Surgery on manifolds with finite fundamental groups

Kharshiladze, A. F.

doi:10.1070/rm1987v042n04abeh001466

Cited by 21 publications

(69 citation statements)

References 17 publications

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“…In this case, diagram (10) can be efficiently used for calculation of obstruction groups and natural mappings in L-theory (see [7,8,14,34]). Moreover, diagram (10) allows one to obtain deep geometric results on representability of elements of Wall groups by normal mappings of closed manifolds (see [16,17,30,35]). (We discuss this in the following section when investigating spectral sequences in surgery theory.…”

Section: Splitting Obstruction Groups and L-spectramentioning

confidence: 99%

Splitting along Submanifolds and L-Spectra

Bak¹,

Muranov²

2004

Journal of Mathematical Sciences

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Abstract. The problem of splitting of homotopy equivalence along a submanifold is closely related to surgeries of submanifolds and exact sequences in surgery theory. We describe possibilities and methods of application of L-spectra for the investigation of the problem of splitting of (simple) homotopy equivalence of manifolds along submanifolds. This approach naturally leads us to commutative diagrams of exact sequences, which play an important role in calculational problems of surgery theory.

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Section: Splitting Obstruction Groups and L-spectramentioning

confidence: 99%

Splitting along Submanifolds and L-Spectra

Bak¹,

Muranov²

2004

Journal of Mathematical Sciences

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“…Here P N (R) is a real projective space of high dimension. Varying the map ψ 1 in its homotopy class we can assume that ψ 1 is transversal to P N −1 (R) ⊂ P N (R) with ψ −1 1 (P N −1 (R)) = X 1 and that X 1 ⊂ X 0 is a Browder-Livesay pair (see [10,12,16,21]). In similar fashion we can now consider a map ψ 2 : X 1 → P N (R) that induces an epimorphism of the fundamental groups with kernel A 2 and with …”

Section: Application To the Browder-livesay Invariantsmentioning

confidence: 99%

“…This invariant is defined only if the Browder-Livesay invariant is trivial. The iterated Browder-Livesay invariants were introduced by Kharshiladze (see [4,9,10]). The elements of L n (B) that do not belong to the image of φ j for some j (and only these elements) are detected by the iterated Browder-Livesay invariants, as immediately follows from [9,10,12].…”

mentioning

confidence: 99%

“…The iterated Browder-Livesay invariants were introduced by Kharshiladze (see [4,9,10]). The elements of L n (B) that do not belong to the image of φ j for some j (and only these elements) are detected by the iterated Browder-Livesay invariants, as immediately follows from [9,10,12]. The fact that such elements cannot be realized by normal maps of closed manifolds was proved by Kharshiladze (see [9,10]) by geometric methods.…”

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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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“…) is defined (see [14] (cx), the last being the definition of the iterated Browder-Livesay invariants. In a similar way denote…”

Section: (V Dv) Of Manifolds With Boundaries and DV = υ Then D(y) mentioning

confidence: 99%