scholarly journals Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Paul Antony Otieno ◽  
Jean Baptiste Gatsinzi ◽  
Vitalis Onyango-Otieno
Keyword(s):  
Cohomology Algebra ◽  
Mapping Spaces ◽  
Rational Cohomology ◽  
Natural Inclusion ◽  
Sullivan Models

We consider the complex Grassmannian Grk,n of k-dimensional subspaces of ℂn. There is a natural inclusion in,r:Grk,n↪Grk,n+r. Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in,r in the space of mappings from Grk,n to Grk,n+r for r≥1 and in particular to show that the cohomology of mapGrn,k,Grn,k+r;in,r contains a truncated algebra ℚx/xr+n+k2−nk, where x=2, for k≥2 and n≥4.

Rational Homotopy Types with the Rational Cohomology Algebra of Stunted Complex Projective Space

1992 ◽  
Vol 44 (6) ◽  
pp. 1241-1261 ◽  
Author(s):  
Gregory Lupton ◽  
Ronald Umble
Keyword(s):  
Projective Space ◽  
Spectral Sequence ◽  
Complex Projective Space ◽  
Classification Theory ◽  
Cohomology Algebra ◽  
Homotopy Equivalence ◽  
Rational Homotopy ◽  
Homotopy Types ◽  
Rational Cohomology ◽  
Complex Projective

AbstractWe consider the number of spaces, up to rational homotopy equivalence, which have rational cohomology algebra isomorphic to that of stunted complex projective space . Using a classification theory due to Schlessinger and Stasheff, we determine the number of rational homotopy types with rational comology algebra isomorphic to , for any given n and k. The necessary computations make use of a spectral sequence introduced by the second named author.

scholarly journals Holomorphic Cohomology of Complex Analytic Linear Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 531-542 ◽  
Author(s):  
G. Hochschild ◽  
G. D. Mostow
Keyword(s):  
Linear Group ◽  
Structure Theory ◽  
Linear Groups ◽  
Dimensional Representation ◽  
Lie Algebra Cohomology ◽  
Analytic Group ◽  
Finite Dimensional Representation ◽  
Semidirect Products ◽  
Finite Dimensional ◽  
Rational Cohomology

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

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 Manifolds over function algebras and mapping spaces

1989 ◽  
Vol 41 (2) ◽  
pp. 263-282 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Mapping Spaces ◽  
Function Algebras

Mapping Spaces and Liftings for Operator Spaces

1994 ◽  
Vol s3-69 (1) ◽  
pp. 171-197 ◽  
Author(s):  
Edward G. Effros ◽  
Zhong-Jin Ruan
Keyword(s):  
Operator Spaces ◽  
Mapping Spaces
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