Equivariant Morse Theory for the Norm-Square of a Moment Map on a Variety

Wilkin, Graeme

doi:10.1093/imrn/rnx286

Cited by 4 publications

(7 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These approaches to the Morse inequalities have been extended in different ways from classical Morse (and Morse-Smale) functions to Morse-Bott and minimally degenerate functions [5,7,13,14,36,48,81], to Novikov inequalities for closed 1-forms [15,16,17,18,60,61,62], to allow to have boundary [2,11,23,50,51] and in suitable circumstances to be non-compact or infinite-dimensional [1,4,15,16,17,18,20,28,30,32,33,34,37,65]; some approaches using stratifications do not require to be a manifold [38,63,76,80], and discrete versions of Morse theory (cf. [35,49,57,58]) have also been studied.…”

Section: Letmentioning

confidence: 99%

Morse Theory without Non-Degeneracy

Kirwan

Penington

2021

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

show abstract

Section: Letmentioning

confidence: 99%

Morse Theory without Non-Degeneracy

Kirwan

Penington

2021

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…In this section we prove Theorem 4.4 which shows that the inclusion A ãÑ X is an equivariant cofibration of stratified spaces. In particular, the result of Corollary 4.5 shows that the homotopy equivalences in the Morse theory of [24] can be chosen to be G-equivariant.…”

Section: Constructing the Equivariant Neighbourhood Deformation Retractmentioning

confidence: 99%

“…An application of our main result is to Morse theory on singular spaces carrying a compact Lie group action (cf. [24]). Given a real analytic manifold M with the action of a Lie group G, a G-invariant closed analytic variety Z Ă M and an invariant Morse function f : M Ñ R satisfying some additional conditions, [24,Thm.…”

mentioning

confidence: 99%

Equivariant control data and neighborhood deformation retractions

Pflaum

Wilkin

2019

Methods and Applications of Analysis

Self Cite

View full text Add to dashboard Cite

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group G which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a G-stratified space carries a system of G-equivariant control data. As an application, we show that if A Ă X is a closed G-stratified subspace which is a union of strata of X, then the inclusion i : A ãÑ X is a G-equivariant cofibration. In particular, this theorem applies whenever X is a G-invariant analytic subspace of an analytic G-manifold M and A ãÑ X is a closed G-invariant analytic subspace of X. Ş0ďtăs U t for all s P p0, 1q. In particular i : A ãÑ X is a G-equivariant cofibration.We then consider the situation where X and A are G-invariant analytic subspaces of an analytic G-manifold M with A Ă X being a closed subspace. Using methods by Wall [22] we show in Theorem 3.1 that X possesses a G-invariant (B) regular stratification such that A is a union of strata. Hence our main result applies to such a G-invariant analytic pair pX, Aq. . G.W. would also like to thank the University of Colorado for their hospitality during this project. Control data compatible with a group actionMather proved in [13] that every (B) regular stratified subspace of a smooth manifold carries a system of control data. In this section we extend his result to the G-equivariant case. To this end we first introduce G-equivariant versions of stratifications, tubular neighborhoods, their isomorphisms and diffeotopies. Afterwards we prove a G-equivariant submersion theorem. This will be used to derive uniqueness and existence results for equivariant tubular neighborhoods. These tools then entail the main result of this section. Equivariant versions of stratifications and tubular neighborhoods.Definition 2.1. Suppose that a compact Lie group G acts on the total space X of a stratified space pX, Sq. The stratification S is called a G-stratification or G-invariant and pX, Sq a G-stratified space if for all g P G and x P X the set germs gS x and S gx coincide and if for each open neighborhood U with an S-inducing decomposition Z the map from a piece R P Z to gR given by the g-action is a diffeomorphism of smooth manifolds. We also say in this situation that the G-action on X is compatible with the stratification.Example 2.2. The orbit type stratification of a G-manifold M is G-invariant since the group action leaves the orbit types of the points of M invariant.Proposition 2.3. Each stratum of a G-stratified space pX, Sq is preserved by the G-action. Moreover, G acts smoothly on the strata of X.Proof. Let x be a point of a stratum S of X. Choose a decomposition Z of an open neighborhood U of x inducing the stratification S over U . Then gU is an open neighborhood of gx, and gZ a

show abstract

“…These approaches to the Morse inequalities have been extended in different ways from classical Morse (and Morse-Smale) functions to Morse-Bott and minimally degenerate functions [5,7,13,14,36,48,79], to Novikov inequalities for closed 1-forms [15,16,17,18,59,60,61], to allow to have boundary [2,11,23,50,51] and in suitable circumstances to be non-compact or infinite-dimensional [1,4,15,16,17,18,20,28,30,32,33,34,63]; some approaches using stratifications do not require to be a manifold [38,62,74,78]. For the main results in this article we will assume that is a (finite-dimensional) compact Riemannian manifold without boundary, though we will briefly consider other situations.…”

Section: Letmentioning

confidence: 99%

“…Morse theory on singular spaces (cf. for example [38] and more recently [74]) and discrete versions of Morse theory (cf. [35,49,56,57]) have also been studied.…”

Section: ∈mentioning

confidence: 99%