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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



A spectral sequence in surgery theory and manifolds with filtrations

Authors: Yu. V. Muranov, D. Repovs and R. Jimenez
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 67 (2006).
Journal: Trans. Moscow Math. Soc. 2006, 261-288
MSC (2000): Primary 57R67; Secondary 19J25, 57Q10, 57Q15
Published electronically: December 27, 2006
MathSciNet review: 2301596
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Abstract: In 1978 Cappell and Shaneson pointed out interesting properties of the Browder-Livesay invariants, which are analogous to the differentials of a certain spectral sequence. Such a spectral sequence was constructed by Hambleton and Kharshiladze in 1991. The main step of the construction of the spectral sequence consists in constructing an infinite filtration of spectra, in which, as is well known, only the first two spectra have a clear geometric meaning. In the present paper a geometric interpretation is given to all the spectra of the filtration in the Hambleton-Kharshiladze construction. Surgery obstruction groups for a system of embedded manifolds are introduced, and it is proved that the spectra realizing these groups coincide with the spectra in the Hambleton-Kharshiladze filtration. The algebraic and geometric properties of these groups and their connections with classical surgery theory are described. An isomorphism between these groups and the Browder-Quinn surgery obstruction groups for stratified manifolds is established. The results obtained are applied to the problem of realization of elements of the Wall groups by normal maps of closed manifolds and to the study of the iterated Browder-Livesay invariants.

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Additional Information

Yu. V. Muranov
Affiliation: Vitebsk State University, Vitebsk, Belarus’

D. Repovs
Affiliation: Institute for Mathematics, Physics, and Mechanics, University of Ljubljana, Slovenia

R. Jimenez
Affiliation: Instituto de Matematicas, National Autonomous University of Mexico (UNAM), Morelos, Mexico

Keywords: Spectral sequence, Browder--Livesay invariants, filtration of spectra, surgery obstruction groups, stratified manifold, Wall groups, fundamental group, simple homotopy equivalence, splitting problem, normal fibration, homotopy group
Published electronically: December 27, 2006
Additional Notes: The first author was supported by the Russian Foundation for Basic Research (grant no. 05–01–00993).
The second author was supported by the MESS Research Programme (no. P1–0292–0101–04).
The third author was supported by grants from CONACyT, DGAPA-UNAM, Fulbright-Garcia Robles, and University of Wisconsin–Madison.
Article copyright: © Copyright 2006 American Mathematical Society

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