2-groups and approximate fibrations

Daverman, Robert J.; Kim, Yongkuk

doi:10.1016/s0166-8641(01)00004-9

Cited by 10 publications

(24 citation statements)

References 14 publications

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“…We note that it is always the case that a space with a non-trivial finite sheeted self cover has a fundamental group with a finite index subgroup that is isomorphic to the entire group. By Daverman [4] the converse is also true, provided one is allowed to pass to high dimensional examples. However, within the category of GBS-groups what is striking is that the converse is true without the need to pass to high dimensions.…”

Section: The Topology Of Gbs-complexesmentioning

confidence: 99%

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Generalized Baumslag–Solitar groups and geometric homomorphisms

Delgado

Robinson

Timm

2011

Journal of Pure and Applied Algebra

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a b s t r a c tA group is called a generalized Baumslag-Solitar group, or GBS-group, if it is the fundamental group of a graph of groups with infinite cyclic vertex and edge groups. A GBS-group is said to be GBS-simple if it has no proper, non-cyclic geometric quotients, i.e., quotients other than Z which arise from geometric homomorphisms. The main result gives a complete description of the GBS-groups which are GBS-simple. This is achieved by constructing a set of natural examples of geometric homomorphisms.

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Section: The Topology Of Gbs-complexesmentioning

confidence: 99%

“…For example, K (2, 4) and BS (1,6) are GBS-simple, but K (4,8) and BS (2,4) are not. Notice that as a consequence of the theorem the property GBS-simple is independent of the chosen maximal subtree.…”

Section: Gbs-simple Groups and Gbs-free Groupsmentioning

confidence: 99%

Generalized Baumslag–Solitar groups and geometric homomorphisms

Delgado

Robinson

Timm

2011

Journal of Pure and Applied Algebra

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“…Here we make a generalization of the definition of a group being hyper-Hopfian [9] in the sense that we omit a condition of the group being finitely presented, so a group G is hyper-Hopfian if every homomorphism ϕ : G → G with ϕ(G) G and G/ϕ(G) cyclic is necessarily an automorphism.…”

Section: Definitions and Notationsmentioning

confidence: 99%

Special manifolds and shape fibrator properties

Vasilevska

2006

Topology and its Applications

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Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S) < 0 are fibrators in the new category. Manifold homotopically determined by π 1 0. Introduction In 1977, Coram and Duvall [3,4] introduced the concept of approximate fibrations as a generalization of both Hurewicz fibrations and cell-like maps. Approximate fibrations are proper mappings that satisfy an approximate version of the homotopy lifting property-the defining property for fibrations. Why study approximate fibrations? Coram and Duvall showed that approximate fibrations have shape theoretic properties analogous to the homotopy theoretic properties of Hurewicz fibrations. For each approximate fibration p : M → B, where M, B are locally compact ANRs, they showed that all fibers of p are fundamental absolute neighborhood retracts [1, p. 266], and moreover that if B is path connected, then any two fibers have the same shape. The most useful property is the existence of an exact sequence involving the homotopy groups of domain, target and shape-theoretical homotopy groups of any point inverse of p. When working with the PL approximate fibration these properties of an approximate fibration reduce to the usual properties of Hurewicz fibration because the fibers are ANRs, so the ith shape homotopy groups are isomorphic to ith homotopy groups. A question that arises is this: how can we detect approximate fibration so we can use these nice properties? Sometimes a proper map defined on an arbitrary manifold of a specific dimension can be recognized as an approximate fibration due to having point inverses all of a certain homotopy type (or shape). So the above question that has been addressed for more than two decades can be restated in a different form: * Tel.: +(

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“…A finitely presented group Γ is said to be hyperhopfian if every homomorphism f : Γ −→ Γ with f (Γ) normal and Γ/f (Γ) cyclic is an isomorphism (onto). Every Hopfian perfect group, every nonabelian group of order pq with distinct primes p and q, and the fundamental group of any closed surface with negative Euler characteristic are examples of hyperhopfian groups [7].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…The most natural objects N to study are s-Hopfian manifolds. All closed s-Hopfian manifolds with either trivial fundamental group or Hopfian fundamental group and nonzero Euler characteristic or hyperhopfian fundamental group are known to be codimension-2 fibrators [6,7,15,16,18]. Surprisingly few nonfibrators are known so far.…”

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confidence: 99%

Necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators

Im¹,

Kim²

2000

Proc. Amer. Math. Soc.

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Abstract. Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p : M → B defined on an (n + 2)-manifold M such that all fibre p −1 (b), b ∈ B, are shape equivalent to N is an approximate fibration. The most natural objects N to study are s-Hopfian manifolds. In this note we give some necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators.

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2-groups and approximate fibrations

Cited by 10 publications

References 14 publications

Generalized Baumslag–Solitar groups and geometric homomorphisms

Generalized Baumslag–Solitar groups and geometric homomorphisms

Special manifolds and shape fibrator properties

Necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators

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