder-Quinn surgery obstruction groups; splitting obstruction groups; manifolds with filtration; surgery on manifolds; surgery obstruction groups for a manifold pair; assembly map; splitting problem; Browder-Livesay group; Browder-Quinn surgery obstruction group; splitting obstruction group; manifold with filtration},

language = {eng},

number = {2},

pages = {551-571},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Surgery on pairs of closed manifolds},

url = {http://eudml.org/doc/37940},

volume = {59},

year = {2009},

}

TY - JOUR

AU - Cavicchioli, Alberto

AU - Muranov, Yuri V.

AU - Spaggiari, Fulvia

TI - Surgery on pairs of closed manifolds

JO - Czechoslovak Mathematical Journal

PY - 2009

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 59

IS - 2

SP - 551

EP - 571

AB - To apply surgery theory to the problem of classifying pairs of closed manifolds, it is necessary to know the subgroup of the group $LP_*$ generated by those elements which are realized by normal maps to a pair of closed manifolds. This closely relates to the surgery problem for a closed manifold and to the computation of the assembly map. In this paper we completely determine such subgroups for many cases of Browder-Livesay pairs of closed manifolds. Moreover, very explicit results are obtained in the case of an elementary fundamental group. Then we generalize them, and obtain several further results about the realization of elements in the Browder-Quinn surgery obstruction groups by means of normal maps to a closed manifold filtered by closed submanifolds.

LA - eng

KW - surgery on manifolds; surgery obstruction groups for a manifold pair; assembly map; splitting problem; Browder-Livesay groups; Browder-Quinn surgery obstruction groups; splitting obstruction groups; manifolds with filtration; surgery on manifolds; surgery obstruction groups for a manifold pair; assembly map; splitting problem; Browder-Livesay group; Browder-Quinn surgery obstruction group; splitting obstruction group; manifold with filtration

