The topology of discrete groups

Baumslag, Gilbert; Dyer, Eldon; Heller, Alex

doi:10.1016/0022-4049(80)90040-7

Cited by 103 publications

(112 citation statements)

References 24 publications

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“…We also use G. Higman's group H which has the properties that it has no non-trivial finite quotients and that it is acyclic (i.e., has the same integral homology as the trivial group) [1,7]. The group G 0 is just an infinite free product of copies of H, and for n > 0, G n may be described as a free product of two groups of type F , amalgamating a common subgroup isomorphic to G n−1 .…”

Section: Introductionmentioning

confidence: 99%

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Some Groups of Finite Homological Type

Leary

Saadetoğlu

2006

Geom Dedicata

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Section: Introductionmentioning

confidence: 99%

“…The proof that H has the stated 'homological properties' was given by Baumslag, Dyer and Heller in [1], where the group H played an important role in their strengthened version of the Kan-Thurston theorem.…”

Section: Corollarymentioning

confidence: 99%

Some Groups of Finite Homological Type

Leary

Saadetoğlu

2006

Geom Dedicata

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“…We now divide the argument into two cases, since it suffices to show that I n+1 (π, w) and C n+1 (π, w) are equal after tensoring with Z (2) and Z[1/2] separately. Theorem 4.2.…”

Section: The Proof Of Theorem a (Localized At 2)mentioning

confidence: 99%

“…(ii) The splitting of L 0 ⊗Z (2) is given by universal cohomology classes ℓ ∈ H 4 * (L 0 ; Z (2) ) and k ∈ H 4 * +2 (L 0 ; Z/2). The domain of the assembly map…”

Section: The Characteristic Class Formulasmentioning

confidence: 99%

Surgery obstructions on closed manifolds and the inertia subgroup

Hambleton

2012

Forum Mathematicum

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Abstract. The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group I n+1 (π, w) and the closed manifold subgroup C n+1 (π, w) are equal in dimensions n + 1 ≥ 6, for any finitely-presented group π and any orientation character w : π → Z/2. This answers a question from [10, p. 107].

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“…Recent work [2,6,9] has exhibited homotopy theories as arising in a variety of contexts not lending themselves to the standard treatments. The present discussion may perhaps serve to reinforce the notion that it may yet be premature to decide just what it is that constitutes the formal structure of such a theory.…”

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confidence: 99%

Homotopy in Functor Categories

Heller¹

1982

Transactions of the American Mathematical Society

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Abstract. If C is a small category enriched over topological spaces the category 5" c of continuous functors from C into topological spaces admits a family of homotopy theories associated with closed subcategories of C. The categories ?T c, for various C, are connected to one another by a functor calculus analogous to the <8>, Horn calculus for modules over rings. The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories. Among the functors so described are homotopy limits and colimits and, more generally, homotopy Kan extensions. A by-product of the method is a generalization to functor categories of E. H. Brown's representability theorem.Introduction. The subject of this investigation is the kind of homotopy theory that can be done in categories 9"c of functors with values in the category of topological (more properly, separated compactly generated) spaces. The index category C is itself supposed to be enriched over 9" so that such categories as, for example, that of G-spaces where G is a topological monoid or group, are subsumed.There are two points which are to be taken into account. First, in such a functor category there are always several notions of homotopy. Two of these have been extensively considered [3,13, 14]; a third, provided by the obvious homotopy congruence, has perhaps been thought too trivial to warrant much interest. In fact there are many more-one, indeed, for each closed subcategory of the index category. Second, these functor categories are related to one another by a functor calculus formally analogous to the " tensor product" and " hom" calculus of module theory. In this calculus appear, in particular, the functors induced by composition as well as their adjoints, the Kan extensions.The objective then is to articulate these notions of homotopy and the functor calculus into a comprehensive system. As usual this is mediated by the introduction of notions of fibration and cofibration. The several "standard" forms in which this may be done-most notably Quillen's "model category" structures [10, cf. also 4]-are perhaps not adequate here. It has, rather, seemed appropriate to associate to each notion of homotopy two each of fibration and cofibration: a kind of bifurcation of function not unfamiliar in such instances as this of generalization.There are to be then weaker and stronger notions of fibration and cofibration. It is the stronger ones which will interact well with the functor calculus. To show that

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The topology of discrete groups

Cited by 103 publications

References 24 publications

Some Groups of Finite Homological Type

Some Groups of Finite Homological Type

Surgery obstructions on closed manifolds and the inertia subgroup

Homotopy in Functor Categories

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