scholarly journals Erratum to “Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements”

2020 ◽  
pp. 1
Author(s):  
Filippo Callegaro ◽  
Michele D’Adderio ◽  
Emanuele Delucchi ◽  
Luca Migliorini ◽  
Roberto Pagaria
Keyword(s):  
Cohomology Algebra

scholarly journals Springer’s Weyl Group Representation via Localization

2017 ◽  
Vol 60 (3) ◽  
pp. 478-483 ◽  
Author(s):  
Jim Carrell ◽  
Kiumars Kaveh
Keyword(s):  
Weyl Group ◽  
Equivariant Cohomology ◽  
Group Representation ◽  
General Theorem ◽  
Torus Action ◽  
Cohomology Algebra ◽  
Reductive Algebraic Group ◽  
Point Set ◽  
Closed Subvariety ◽  
Springer Variety

AbstractLet G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra 𝔤. The Springer variety Bx is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of 𝔤 containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of Bx even though W rarely acts on Bx itself. Well-known constructions of this action due to Springer and others use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H*S (Bx), where S is a certain algebraic subtorus of G. This action descends to H*(Bx) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H*S (Bx) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the ûxed point set of a torus action on a space lifts to the full cohomology algebra of the space.

scholarly journals Extreme nonuniqueness of end-sum

2020 ◽  
pp. 1-43
Author(s):  
Jack S. Calcut ◽  
Craig R. Guilbault ◽  
Patrick V. Haggerty
Keyword(s):  
Abelian Groups ◽  
Cohomology Algebra ◽  
Homotopy Types ◽  
Proper Homotopy ◽  
Open Formula

We give explicit examples of pairs of one-ended, open [Formula: see text]-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

scholarly journals Pro-$p$ groups with few relations and universal Koszulity

2021 ◽  
Vol 127 (1) ◽  
pp. 28-42
Author(s):  
Claudio Quadrelli
Keyword(s):  
Galois Groups ◽  
Cohomology Algebra ◽  
Defining Relations

Let $p$ be a prime. We show that if a pro-$p$ group with at most $2$ defining relations has quadratic $\mathbb{F}_p$-cohomology algebra, then this algebra is universally Koszul. This proves the “Universal Koszulity Conjecture” formulated by J. Miná{č} et al. in the case of maximal pro-$p$ Galois groups of fields with at most $2$ defining relations.

scholarly journals Cohomology of p-groups of nilpotency class smaller than p

2018 ◽  
Vol 21 (2) ◽  
pp. 337-350 ◽  
Author(s):  
Antonio Díaz Ramos ◽  
Oihana Garaialde Ocaña ◽  
Jon González-Sánchez
Keyword(s):  
Prime Number ◽  
Nilpotency Class ◽  
Cohomology Algebra

AbstractLetpbe a prime number, letdbe an integer and letGbe ad-generated finitep-group of nilpotency class smaller thanp. Then the number of possible isomorphism types for the modpcohomology algebra{H^{*}(G;{\mathbb{F}}_{p})}is bounded in terms ofpandd.

scholarly journals The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces

2013 ◽  
Vol 149 (3) ◽  
pp. 481-494 ◽  
Author(s):  
François Charles ◽  
Eyal Markman
Keyword(s):  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Cohomology Algebra ◽  
Hilbert Schemes ◽  
Projective Varieties ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Symplectic Varieties

AbstractWe prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky’s theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.

An n-dimensional Klein bottle

2019 ◽  
Vol 149 (5) ◽  
pp. 1207-1221
Author(s):  
Donald M. Davis
Keyword(s):  
Euclidean Space ◽  
Homotopy Type ◽  
Klein Bottle ◽  
Stable Homotopy ◽  
Cohomology Algebra ◽  
Topological Complexity ◽  
Integral Cohomology ◽  
Planar Polygon

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

scholarly journals UnstableK-cohomology algebra is filteredλ-ring

2003 ◽  
Vol 2003 (10) ◽  
pp. 593-605 ◽  
Author(s):  
Donald Yau
Keyword(s):  
Cohomology Theory ◽  
Cohomology Algebra ◽  
Precise Formulation ◽  
Generalized Cohomology

Boardman, Johnson, and Wilson (1995) gave a precise formulation for an unstable algebra over a generalized cohomology theory. Modifying their definition slightly in the case of complexK-theory by taking into account its periodicity, we prove that an unstable algebra for complexK-theory is precisely a filteredλ-ring, and vice versa.

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