Surgery on triples of manifolds

Muranov, Yu. V.; Repovš, Dušan; Spaggiari, Fulvia

doi:10.1070/sm2003v194n08abeh000764

Cited by 9 publications

(26 citation statements)

References 16 publications

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“…The notion of s-triangulation of the filtration X naturally generalizes the notion of s-triangulation of a manifold and of a manifold pair in [30]. In particular, we prove that for a manifold triple Z ⊂ Y ⊂ X the surgery obstruction groups LT * (X, Y, Z) in [8] coincide with the Browder-Quinn groups L BQ * of the stratified manifold Z ⊂ Y ⊂ X (see [14,35]). …”

Section: Introductionmentioning

confidence: 94%

“…Also in § 3, we introduce structure sets for the filtration (1.11) that generalize the structure sets S n+1 (X, Y, ξ) and S n+1 (X), and we study their properties. Some results for the case of a manifold triple were obtained in [8,27,28].…”

Section: Introductionmentioning

confidence: 99%

“…The groups LM k * (X ) coincide with the Browder-Quinn stratified L-groups L BQ * (X ) (see [14,35]) up to a shift of dimension * . The spectrum LM 0 coincides with the spectrum L(π 1 (X)), the spectrum LM 1 coincides with the spectrum LP (F ) for the pair (X 0 , X 1 ) (see [30,34]), and the spectrum LM 2 coincides with the spectrum LT for the triple (X 0 , X 1 , X 2 ) (see [8,27,28]). …”

Section: Introductionmentioning

confidence: 99%

“…The surgery obstruction groups LT n−q−q (X, Y, Z) for a manifold triple were introduced in [8]. These groups are realized at the spectrum level by LT (X, Y, Z).…”

Section: Introductionmentioning

confidence: 99%

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A spectral sequence in surgery theory and manifolds with filtrations

Muranov

Repovš

Jiménez

2006

Trans. Moscow Math. Soc.

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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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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The surgery obstruction groups LT n−q−q (X, Y, Z) for a manifold triple were introduced in [8]. These groups are realized at the spectrum level by LT (X, Y, Z).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A spectral sequence in surgery theory and manifolds with filtrations

Muranov

Repovš

Jiménez

2006

Trans. Moscow Math. Soc.

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“…This permits the investigation of the splitting obstruction groups and the surgery obstruction groups for a manifold pair and for a triple of manifolds, obtaining explicit computations in many cases (see [3], [12], [13], and [20]). Following the classical book of Wall (see [24, p. 116]), we introduce a mixed type of structure on a pair (X, ∂X), where X is a compact topological n-manifold.…”

Section: Introductionmentioning

confidence: 99%

Mixed Structures on a Manifold With Boundary

2006

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Abstract. For a closed topological n-manifold X, the surgery exact sequence contains the set of manifold structures and the set of tangential structures of X. In the case of a compact topological n-manifold with boundary (X, ∂X), the classical surgery theory usually considers two different types of structures. The first one concerns structures whose restrictions are fixed on the boundary. The second one uses two similar structures on the manifold pair. In his classical book, Wall mentioned the possibility of introducing a mixed type of structure on a manifold with boundary. Following this suggestion, we introduce mixed structures on a topological manifold with boundary, and describe their properties. Then we obtain connections between these structures and the classical ones, and prove that they fit in some surgery exact sequences. The relationships can be described by using certain braids of exact sequences. Finally, we discuss explicitly several geometric examples.2000 Mathematics Subject Classification. Primary 57R67, 57Q10 Secondary 57R10, 55U35, 18F25.

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