On the regular genus of 5-manifolds with free fundamental group

2003 ◽  
Vol 15 (3) ◽  
Author(s):  
Maria Rita Casali
Keyword(s):  
Fundamental Group ◽  
Regular Genus

scholarly journals Classifying Pl 5-Manifolds by Regular Genus: The Boundary Case

1997 ◽  
Vol 49 (2) ◽  
pp. 193-211 ◽  
Author(s):  
Maria Rita Casali
Keyword(s):  
Fundamental Group ◽  
Complete Classification ◽  
Combinatorial Approach ◽  
3 Dimensional ◽  
Combinatorial Invariant ◽  
Boundary Case ◽  
Regular Genus ◽  
Poincare Conjecture ◽  
Image Position

AbstractIn the present paper, we face the problem of classifying classes of orientable PL 5-manifolds M5 with h ≥ 1 boundary components, by making use of a combinatorial invariant called regular genusG(M5). In particular, a complete classification up to regular genus five is obtained: where denotes the regular genus of the boundary ∂M5 and denotes the connected sumof h ≥ 1 orientable 5-dimensional handlebodies 𝕐αi of genus αi ≥ 0 (i = 1, . . . ,h), so that .Moreover, we give the following characterizations of orientable PL 5-manifolds M5 with boundary satisfying particular conditions related to the “gap” between G(M5) and either G(∂M5) or the rank of their fundamental group rk(π1(M5)): Further, the paper explains how the above results (together with other known properties of regular genus of PL manifolds) may lead to a combinatorial approach to 3-dimensional Poincaré Conjecture.

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.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

scholarly journals A criterion for residual 𝑝-finiteness of arbitrary graphs of finite 𝑝-groups

2019 ◽  
Vol 22 (5) ◽  
pp. 837-844
Author(s):  
Gareth Wilkes
Keyword(s):  
Fundamental Group ◽  
Infinite Graphs ◽  
Graphs Of Groups ◽  
Arbitrary Graphs

Abstract We establish conditions under which the fundamental group of a graph of finite p-groups is necessarily residually p-finite. The technique of proof is independent of previously established results of this type, and the result is also valid for infinite graphs of groups.

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