On Two-Dimensional Aspherical Complexes

Cockcroft, W. H.

doi:10.1112/plms/s3-4.1.375

Cited by 52 publications

(45 citation statements)

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“…Let e 2 be the 2-cell containing the base point of Y. d{e 2 ) is the homology class represented by the loop on the boundary. But p* of this loop is the class of the attaching map R.…”

Section: Transformations Hence H 2 (X Y) S Z(g) As G-modules; and mentioning

confidence: 99%

Some small aspherical spaces

Dyer¹,

Vasquez

1973

J. Aust. Math. Soc.

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Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

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“…Let e 2 be the 2-cell containing the base point of Y. d{e 2 ) is the homology class represented by the loop on the boundary. But p* of this loop is the class of the attaching map R.…”

Section: Transformations Hence H 2 (X Y) S Z(g) As G-modules; and mentioning

confidence: 99%

Some small aspherical spaces

Dyer¹,

Vasquez

1973

J. Aust. Math. Soc.

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“…Then 2 is torsion-free ( [11], Theorem 4.2, p. 266). This implies that π is torsion-free as well so that q = 1; thus X and Y are aspherical (see [10], [1], or [4]). Since by hypothesis 7t λ {X) -Ξ f& π = TC^Y), they have the same homotopy type and in fact there is a homotopy equivalence between X and Y inducing any isomorphism a: Ξ -> π.…”

Section: Proof Since C*(π) Is Free Overmentioning

confidence: 99%

“…If the single relator R is not a proper power, it is known that the cellular model C{&) is aspherical (see [10], [1], or [4]), hence it is determined up to homotopy type by its fundamental group. If the single relator R is a proper power, C(&) is not aspherical, nevertheless we are able to prove the following: THEOREM …”

Section: On 2-dimensional Ctf-complexes With a Single 2-cell Sushil Jmentioning

confidence: 99%

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On 2-dimensionalCW-complexes with a single 2-cell

Jajodia¹

1979

Pacific J. Math.

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“…Remark 2.8. It is well-known that if the one-relator G is torsion-free then the universal coverK P of K P is a contractible 2-dimensional CW -complex, by a result of Dyer and Vasquez [12] which can be thought of as a geometric version of Lyndon's Identity Theorem [24] (see also [10]). Therefore, K P is also contractible and C 1 ⊂ C 2 · · · ⊂ K P is a nice filtration consisting of compact simply connected acyclic, and hence contractible, subcomplexes.…”

Section: Lemma 24 Let X Be a 2-dimensional Simply Connected Cw-compmentioning

confidence: 99%

One-relator groups and proper 3-realizability

Cárdenas¹,

Lasheras²,

Quintero³

et al. 2009

Rev. Mat. Iberoam.

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How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π 1 (K) ∼ = G whose universal coverK has the proper homotopy type of a PL 3-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly 3-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.

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On Two-Dimensional Aspherical Complexes

Cited by 52 publications

References 0 publications

Some small aspherical spaces

Some small aspherical spaces

On 2-dimensionalCW-complexes with a single 2-cell

One-relator groups and proper 3-realizability

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