UR - http://eudml.org/doc/37940

ER -

## References

top- Akhmetiev, P. M., Cavicchioli, A., Repovš, D., 10.1017/S0013091500020666, Proc. Edinb. Math. Soc. 43 (2000), 15-25. (2000) MR1744696DOI10.1017/S0013091500020666
- Bak, A., Muranov, Yu. V., Splitting along submanifolds and $\mathbb{L}$-spectra, Sovrem. Mat. Prilozh., Topol., Anal. Smezh. Vopr. (2003), 3-18 Russian English transl. J. Math. Sci. 123 (2004), 4169-4184. (2004) Zbl1078.57030MR2157601
- Bak, A., Muranov, Yu. V., Normal invariants of manifold pairs and assembly maps, Mat. Sb. 197 (2006), 3-24 Russian English transl. Sb. Math. 197 (2006), 791-811. (2006) Zbl1148.57042MR2477279
- Bak, A., Muranov, Yu. V., Splitting along a submanifold with filtration, In preparation.
- Browder, W., Livesay, G. R., 10.1090/S0002-9904-1967-11700-2, Bull. Am. Math. Soc. 73 (1967), 242-245. (1967) Zbl0156.21903MR0206965DOI10.1090/S0002-9904-1967-11700-2
- Browder, W., Quinn, F., A Surgery Theory for G-manifolds and Stratified Sets, Proc. Int. Conf. Manifolds, Tokyo 1973 Univ. of Tokyo Press Tokyo (1975), 27-36. (1975) Zbl0343.57017MR0375348
- Cappell, S. E., Shaneson, J. L., 10.1007/BFb0088095, Lect. Notes Math. 763 (1979), 395-447. (1979) Zbl0416.57020MR0561231DOI10.1007/BFb0088095
- Cavicchioli, A., Muranov, Yu. V., Spaggiari, F., Relative groups in surgery theory, Bull. Belg. Math. Soc. 12 (2005), 109-135. (2005) Zbl1072.57025MR2134861
- Cavicchioli, A., Muranov, Yu. V., Spaggiari, F., 10.1017/S0017089505002934, Glasg. Math. J. 48 (2006), 125-143. (2006) Zbl1110.57019MR2224934DOI10.1017/S0017089505002934
- Cavicchioli, A., Muranov, Yu. V., Spaggiari, F., Assembly maps and realization of splitting obstructions, (to appear). MR2563187
- Hambleton, I., 10.1007/BFb0061900, Lect. Notes Math. 967 (1982), 101-131. (1982) Zbl0503.57018MR0689390DOI10.1007/BFb0061900
- Hambleton, I., Ranicki, A., Taylor, L., 10.1016/0022-4049(87)90057-0, J. Pure Appl. Algebra 47 (1987), 131-154. (1987) Zbl0638.18003MR0906966DOI10.1016/0022-4049(87)90057-0
- Hambleton, I., Milgram, J., Taylor, L., Williams, B., 10.1112/plms/s3-56.2.349, Proc. Lond. Math. Soc. 56 (1988), 349-379. (1988) Zbl0665.57026MR0922660DOI10.1112/plms/s3-56.2.349
- Hambleton, I., Kharshiladze, A. F., A spectral sequence in surgery theory, Mat. Sb. (1992), 183 3-14 Russian English transl. Russ. Acad. Sci. Sb. Math. 77 (1994), 1-9. (1994) MR1198831
- Kharshiladze, A. F., Iterated Browder-Livesay invariants and oozing problem, Mat. Zametki 41 (1987), 557-563 Russian English transl. Math. Notes 41 (1987), 312-315. (1987) MR0897701
- Kharshiladze, A. F., Surgery on manifolds with finite fundamental groups, Uspechi Mat. Nauk 42 (1987), 55-85 Russian English transl. Russ. Math. Surv. 42 (1987), 65-103. (1987) Zbl0671.57020MR0912061
- Jimenez, R., Muranov, Yu. V., Repovš, D., Splitting along a submanifold pair, -theory (to appear). MR2456107
- Medrano, S. Lopez de, Involutions on Manifolds, Springer Berlin-Heidelberg-New York (1971). (1971) MR0298698
- Muranov, Yu. V., Splitting problem, Trudi MIRAN 212 (1996), 123-146 Russian English transl. Proc. Steklov Inst. Math. 212 (1996), 115-137. (1996) Zbl0898.57014MR1635031
- Muranov, Yu. V., Jimenez, R., Transfer maps for triples of manifolds, Matem. Zametki 79 (2006), 420-433 Russian English translation Math. Notes 79 (2006), 387-398. (2006) MR2251365
- Muranov, Yu. V., Kharshiladze, A. F., Browder-Livesay groups of abelian 2-groups, Mat. Sb. 181 (1990), 1061-1098 Russian English transl. Math. USSR Sb. 70 (1991), 499-540. (1991) Zbl0732.55003MR1076143
- Muranov, Yu. V., Repovš, D., Jimenez, R., A spectral sequence in surgery theory and manifolds with filtrations, Trudy MMO (2006), 67 294-325 Russian English transl. Trans. Mosc. Math. Soc. 2006 (2006), 261-288. (2006) MR2301596
- Muranov, Yu. V., Repovš, D., Spaggiari, F., Surgery on triples of manifolds, Mat. Sb. 194 (2003), 139-160 Russian English transl. Sb. Math. 194 (2003), 1251-1271. (2003) MR2034535
- Ranicki, A. A., 10.1007/BFb0088091, Lect. Notes Math. 763 (1979), 275-316. (1979) Zbl0428.57012MR0561227DOI10.1007/BFb0088091
- Ranicki, A. A., Exact Sequences in the Algebraic Theory of Surgery. Math. Notes 26, Princeton Univ. Press Princeton (1981). (1981) MR0620795
- Ranicki, A. A., 10.4153/CJM-1987-017-x, Can. J. Math. (1987), 39 245-364. (1987) Zbl0635.57017MR0899842DOI10.4153/CJM-1987-017-x
- Ranicki, A. A., Algebraic $L$-Theory and Topological Manifolds. Cambridge Tracts in Math. 102, Cambridge University Press Cambridge (1992). (1992) MR1211640
- Switzer, R., Algebraic Topology--Homotopy and Homology. Grund. Math. Wiss. 212, Springer Berlin-Heidelberg-New York (1975). (1975) MR0385836
- Wall, C. T. C., Surgery on Compact Manifolds, Academic Press London-New York (1970), Second Edition A. A. Ranicki Am. Math. Soc. Providence, 1999. (1970) Zbl0219.57024MR0431216
- Wall, C. T. C., Formulae for surgery obstructions, Topology 15 (1976), 182-210. corrigendum ibid. 16 (1977), 495-496. (1977) Zbl0377.57006MR0488092
- Wall, C. T. C., 10.2307/1971019, Ann. Math. 103 (1976), 1-80. (1976) Zbl0328.18006MR0432737DOI10.2307/1971019
- Weinberger, S., The Topological Classification of Stratified Spaces, The University of Chicago Press Chicago--London (1994). (1994) Zbl0826.57001MR1308714

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.