scholarly journals LOOP GROUPS, STRING CLASSES AND EQUIVARIANT COHOMOLOGY

2011 ◽  
Vol 90 (1) ◽  
pp. 109-127 ◽  
Author(s):  
RAYMOND F. VOZZO
Keyword(s):  
Lie Group ◽  
Differential Form ◽  
Equivariant Cohomology ◽  
Loop Group ◽  
Characteristic Classes ◽  
Loop Groups ◽  
Compact Lie Group ◽  
String Class ◽  
String Classes ◽  
Odd Dimensions

AbstractWe give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG-bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.

The Cartan Model in General

2020 ◽  
pp. 167-180
Author(s):  
Loring W. Tu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Differential Form ◽  
Crucial Role ◽  
Equivariant Cohomology ◽  
Circle Action ◽  
Lie Group Action ◽  
Cartan Model ◽  
Connected Lie Group

This chapter looks at the Cartan model. Specifically, it generalizes the Cartan model from a circle action to a connected Lie group action. The chapter assumes the Lie group to be connected, because the condition that LX α‎ = 0 is sufficient for a differential form α‎ on M to be invariant holds only for a connected Lie group. It also considers the theorem that marks the transition from the Weil model to the Cartan model. It is due to Henri Cartan, who played a crucial role in the development of equivariant cohomology. The chapter then studies the Weil-Cartan isomorphism.

scholarly journals The topological nilpotence degree of a Noetherian unstable algebra

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Drew Heard
Keyword(s):  
Lie Group ◽  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Cohomological Dimension ◽  
Steenrod Algebra ◽  
Compact Groups ◽  
Compact Lie Group ◽  
Profinite Groups ◽  
Minimal Depth ◽  
Virtual Cohomological Dimension

AbstractWe investigate the topological nilpotence degree, in the sense of Henn–Lannes–Schwartz, of a connected Noetherian unstable algebra R. When R is the mod p cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian p-subgroups. By replacing centralizers of elementary abelian p-subgroups with components of Lannes’ T-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn’s result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn’s result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac–Moody groups. In fact, our results apply much more generally, for example, we establish results for p-local compact groups in the sense of Broto–Levi–Oliver, for connected H-spaces with Noetherian mod p cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson’s depth conjecture in the case of a Noetherian unstable algebra of minimal depth.

scholarly journals On the vanishing problem of string classes

1996 ◽  
Vol 61 (2) ◽  
pp. 258-266 ◽  
Author(s):  
Katsuhiko Kuribayashi
Keyword(s):  
Central Extension ◽  
Loop Group ◽  
Ring Structure ◽  
Structure Group ◽  
Universal Central Extension ◽  
String Class ◽  
String Classes

AbstractThe ordinary string class is an obstruction to lift the structure group LSpin(n) of a loop group bundle LQ → LM to the universal central extension of LSpin(n) by the circle. The vanishing problem of the ordinary string class and generalized string classes are considered from the viewpoint of the ring structure of the cohomology H*(M; R).

scholarly journals A classification of equivariant gerbe connections

2019 ◽  
Vol 21 (02) ◽  
pp. 1850001
Author(s):  
Byungdo Park ◽  
Corbett Redden
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Existence And Uniqueness ◽  
Smooth Manifold ◽  
Equivariant Cohomology ◽  
Compact Lie Group ◽  
Semisimple Lie Groups ◽  
Isomorphism Classes ◽  
Bundle Gerbe

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

An equivariant version of the Kuhn–Schwartz non-realizability theorem

2002 ◽  
Vol 133 (1) ◽  
pp. 117-124
Author(s):  
DORRA BOURGUIBA ◽  
DAGMAR M. MEYER
Keyword(s):  
Topological Space ◽  
Lie Group ◽  
Equivariant Cohomology ◽  
Steenrod Algebra ◽  
Module Structure ◽  
Compact Lie Group ◽  
Finitely Generated ◽  
Equivariant Version

The Kuhn–Schwartz non-realizability theorem states the following: if the mod p cohomology of a topological space is finitely generated as a module over the Steenrod algebra [Ascr ] then it is finite. We generalize this result to the category of G-spaces, where G is a compact Lie group, by considering the equivariant cohomology of a G-space as an object in the category of all [Ascr ]-modules with a compatible H*(BG; [ ]p)-module structure.

scholarly journals FORMAL FROBENIUS MANIFOLD STRUCTURE ON EQUIVARIANT COHOMOLOGY

1999 ◽  
Vol 01 (04) ◽  
pp. 535-552 ◽  
Author(s):  
HUAI-DONG CAO ◽  
JIAN ZHOU
Keyword(s):  
Lie Group ◽  
Equivariant Cohomology ◽  
Kähler Manifold ◽  
Frobenius Manifold ◽  
Algebra Structure ◽  
Compact Lie Group ◽  
Hamiltonian Action ◽  
Manifold Structure ◽  
Kahler Manifold ◽  
Cartan Model

For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.

On the construction of G-spaces and applications to homogeneous spaces

1970 ◽  
Vol 68 (2) ◽  
pp. 321-327 ◽  
Author(s):  
Daniel Henry Gottlieb
Keyword(s):  
Lie Group ◽  
Homogeneous Spaces ◽  
Maximal Rank ◽  
Compact Lie Group ◽  
Connected Subgroup ◽  
Odd Dimensions

In (3), the author defined the notion of a G-space. A G-space is a weaker notion than that of an H-space. The main purpose of this paper is to present various means of constructing G-spaces. As an application of some of the techniques of (3) and of this paper (though not an application of the concept of G-space) we shall prove the following theorem:Theorem. Let G be a connected compact Lie group and let H be a connected subgroup of maximal rank. Then H3(G/H; Z) = 0. In fact, the Hurewicz homomorphism is trivial for odd dimensions.

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

scholarly journals Stratifications of equivariant varieties

1977 ◽  
Vol 16 (2) ◽  
pp. 279-295 ◽  
Author(s):  
M.J. Field
Keyword(s):  
Lie Group ◽  
General Position ◽  
Higher Order ◽  
Order Conditions ◽  
Compact Lie Group ◽  
Algebraic Varieties ◽  
Equivariant Maps

Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

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