scholarly journals Triangulations with few vertices of manifolds with non-free fundamental group

2019 ◽  
Vol 149 (6) ◽  
pp. 1453-1463
Author(s):  
Petar Pavešić
Keyword(s):  
Fundamental Group ◽  
Lower Bounds ◽  
Category Theory ◽  
Homology Sphere ◽  
Dimensional Manifold ◽  
Lusternik Schnirelmann ◽  
Few Vertices

AbstractWe study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3d + 1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with less than 3d vertices is PL-homeomorphic to Sd. Another important consequence is that every triangulation with small links of M is combinatorial.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

2007 ◽  
Vol 142 (2) ◽  
pp. 259-268 ◽  
Author(s):  
YUYA KODA
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Alexander Polynomial ◽  
Homology Sphere ◽  
Cyclic Covering ◽  
Rational Homology ◽  
Branched Coverings ◽  
Branched Covering ◽  
Rational Homology Spheres ◽  
Rational Homology Sphere

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

scholarly journals LS-category of moment-angle manifolds and higher order Massey products

2021 ◽  
Vol 33 (5) ◽  
pp. 1179-1205
Author(s):  
Piotr Beben ◽  
Jelena Grbić
Keyword(s):  
Simplicial Complex ◽  
Structural Properties ◽  
Lower Bounds ◽  
Higher Order ◽  
Hyperbolic Manifolds ◽  
Upper And Lower Bounds ◽  
Connected Sum ◽  
Massey Products ◽  
Higher Dimensional ◽  
Lusternik Schnirelmann

Abstract Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

scholarly journals Casson's invariant and surgery on knots

1992 ◽  
Vol 35 (3) ◽  
pp. 383-395 ◽  
Author(s):  
C. D. Frohman ◽  
D. D. Long
Keyword(s):  
Fundamental Group ◽  
Irreducible Representations ◽  
Homology Sphere

We show that given a knot in a homology sphere there is a sequence of invariants with the property that if the nth invariant does not vanish, then this implies the existence of a family of irreducible representations of the fundamental group of the complement of the knot into SU(n).

A group with zero homology

1968 ◽  
Vol 64 (3) ◽  
pp. 599-602 ◽  
Author(s):  
D. B. A Epstein
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Dimensional Manifold ◽  
Finitely Generated ◽  
3 Dimensional ◽  
Identity Group ◽  
Dimension 2

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

Floer Homology for Knots and SU(2)-Representations for Knot Complements and Cyclic Branched Covers

2000 ◽  
Vol 52 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Olivier Collin
Keyword(s):  
Fundamental Group ◽  
Existence Result ◽  
Floer Homology ◽  
Homology Sphere ◽  
Branched Covers ◽  
Underlying Space ◽  
Knot Complements

AbstractIn this article, using 3-orbifolds singular along a knot with underlying space a homology sphere Y3, the question of existence of non-trivial and non-abelian SU(2)-representations of the fundamental group of cyclic branched covers of Y3 along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian SU(2)-representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian SU(2)-representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in S3.

scholarly journals Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Biplab Basak ◽  
Manisha Binjola
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Lower Bounds ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Regular Genus

AbstractLet 𝑀 be a connected compact PL 4-manifold with boundary. In this article, we give several lower bounds for regular genus and gem-complexity of the manifold 𝑀. In particular, we prove that if 𝑀 is a connected compact 4-manifold with ℎ boundary components, then its gem-complexity k(M) satisfies the inequalities k(M)\geq 3\chi(M)+7m+7h-10 and k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9, and its regular genus \mathcal{G}(M) satisfies the inequalities \mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and \mathcal{G}(M)\geq\mathcal{G}(\partial M)+2\chi(M)+2m+2h-4, where 𝑚 is the rank of the fundamental group of the manifold 𝑀. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds is also shown for a large class of PL 4-manifolds with boundary.

Spherical Mean and the Fundamental Group

1991 ◽  
Vol 34 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Random Walks ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Spherical Means ◽  
Universal Covering Space ◽  
Spherical Mean

AbstractWe investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.

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