scholarly journals The Reeb Graph of a Map Germ from ℝ3 to ℝ2 with Isolated Zeros

2016 ◽  
Vol 60 (2) ◽  
pp. 319-348 ◽  
Author(s):  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa ◽  
Juan J. Nuño-Ballesteros
Keyword(s):  
Structure Theorem ◽  
Topological Type ◽  
Topological Invariant ◽  
Topological Equivalence ◽  
Stable Maps ◽  
Reeb Graph ◽  
Cone Structure ◽  
Topological Description ◽  
Stable Map

AbstractWe consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

The Cone Structure Theorem

2020 ◽  
Author(s):  
Erica Boizan Batista ◽  
João Carlos Ferreira Costa ◽  
Juan José Nuño-Ballesteros
Keyword(s):  
Structure Theorem ◽  
Topological Classification ◽  
Topological Equivalence ◽  
Link Diagram ◽  
Cone Structure

Abstract We consider the topological classification of finitely determined map germs $f:(\mathbb{R}^n,0)\to (\mathbb{R}^p,0)$ with $f^{-1}(0)\neq \{0\}$. Associated with $f$ we have a link diagram, which is well defined up to topological equivalence. We prove that $f$ is topologically $\mathcal{A}$-equivalent to the generalized cone of its link diagram.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

scholarly journals Extracting topological information from momentum space propagators

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
Fabrizio Canfora ◽  
David Dudal ◽  
Alex Giacomini ◽  
Igor F. Justo ◽  
Pablo Pais ◽  
...  
Keyword(s):  
Gauge Invariance ◽  
Momentum Space ◽  
Chiral Symmetry Breaking ◽  
Topological Invariant ◽  
Analytic Structure ◽  
Topological Information ◽  
Gauge Transformations ◽  
Small Gauge ◽  
Topological Number

Abstract A new topological invariant quantity, sensitive to the analytic structure of both fermionic and bosonic propagators, is proposed. The gauge invariance of our construct is guaranteed for at least small gauge transformations. A generalization compatible with the presence of complex poles is introduced and applied to the classification of propagators typically emerging from non-perturbative considerations. We present partial evidence that the topological number can be used to detect chiral symmetry breaking or deconfinement.

scholarly journals On the geometry of multisymplectic manifolds

1999 ◽  
Vol 66 (3) ◽  
pp. 303-330 ◽  
Author(s):  
F. Cantrijn ◽  
A. Ibort ◽  
M. De León
Keyword(s):  
Differential Form ◽  
Structure Theorem ◽  
General Structure ◽  
Linear Case ◽  
Characteristic Distribution ◽  
Basic Properties ◽  
Lagrangian Fibration ◽  
Zero Characteristic

AbstractA multisymplectic structure on a manifold is defined by a closed differential form with zero characteristic distribution. Starting from the linear case, some of the basic properties of multisymplectic structures are described. Various examples of multisymplectic manifolds are considered, and special attention is paid to the canonical multisymplectic structure living on a bundle of exterior k-forms on a manifold. For a class of multisymplectic manifolds admitting a ‘Lagrangian’ fibration, a general structure theorem is given which, in particular, leads to a classification of these manifolds in terms of a prescribed family of cohomology classes.

A topological invariant for stable map germs

1976 ◽  
Vol 32 (2) ◽  
pp. 103-132 ◽  
Author(s):  
James Damon ◽  
Andr� Galligo
Keyword(s):  
Topological Invariant ◽  
Stable Map

Cr [Kscr ]-versality of the graph deformation of a Cr stable map-germ

1999 ◽  
Vol 127 (1) ◽  
pp. 93-97 ◽  
Author(s):  
TAKASHI NISHIMURA
Keyword(s):  
Stable Map

In his celebrated paper [7], Martinet showed the equivalence between the infinitesimal versality and the versality for [Ascr ]- and [Kscr ]-morphisms. By using this theorem, he obtained the following Theorem 1·1 which played one of the key roles for the classification of C∞ stable map-germs ([1, 7]).

scholarly journals Interplay of symmetry and topology in science

2014 ◽  
Vol 70 (a1) ◽  
pp. C1419-C1419
Author(s):  
Boris Zhilinskii
Keyword(s):  
Classical System ◽  
Quantum Systems ◽  
Qualitative Theory ◽  
Topological Invariant ◽  
Particle Systems ◽  
Qualitative Description ◽  
Energy Bands ◽  
And Topology ◽  
Finite Particle

Qualitative methods in natural science are based mainly on simultaneous use of symmetry and topology arguments. The idea of the present talk is to demonstrate how the corresponding mathematical tools (based on symmetry and topology arguments) initially applied to describe classification of different phases of matter and transitions between them are extended to construct qualitative theory of finite particle systems and more general dynamical systems. I start with reminding basic notions and tools associated with application of group action ideas to physics as initiated and developed by Louis Michel (1923-1999) [1,2]. Then geometric combinatorial and topological ideas are used to give qualitative description of singularities of dynamical integrable classical system and their quantum analogs. Quantum monodromy and its various generalizations as well as description of energy bands of isolated finite particle quantum systems in terms of topological invariant, Chern number [3], will be discussed on concrete molecular and atomic examples.

Topological Equivalence Classification of Balanced Linearly Separable Boolean Functions on n-Dimensional Hypercube

2021 ◽  
Vol 31 (02) ◽  
pp. 2150031
Author(s):  
Qinbin He ◽  
Fangyue Chen ◽  
Wei Jin
Keyword(s):  
Spatial Structure ◽  
Boolean Functions ◽  
Conformal Transformation ◽  
Classification Algorithm ◽  
Equivalence Classes ◽  
Topological Equivalence ◽  
Novel Algorithm

The concept of conformal transformation is proposed through the study of the spatial structure of [Formula: see text]-dimensional hypercubes. Based on conformal transformation, a novel algorithm, called topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the topological equivalence classes of all balanced linearly separable Boolean functions and the number of Boolean functions in each of the topological equivalence classes are obtained. In addition, the properties of conformal transformation also show an application prospect for decomposing nonlinearly separable Boolean functions.

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