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On realization of splitting obstructions in Browder-Livesay groups for closed manifold pairs

Published online by Cambridge University Press:  20 January 2009

P. M. Akhmetiev ,
A. Cavicchioli  and
D. Repovš
P. M. Akhmetiev
Affiliation:
Izmiran, 142092 Troitsk, Moscow Region, Russia ([email protected])
A. Cavicchioli
Affiliation:
Dipartimento di Matematica, Università degli studi di Modena, Via Campi 213/B, 41100 Modena, Italy ([email protected])
D. Repovš
Affiliation:
Institute of Mathematics, Physics and Mechanics, University of Ljubljana, 1001 Ljubljana, POB 2964, Slovenia ([email protected])
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Abstract

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We consider some new types of realization problem for obstructions in the Browder-Livesay groups by homotopy equivalences of closed manifold pairs. We give several examples of calculations. We also consider relations with classical surgery problems.

Keywords

K-theoryBrowder-Livesay groupssplitting obstruction
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 15 - 25
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Akhmetiev, P. M. and Muranov, Yu. V., Obstructions to the splitting of manifolds with infinite fundamental group, Mat. Zametki 60 (1996), 163–175 (in Russian). (English transl, in Math. Notes 60 (1996), 121–129.)Google Scholar
2.Browder, W. and Livesay, G. R., Fixed point free involutions on homotopy spheres, Bull. Am. Math. Soc. 73 (1976), 242–245.CrossRefGoogle Scholar
3.Cappell, S. E., Mayer-Vietoris sequences in Hermitian K-theory, Algebraic K-theory, III: Hermitian K-theory and geometric applications, in Proc. Conf. Battelle Memorial Inst., Seattle, WA, 1972, pp. 478–512 (Springer, Berlin, 1972).Google Scholar
4.Cappell, S. E., Groups of singular Hermitian forms, in Proc. Conf. Battelle Memorial Inst., Seattle, WA, 1972, pp. 513–525 (Springer, Berlin, 1972).Google Scholar
5.Cappell, S. E. and Shaneson, J. L., A counterexample on the oozing problem for closed manifolds, in Algebraic topology (Aarhus, 1978), in Proc. Symp. Univ. Aarhus, Aarhus, 1978, Lecture Notes in Mathematics, vol. 763, pp. 627–634 (Springer, Berlin, 1979).Google Scholar
6.Hambleton, I., Milgram, R. J., Taylor, L. and Williams, B., Surgery with finite fundamental group, Proc. Lond. Math. Soc. 56 (1988), 349–379.Google Scholar
7.Kharshiladze, A. F., Hermitian K-theory and quadratic extensions, Trudy Moskov. Mat. Obšč. 41 (1980), 3–36 (in Russian).Google Scholar
8.Kharshiladze, A. F., Surgery on manifolds with finite fundamental groups, Usp. Mat. Nauk 42 (4) (1987), 55–85 (in Russian).Google Scholar
9.de Medrano, S. L., Involutions on manifolds (Springer, Berlin, 1971).Google Scholar
10.Muranov, Yu. V. and Repovš, D., Groups of obstructions to surgery and splitting for a manifold pair, Mat. Sbornik 188 (3) (1997), 127–142 (in Russian). (English transl. Math. Sbornik 188 (3) (1997), 449–463.)Google Scholar
11.Ranicki, A. A., Exact sequences in the algebraic theory of surgery, Mathematical Notes, vol. 26 (Princeton University Press, 1981).Google Scholar
12.Ranicki, A. A., Algebraic L-theory and topological manifolds (Cambridge University Press, 1992).Google Scholar
13.Shaneson, J. L., Wall's surgery obstruction groups for G × ℤ, Ann. Math. (2) 90 (1969), 296–334.CrossRefGoogle Scholar
14.Wall, C. T. C., Surgery on compact manifolds, London Mathematical Society Monographs, vol. 1 (Academic Press, New York, 1970).Google Scholar