Homology and central series of groups

Stallings, John R.

doi:10.1016/0021-8693(65)90017-7

Cited by 410 publications

(308 citation statements)

References 5 publications

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“…Stallings Theorem and representations through p-groups. We first construct the bijections in Theorem 3.1 using Stallings' theorem [Sta65].…”

Section: ψ Is Admissible If and Only If ψ(ψ) Is Admissible And In Thmentioning

confidence: 99%

“…Finally note that γ descends to a map F/F q → π/π q which we denote by γ. We now have the following commutative diagram, where the vertical maps are the obvious projection maps: [Sta65] applied to ψ and from the commutativity of the diagram that the map ϕ • γ is an isomorphism. On the other hand it follows from Milnor's theorem [Mi57, Theorem 4] that γ is surjective.…”

Section: Link Concordance and Twisted Torsionmentioning

confidence: 99%

“…Let W be the exterior of a concordance, E = E L , and E ′ = E L ′ . Since H * (E) ∼ = H * (W ) ∼ = H * (E ′ ) we can apply Stallings' theorem [Sta65] to conclude that the inclusion maps induce isomorphisms…”

Section: 2mentioning

confidence: 99%

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Twisted torsion invariants and link concordance

Choon

Friedl

2011

Forum Mathematicum

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Abstract. The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor to the non-acyclic case. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.

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“…Stallings Theorem and representations through p-groups. We first construct the bijections in Theorem 3.1 using Stallings' theorem [Sta65].…”

Section: ψ Is Admissible If and Only If ψ(ψ) Is Admissible And In Thmentioning

confidence: 99%

Section: Link Concordance and Twisted Torsionmentioning

confidence: 99%

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Twisted torsion invariants and link concordance

Choon

Friedl

2011

Forum Mathematicum

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“…If Σ = X ∪ M Y with H 2 (X; Z) = 0 and β = β 1 (M ; Q), the inclusions of M and of ∨ β S 1 into X induce isomorphisms on all the rational lower central series quotients of the fundamental groups [17]. Hence these quotients are those of the free group F (β).…”

Section: Lemma 42 Let X Be a Compact 4-manifold With Connected Bounmentioning

confidence: 99%

Embedding 3-manifolds with circle actions

Hillman¹

2009

Proc. Amer. Math. Soc.

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Abstract. Constraints on the Seifert invariants of orientable 3-manifolds M which admit fixed-point free S 1 -actions and embed in R 4 are given. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold B unless H 1 (M ; Z) is torsion free, in which case it can take at most one nonzero value (up to sign). An H 2 × E 1 -manifold M with base orbifold B = S 2 (α 1 , . . . , α r ) where all cone point orders are odd embeds in R 4 if and only if its Seifert data S is skew-symmetric.The question of which closed 3-manifolds M (other than homology spheres) embed in R 4 has received surprisingly little attention. (The relevant papers known to us are [2,3,[5][6][7][8][9][10][11].) In particular, it is not yet known which Seifert fibred 3-manifolds embed, although in many other respects this is a well-understood class of spaces, with natural parametrizations in terms of Seifert data. It was shown earlier that if M embeds in R 4 , then it must be orientable and the torsion subgroup T (M ) of H 1 (M ; Z) must be a direct double: T (M ) ∼ = U ⊕ U for some finite abelian group U [8]. Moreover, the linking pairing M on T (M ) must be hyperbolic [13]. Most of the known constructions give smooth embeddings, but these conditions must also hold if M embeds as a TOP locally flat submanifold.The first two sections establish our notation and summarize some basic facts about compact 3-manifolds in R 4 . In §3 we shall observe that T (M ) being a direct double imposes strong constraints on the Seifert data of orientable 3-manifolds which admit fixed-point free S 1 -actions and which embed in R 4 . The present simple argument does not work for orientable Seifert fibred 3-manifolds with nonorientable base orbifolds. These admit no compatible S 1 -action. However in [3] we used the Z/2Z-index theorem to constrain the Euler invariants for such 3-manifolds. As a consequence we were able to settle there the question of embeddability for manifolds having one of the geometries S 3 , E 3 , Nil 3 , S 2 × E 1 or Sol 3 . When the Seifert data is "skew-symmetric" (i.e., is a set of complementary pairs) and all cone point orders are odd, the corresponding Seifert manifold embeds smoothly [3]. Such a manifold has generalized Euler invariant ε = 0 and so is geometric of type E 3 , S 2 × E

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“…In general (i.e. when the spheres are knotted) the above assertion is proved using the Stallings theorem on the lower central series of groups [St65,FKT94], cf. below.…”

Section: Sketch Of the Freedman-krushkal-teichner Constructionmentioning

confidence: 99%

Embedding and knotting of manifolds in Euclidean spaces

Skopenkov¹

2007

Surveys in Contemporary Mathematics

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Abstract. A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and intuitive part of the constructions and the arguments. In particular, we show how abstract algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities.More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to non-specialists. We outline a general approach to embeddings via the classical van Kampen-Shapiro-Wu-Haefliger-Weber 'deleted product' obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spież and the author): a generalization the Haefliger-Weber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range.

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Homology and central series of groups

Cited by 410 publications

References 5 publications

Twisted torsion invariants and link concordance

Twisted torsion invariants and link concordance

Embedding 3-manifolds with circle actions

Embedding and knotting of manifolds in Euclidean spaces

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