(Singular) 3-Manifolds

Hog-Angeloni, Cynthia; Sieradski, Allan J.; Metzler, Wolfgang

doi:10.1017/cbo9780511629358.010

Cited by 5 publications

(7 citation statements)

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“…Proposition 1.4 (Lemma 2.5-I in [6]). Two n-dimensional CW-pairs (X, Y ) and (X ′ , Y ) are n-equivalent if and only if there is a map X → X ′ which is homotopic to a to a sequence of expansions and collapses of dimension at most n + 1 and induces the identity on the common subcomplex Y .…”

Section: N-deformation and Simple Homotopymentioning

confidence: 99%

“…Such terminology is not universally accepted. Indeed, due to the proposition bellow, what we call here n-deformation is called a (n + 1)-deformation in [6] and [19].…”

Section: N-deformation and Simple Homotopymentioning

confidence: 99%

“…We remind some basic definitions and facts about CW-complexes. The reader can find all details and a complete reference list in [6].…”

Section: Preliminaries On 2-dimensional Cw-complexesmentioning

confidence: 99%

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HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

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The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1-and 2-handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T 4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T 4 two subsets are identified : T 3 ⊂ T 4 , which produces invariants of 4-thickenings normalizable to invariants of the boundary, and T 2 ⊂ T 4 , which produces invariants of 4-thickenings depending only on the 2-dimensional spine and the second Whitney number of the 4-thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR-type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

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Section: N-deformation and Simple Homotopymentioning

confidence: 99%

“…Such terminology is not universally accepted. Indeed, due to the proposition bellow, what we call here n-deformation is called a (n + 1)-deformation in [6] and [19].…”

Section: N-deformation and Simple Homotopymentioning

confidence: 99%

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HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

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“…The Andrews-Curtis Conjecture, stated topologically, says that every finite 2-complex with trivial fundamental group 3-deforms to a wedge of spheres [10]. For Euler characteristic 1, this would mean such a 2-complex would 3-deform to a point.…”

Section: Introductionmentioning

confidence: 99%

“…For Euler characteristic 1, this would mean such a 2-complex would 3-deform to a point. A counter-example to Andrews-Curtis which is also a spine of a 3-manifold would give a counter-example to the Poincare Conjecture [10].…”

Section: Introductionmentioning

confidence: 99%

Classifications of 2-complexes whose finite fundamental group is that of a 3-manifold

Beyl

Latiolais

Waller

1997

Proceedings of the Edinburgh Mathematical Society

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We consider spines of spherical space forms; i.e., spines of closed oriented 3-manifolds whose universal cover is the 3-sphere. We give sufficient conditions for such spines to be homotopy or simple homotopy equivalent to 2-complexes with the same fundamental group G and minimal Euler characteristic 1. If the group ring ZG satisfies stably-free cancellation, then any such 2-complex is homotopy equivalent to a spine of a 3-manifold. If K,(ZG) is represented by units and K is homotopy equivalent to a spine X, then K and X are simple homotopy equivalent. We exhibit several infinite families of non-abelian groups G for which these conditions apply.

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Three-Dimensional Alexandrov Spaces: A Survey

Galaz-García

Núñez-Zimbrón

2022

Recent Advances in Alexandrov Geometry

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We survey several results concerning the geometry and topology of threedimensional Alexandrov spaces with the aim of providing a panoramic and up-to-date view of the subject. In particular we present the classification of positively and nonnegatively curved spaces, the geometrization theorem, a discussion of known results for simply-connected and aspherical spaces, the equivariant and topological classifications of closed three-dimensional Alexandrov spaces with isometric compact Lie group actions, and recent developments on collapsing theory.

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(Singular) 3-Manifolds

Cited by 5 publications

References 0 publications

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

HKR-type invariants of 4–thickenings of 2–dimensional CW complexes

Classifications of 2-complexes whose finite fundamental group is that of a 3-manifold

Three-Dimensional Alexandrov Spaces: A Survey

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