Geometric Aspects of Two-Dimensional Complexes

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

doi:10.1017/cbo9780511629358.003

Cited by 22 publications

(20 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We broadly define a model simplicial complex to be one that is constructed in a standard way (i.e., by a fix algorithm) from a presentation of a group, and whose fundamental group is isomorphic with the presented group. This is in contrast to the common definition, for instance in [7], and more suited to our purposes. In the literature, there exist proofs of the fact that model 2-complexes embed in R 4 .…”

Section: Introductionmentioning

confidence: 88%

See 1 more Smart Citation

Small Model $2$-Complexes in $4$-space and Applications

Parsa

2015

Preprint

View full text Add to dashboard Cite

We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) 2-complexes. We show, as an extension of an earlier work, that computing first homology of 2-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usual procedures for computing homology cannot be improved other than by matrix methods. This is true even if the complex is in the euclidean 4-space. For this purpose, we use 2-complexes built in a standard way from group presentations, called model 2-complexes. Model complexes have fundamental group isomorphic with the group defined by the presentation. We show that there are model complexes of size in the order of the bit-complexity of the presentation that can be realized linearly in 4-space. We further derive some applications of this result regarding embeddability problems in the euclidean 4-space.

show abstract

Section: Introductionmentioning

confidence: 88%

“…In [8], together with giving another proof of embeddability of model complexes, it is shown that acyclic 2-complexes can be embedded with contractible complement, see also references in it. For a survey on possible thickenings of 2-complexes in various dimensions see [7].…”

Section: Introductionmentioning

confidence: 99%

Small Model $2$-Complexes in $4$-space and Applications

Parsa

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…This conjecture is expected to be false and different proposals for counterexamples have been made, but there seem to be a lack of tools for actually detecting them as such. An extensive reference for all the problems connected with the Andrews-Curtis conjecture is [6].…”

Section: 1mentioning

confidence: 99%

“…, R m , depends on the choices made, but this dependence can be explicitly described. In [6] (theorem 2.4), it is shown that the correspondence P →P induces a bijection between the 2-deformation types of connected 2-dimensional CW complexes and the equivalence classes of finite presentations under the following moves:…”

Section: 1mentioning

confidence: 99%

See 1 more Smart Citation

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

Bobtcheva

Messia

2003

Algebr. Geom. Topol.

View full text Add to dashboard Cite

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.

show abstract

Three-Dimensional Alexandrov Spaces: A Survey

Galaz-García

Núñez-Zimbrón

2022

Recent Advances in Alexandrov Geometry

View full text Add to dashboard Cite

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.

show abstract

Geometric Aspects of Two-Dimensional Complexes

Cited by 22 publications

References 0 publications

Small Model $2$-Complexes in $4$-space and Applications

Small Model $2$-Complexes in $4$-space and Applications

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

Three-Dimensional Alexandrov Spaces: A Survey

Contact Info

Product

Resources

About