Splitting along Submanifolds and L-Spectra

Bak, Anthony; Muranov, Yu. V.

doi:10.1023/b:joth.0000039816.14895.35

Cited by 14 publications

(47 citation statements)

References 28 publications

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“…In this section we present some preliminary results concerning surgery on topological manifolds and the use of L-spectra (see [1,8,19,22,29,30,33]). We give the requisite definitions and prove several technical results.…”

Section: Technical Resultsmentioning

confidence: 99%

“…Note that the two lower rows represent (1.7) for the manifold pair (Y, Z). Diagram (2.15) is realized at the spectrum level (see [1,8]). In particular, the composite…”

Section: Proposition 25 Suppose That a T-triangulationmentioning

confidence: 99%

“…(see [1,8,29,30]). We denote the cofibre of the map in (3.18) by S 1 (X ) = S(X, Y, ξ) with the homotopy groups…”

Section: A Spectral Sequence In Surgery Theory 275mentioning

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: Technical Resultsmentioning

confidence: 99%

“…Note that the two lower rows represent (1.7) for the manifold pair (Y, Z). Diagram (2.15) is realized at the spectrum level (see [1,8]). In particular, the composite…”

Section: Proposition 25 Suppose That a T-triangulationmentioning

confidence: 99%

See 1 more Smart Citation

A spectral sequence in surgery theory and manifolds with filtrations

Muranov

Repovš

Jiménez

2006

Trans. Moscow Math. Soc.

Self Cite

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“…In Section 4 we present the necessary material on the realization of the structure sets introduced above, on the spectrum level ( [1,2,4,5,6,8], and [9]). From this realization and the results of the previous sections, we derive the well-known basic diagrams of exact sequences for the structure sets (cf.…”

Section: Introductionmentioning

confidence: 99%

On structure sets of manifold pairs

Cencelj

Muranov

Repovš

2009

Homology, Homotopy and Applications

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In this paper we systematically describe relations between various structure sets which arise naturally for pairs of compact topological manifolds with boundary. Our consideration is based on a deep analogy between the case of a compact manifold with boundary and the case of a closed manifold pair. This approach also gives a possibility to construct the obstruction groups for natural maps of various structure sets and to investigate their properties.

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“…By using the realizations of exact sequences (1.1)-(1.3) on the spectra level (see [2], [3], [5], [14] and [15]) we obtain a spectrum ‫,‪(X‬ޓޔ‬ ∂X) with homotopy groups…”

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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Splitting along Submanifolds and L-Spectra

Cited by 14 publications

References 28 publications

A spectral sequence in surgery theory and manifolds with filtrations

A spectral sequence in surgery theory and manifolds with filtrations

On structure sets of manifold pairs

Mixed Structures on a Manifold With Boundary

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