Fundamental classes of 3-manifold groups representations in SL(4,R)

2017 ◽  
Vol 26 (07) ◽  
pp. 1750036
Author(s):  
Thilo Kuessner
Keyword(s):  
Connected Components ◽  
Fundamental Groups ◽  
Fundamental Class ◽  
Character Varieties ◽  
Chern Simons

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of [Formula: see text]-manifolds to [Formula: see text] that factor over [Formula: see text], in particular for those factoring over the isomorphism [Formula: see text]. We also discuss consequences for the number of connected components of [Formula: see text]-character varieties, and we show that there are knots with arbitrarily many components of vanishing Chern–Simons invariant in their [Formula: see text]-character varieties.

Complexity of shadows and traversing flows in terms of the simplicial volume

2016 ◽  
Vol 08 (03) ◽  
pp. 501-543 ◽  
Author(s):  
Gabriel Katz
Keyword(s):  
Compact Manifolds ◽  
Connected Components ◽  
Fundamental Groups ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Simplicial Volume ◽  
Localization Technique ◽  
Close Relatives ◽  
Universal Bounds

We combine Gromov’s amenable localization technique with the Poincaré duality to study the traversally generic vector flows on smooth compact manifolds [Formula: see text] with boundary. Such flows generate well-understood stratifications of [Formula: see text] by the trajectories that are tangent to the boundary in a particular canonical fashion. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension. These universal bounds are basically expressed in terms of the normed homology of the fundamental groups [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the double of [Formula: see text]. The norm here is the Gromov simplicial semi-norm in homology. It turns out that some close relatives of the normed homology spaces [Formula: see text], [Formula: see text] form obstructions to the existence of [Formula: see text]-convex traversally generic vector flows on [Formula: see text].

scholarly journals Compact connected components in relative character varieties of punctured spheres

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Nicolas Tholozan ◽  
Jérémy Toulisse
Keyword(s):  
Lie Groups ◽  
Invariant Theory ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Connected Components ◽  
Higgs Bundles ◽  
Geometric Invariant ◽  
Projective Varieties ◽  
Elliptic Element ◽  
Character Varieties

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

scholarly journals On the topology of a boolean representable simplicial complex

2017 ◽  
Vol 27 (01) ◽  
pp. 121-156 ◽  
Author(s):  
Stuart Margolis ◽  
John Rhodes ◽  
Pedro V. Silva
Keyword(s):  
Simplicial Complex ◽  
Homotopy Type ◽  
Sufficient Conditions ◽  
Simplicial Complexes ◽  
Connected Components ◽  
Fundamental Groups ◽  
Complexity Bounds ◽  
Dimension Formula ◽  
Dimension 2 ◽  
Necessary And Sufficient

It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension [Formula: see text]. In the case of dimension 2, it is shown that BRSC have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also, in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen–Macaulay are determined. Complexity bounds are provided for all the algorithms involved.

scholarly journals On model-theoretic connected components in some group extensions

2015 ◽  
Vol 15 (02) ◽  
pp. 1550009 ◽  
Author(s):  
Jakub Gismatullin ◽  
Krzysztof Krupiński
Keyword(s):  
Free Groups ◽  
Invariant Subgroup ◽  
Universal Cover ◽  
Connected Components ◽  
Fundamental Groups ◽  
Bounded Cohomology ◽  
Central Extensions ◽  
Finite Image ◽  
Genus Formula ◽  
Orientable Surfaces

We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new classes of examples of groups whose smallest type-definable subgroup of bounded index differs from the smallest invariant subgroup of bounded index. This includes the first known example of a group with this property found by Conversano and Pillay, namely the universal cover of [Formula: see text] (interpreted in a monster model), as well as various examples of different nature, e.g. some central extensions of free groups or of fundamental groups of closed orientable surfaces. As a corollary, we get that both non-abelian free groups and fundamental groups of closed orientable surfaces of genus [Formula: see text], expanded by predicates for all subsets, have this property, too. We also obtain a variant of the example of Conversano and Pillay for [Formula: see text] instead of [Formula: see text], which (as most of our examples) was not accessible by the previously known methods.

scholarly journals Twisted spectral correspondence and torus knots

2020 ◽  
Vol 29 (06) ◽  
pp. 2050040
Author(s):  
Wu-Yen Chuang ◽  
Duiliu-Emanuel Diaconescu ◽  
Ron Donagi ◽  
Satoshi Nawata ◽  
Tony Pantev
Keyword(s):  
Conjugacy Classes ◽  
Present Approach ◽  
Higgs Bundles ◽  
Torus Knots ◽  
Cohomological Invariants ◽  
Character Varieties ◽  
Spectral Correspondence ◽  
Marked Points ◽  
Chern Simons

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to [Formula: see text] torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

scholarly journals Fundamental groups of character varieties: surfaces and tori

2015 ◽  
Vol 281 (1-2) ◽  
pp. 415-425 ◽  
Author(s):  
Indranil Biswas ◽  
Sean Lawton ◽  
Daniel Ramras
Keyword(s):  
Fundamental Groups ◽  
Character Varieties

Chern-Simons theory for electrons in high Landau levels

1999 ◽  
Vol 09 (PR10) ◽  
pp. Pr10-223-Pr10-225
Author(s):  
S. Scheidl ◽  
B. Rosenow
Keyword(s):  
Landau Levels ◽  
Simons Theory ◽  
Chern Simons ◽  
Chern Simons Theory

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.

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