scholarly journals Examples of Surfaces with Canonical Map of Maximal Degree

2021 ◽  
Author(s):  
Ching-Jui Lai ◽  
Sai-Kee Yeung
Keyword(s):  
Maximal Degree ◽  
Canonical Map

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 RINGS OF DEFINABLE SCALARS OF VERMA MODULES

2007 ◽  
Vol 06 (05) ◽  
pp. 779-787 ◽  
Author(s):  
SONIA L'INNOCENTE ◽  
MIKE PREST
Keyword(s):  
Lie Algebra ◽  
Model Theory ◽  
Regular Ring ◽  
Verma Module ◽  
Closed Field ◽  
Verma Modules ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Canonical Map ◽  
Von Neumann Regular

Let M be a Verma module over the Lie algebra, sl 2(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U( sl 2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

scholarly journals The solution of length three equations over groups

1983 ◽  
Vol 26 (1) ◽  
pp. 89-96 ◽  
Author(s):  
James Howie
Keyword(s):  
Free Product ◽  
Cyclic Group ◽  
Normal Closure ◽  
Infinite Cyclic Group ◽  
Canonical Map ◽  
Equations Over Groups

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 462-469
Author(s):  
Bernard Alechenu ◽  
Babayo Muhammed Abdullahi ◽  
Daniel Eneojo Emmanuel
Keyword(s):  
Abelian Group ◽  
Complex Analysis ◽  
Factor Group ◽  
Roots Of Unity ◽  
Composition Series ◽  
Isomorphism Theorem ◽  
Polynomial Equations ◽  
Canonical Map ◽  
The Third ◽  
Recursive Set

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through  the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.    

scholarly journals Non-defectivity of Grassmannian bundles over a curve

2016 ◽  
Vol 27 (07) ◽  
pp. 1640002 ◽  
Author(s):  
Insong Choe ◽  
George H. Hitching
Keyword(s):  
Vector Bundles ◽  
Manuscripta Math ◽  
Maximal Degree ◽  
Geometric Proof ◽  
Secant Varieties ◽  
Grassmann Bundle ◽  
Positive Rank

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

scholarly journals Clique coverings of graphs V: maximal-clique partitions

1982 ◽  
Vol 25 (3) ◽  
pp. 337-356 ◽  
Author(s):  
N.J. Pullman ◽  
H. Shank ◽  
W.D. Wallis
Keyword(s):  
Maximal Clique ◽  
Maximal Degree ◽  
Open Problems ◽  
Partition Number

A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed – for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions - and discuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented.

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