scholarly journals TOPOLOGY OF THE REPRESENTATION VARIETIES WITH BOREL MOLD FOR UNSTABLE CASES

2011 ◽  
Vol 91 (1) ◽  
pp. 55-87 ◽  
Author(s):  
KAZUNORI NAKAMOTO ◽  
TAKESHI TORII
Keyword(s):  
Configuration Space ◽  
Cohomology Group ◽  
Cohomology Ring ◽  
Picard Group ◽  
Homotopy Equivalent ◽  
Representation Variety ◽  
Stable Case ◽  
Representation Varieties

AbstractIn this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and $H^{\ast }(F_{n}(\mathbb {C}^m)) \otimes H^{\ast }(\mathrm {Flag}(\mathbb { C}^n)) \otimes \Lambda (s_{1}, \ldots , s_{n-1})$ are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

A toral configuration space and regular semisimple conjugacy classes

1995 ◽  
Vol 118 (1) ◽  
pp. 105-113 ◽  
Author(s):  
G. I. Lehrer
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Cohomology Ring ◽  
General Setting ◽  
De Rham Cohomology ◽  
Pure Braid Group ◽  
De Rham ◽  
Image Position

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration spaceThe symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

scholarly journals THE RELATIVE PICARD GROUP OF A COMODULE ALGEBRA AND HARRISON COHOMOLOGY

2005 ◽  
Vol 48 (3) ◽  
pp. 557-569 ◽  
Author(s):  
S. Caenepeel ◽  
T. Guédénon
Keyword(s):  
Hopf Algebra ◽  
Quotient Group ◽  
Cohomology Group ◽  
Picard Group ◽  
Base Extension ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
K Theory ◽  
Algebra Over A Field

AbstractLet $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible group-like elements of the coring $A\otimes H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a cosemisimple Hopf algebra over a field.

scholarly journals On distinguishing spaces not homotopy-equivalent

1994 ◽  
Vol 49 (1) ◽  
pp. 117-119
Author(s):  
M.H. Eggar
Keyword(s):  
Cohomology Ring ◽  
Product Structure ◽  
Homotopy Equivalent ◽  
Topological Spaces ◽  
Homotopy Groups ◽  
Whitehead Product ◽  
Homology Groups

A non-pathological example is given of two topological spaces which have isomorphic homotopy groups, homology groups and cohomology ring and which cannot be distinguished from each other by the Whitehead product structure. A family of examples can be constructed likewise.

scholarly journals Spider evaluation and representations of web groups

2019 ◽  
Vol 28 (04) ◽  
pp. 1950021
Author(s):  
Charles Frohman
Keyword(s):  
Irreducible Component ◽  
Rooted Tree ◽  
Fundamental Groups ◽  
Text Representation ◽  
Poincaré Polynomial ◽  
Representation Variety ◽  
Representation Varieties ◽  
The Web

The topology of [Formula: see text]-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to [Formula: see text] are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic [Formula: see text] from the root of the tree to the tip of a leaf an irreducible component [Formula: see text] of the representation variety of the web, and a graded subalgebra [Formula: see text] of [Formula: see text]. The spider evaluation of geodesic [Formula: see text] is the symmetrized Poincaré polynomial of [Formula: see text]. The spider evaluation of the web is the sum of the symmetrized Poincaré polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves.

Non-Formality of Planar Configuration Spaces in Characteristic 2

2018 ◽  
Vol 2020 (10) ◽  
pp. 3100-3129 ◽  
Author(s):  
Paolo Salvatore
Keyword(s):  
Configuration Space ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Configuration Spaces ◽  
Planar Configuration ◽  
Characteristic 2 ◽  
Obstruction Class

Abstract We prove that the ordered configuration space of four points or more in the plane has a nonformal singular cochain algebra in characteristic 2. This is proved by constructing an explicit nontrivial obstruction class in the Hochschild cohomology of the cohomology ring of the configuration space, by means of the Barratt–Eccles–Smith simplicial model. We also show that if the number of points does not exceed its dimension then a Euclidean configuration space is intrinsically formal over any ring.

THE ℚ-PICARD GROUP OF THE MODULI SPACE OF CURVES IN POSITIVE CHARACTERISTIC

2001 ◽  
Vol 12 (05) ◽  
pp. 519-534 ◽  
Author(s):  
ATSUSHI MORIWAKI
Keyword(s):  
Moduli Space ◽  
Cohomology Group ◽  
Positive Characteristic ◽  
Picard Group ◽  
Closed Field ◽  
Moduli Space Of Curves ◽  
Algebraically Closed Field ◽  
Stable Curves ◽  
Étale Cohomology ◽  
Cycle Map

In this note, we prove that the ℚ-Picard group of the moduli space of n-pointed stable curves of genus g over an algebraically closed field is generated by the tautological classes. We also prove that the cycle map to the second étale cohomology group is bijective.

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

Prognosticating the spread of covid-19 pandemic based on optimal arima estimators

2020 ◽  
Vol 20 ◽  
Author(s):  
Venuka Sandhir ◽  
Vinod Kumar ◽  
Vikash Kumar
Keyword(s):  
South Africa ◽  
Statistical Analysis ◽  
Moving Average ◽  
Arima Model ◽  
Information Criterion ◽  
Stable Case ◽  
Johns Hopkins University ◽  
Modelling Techniques ◽  
Auto Regressive ◽  
Global Threat

Background: COVID-19 cases have been reported as a global threat and several studies are being conducted using various modelling techniques to evaluate patterns of disease dispersion in the upcoming weeks. Here we propose a simple statistical model that could be used to predict the epidemiological extent of community spread of COVID-19from the explicit data based on optimal ARIMA model estimators. Methods: Raw data was retrieved on confirmed cases of COVID-19 from Johns Hopkins University (https://github.com/CSSEGISandData/COVID-19) and Auto-Regressive Integrated Moving Average (ARIMA) model was fitted based on cumulative daily figures of confirmed cases aggregated globally for ten major countries to predict their incidence trend. Statistical analysis was completed by using R 3.5.3 software. Results: The optimal ARIMA model having the lowest Akaike information criterion (AIC) value for US (0,2,0); Spain (1,2,0); France (0,2,1); Germany (3,2,2); Iran (1,2,1); China (0,2,1); Russia (3,2,1); India (2,2,2); Australia (1,2,0) and South Africa (0,2,2) imparted the nowcasting of trends for the upcoming weeks. These parameters are (p, d, q) where p refers to number of autoregressive terms, d refers to number of times the series has to be differenced before it becomes stationary, and q refers to number of moving average terms. Results obtained from ARIMA model showed significant decrease cases in Australia; stable case for China and rising cases has been observed in other countries. Conclusion: This study tried their best at predicting the possible proliferate of COVID-19, although spreading significantly depends upon the various control and measurement policy taken by each country.

scholarly journals Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

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