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Article

Cavicchioli, Alberto ; Muranov, Yuri V. ; Spaggiari, Fulvia
Surgery on pairs of closed manifolds. (English). Czechoslovak Mathematical Journal, vol. 59 (2009), issue 2, pp. 551-571
MSC: 18F25, 19J25, 55T99, 57R67, 58A35 | MR 2532390 | Zbl 1224.57014
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
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.
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