| Title:
|
Surgery on pairs of closed manifolds (English) |
| Author:
|
Cavicchioli, Alberto |
| Author:
|
Muranov, Yuri V. |
| Author:
|
Spaggiari, Fulvia |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
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59 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
551-571 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
surgery on manifolds |
| Keyword:
|
surgery obstruction groups for a manifold pair |
| Keyword:
|
assembly map |
| Keyword:
|
splitting problem |
| Keyword:
|
Browder-Livesay groups |
| Keyword:
|
Browder-Quinn surgery obstruction groups |
| Keyword:
|
splitting obstruction groups |
| Keyword:
|
manifolds with filtration |
| MSC:
|
18F25 |
| MSC:
|
19J25 |
| MSC:
|
55T99 |
| MSC:
|
57R67 |
| MSC:
|
58A35 |
| idZBL:
|
Zbl 1224.57014 |
| idMR:
|
MR2532390 |
| . |
| Date available:
|
2010-07-20T15:24:50Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140497 |
| . |
| Reference:
|
[1] Akhmetiev, P. M., Cavicchioli, A., Repovš, D.: On realization of splitting obstructions in Browder-Livesay groups for closed manifold pairs.Proc. Edinb. Math. Soc. 43 (2000), 15-25. MR 1744696, 10.1017/S0013091500020666 |
| Reference:
|
[2] Bak, A., Muranov, Yu. V.: Splitting along submanifolds and $\Bbb L$-spectra.Sovrem. Mat. Prilozh., Topol., Anal. Smezh. Vopr. (2003), 3-18 Russian English transl. J. Math. Sci. 123 (2004), 4169-4184. Zbl 1078.57030, MR 2157601 |
| Reference:
|
[3] 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. Zbl 1148.57042, MR 2477279 |
| Reference:
|
[4] Bak, A., Muranov, Yu. V.: Splitting along a submanifold with filtration.In preparation. |
| Reference:
|
[5] Browder, W., Livesay, G. R.: Fixed point free involutions on homotopy spheres.Bull. Am. Math. Soc. 73 (1967), 242-245. Zbl 0156.21903, MR 0206965, 10.1090/S0002-9904-1967-11700-2 |
| Reference:
|
[6] 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. Zbl 0343.57017, MR 0375348 |
| Reference:
|
[7] Cappell, S. E., Shaneson, J. L.: Pseudo-free actions I.Lect. Notes Math. 763 (1979), 395-447. Zbl 0416.57020, MR 0561231, 10.1007/BFb0088095 |
| Reference:
|
[8] Cavicchioli, A., Muranov, Yu. V., Spaggiari, F.: Relative groups in surgery theory.Bull. Belg. Math. Soc. 12 (2005), 109-135. Zbl 1072.57025, MR 2134861, 10.36045/bbms/1113318134 |
| Reference:
|
[9] Cavicchioli, A., Muranov, Yu. V., Spaggiari, F.: Mixed structures on a manifold with boundary.Glasg. Math. J. 48 (2006), 125-143. Zbl 1110.57019, MR 2224934, 10.1017/S0017089505002934 |
| Reference:
|
[10] Cavicchioli, A., Muranov, Yu. V., Spaggiari, F.: Assembly maps and realization of splitting obstructions.(to appear). MR 2563187 |
| Reference:
|
[11] Hambleton, I.: Projective surgery obstructions on closed manifolds.Lect. Notes Math. 967 (1982), 101-131. Zbl 0503.57018, MR 0689390, 10.1007/BFb0061900 |
| Reference:
|
[12] Hambleton, I., Ranicki, A., Taylor, L.: Round $L$-theory.J. Pure Appl. Algebra 47 (1987), 131-154. Zbl 0638.18003, MR 0906966, 10.1016/0022-4049(87)90057-0 |
| Reference:
|
[13] Hambleton, I., Milgram, J., Taylor, L., Williams, B.: Surgery with finite fundamental group.Proc. Lond. Math. Soc. 56 (1988), 349-379. Zbl 0665.57026, MR 0922660, 10.1112/plms/s3-56.2.349 |
| Reference:
|
[14] 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. MR 1198831 |
| Reference:
|
[15] 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. MR 0897701 |
| Reference:
|
[16] 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. Zbl 0671.57020, MR 0912061 |
| Reference:
|
[17] Jimenez, R., Muranov, Yu. V., Repovš, D.: Splitting along a submanifold pair.$K$-theory (to appear). MR 2456107 |
| Reference:
|
[18] Medrano, S. Lopez de: Involutions on Manifolds.Springer Berlin-Heidelberg-New York (1971). MR 0298698 |
| Reference:
|
[19] Muranov, Yu. V.: Splitting problem.Trudi MIRAN 212 (1996), 123-146 Russian English transl. Proc. Steklov Inst. Math. 212 (1996), 115-137. Zbl 0898.57014, MR 1635031 |
| Reference:
|
[20] 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. MR 2251365 |
| Reference:
|
[21] 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. Zbl 0732.55003, MR 1076143 |
| Reference:
|
[22] 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. MR 2301596 |
| Reference:
|
[23] 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. MR 2034535 |
| Reference:
|
[24] Ranicki, A. A.: The total surgery obstruction.Lect. Notes Math. 763 (1979), 275-316. Zbl 0428.57012, MR 0561227, 10.1007/BFb0088091 |
| Reference:
|
[25] Ranicki, A. A.: Exact Sequences in the Algebraic Theory of Surgery. Math. Notes 26.Princeton Univ. Press Princeton (1981). MR 0620795 |
| Reference:
|
[26] Ranicki, A. A.: The $L$-theory of twisted quadratic extensions.Can. J. Math. (1987), 39 245-364. Zbl 0635.57017, MR 0899842, 10.4153/CJM-1987-017-x |
| Reference:
|
[27] Ranicki, A. A.: Algebraic $L$-Theory and Topological Manifolds. Cambridge Tracts in Math. 102.Cambridge University Press Cambridge (1992). MR 1211640 |
| Reference:
|
[28] Switzer, R.: Algebraic Topology--Homotopy and Homology. Grund. Math. Wiss. 212.Springer Berlin-Heidelberg-New York (1975). MR 0385836 |
| Reference:
|
[29] Wall, C. T. C.: Surgery on Compact Manifolds.Academic Press London-New York (1970), Second Edition A. A. Ranicki Am. Math. Soc. Providence, 1999. Zbl 0219.57024, MR 0431216 |
| Reference:
|
[30] Wall, C. T. C.: Formulae for surgery obstructions.Topology 15 (1976), 182-210. corrigendum ibid. 16 (1977), 495-496. Zbl 0377.57006, MR 0488092 |
| Reference:
|
[31] Wall, C. T. C.: Classification of hermitian forms. VI. Group rings.Ann. Math. 103 (1976), 1-80. Zbl 0328.18006, MR 0432737, 10.2307/1971019 |
| Reference:
|
[32] Weinberger, S.: The Topological Classification of Stratified Spaces.The University of Chicago Press Chicago--London (1994). Zbl 0826.57001, MR 1308714 |
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