Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions

Kankaanrinta, Marja

doi:10.2140/agt.2007.7.1

Cited by 22 publications

(23 citation statements)

References 13 publications

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“…By [Kan,Theorem 3.5] there exists a G-invariant open neighborhood U of ∂M and a Gequivariant diffeomorphism ψ : ∂M × (−∞, 0] → U such that ψ(y, 0) = y for every y ∈ ∂M and the action of G is given by…”

Section: An Equivariant Collarmentioning

confidence: 99%

Equivariant APS index for Dirac operators of non-product type near the boundary

Braverman¹,

Maschler²

2019

Indiana Univ. Math. J.

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We consider a generalized Atiyah-Patodi-Singer boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type operators. In particular, we get a formula for the equivariant signature of a local system over a manifold with boundary. In case of a trivial local system, our formula can be viewed as a new way to compute the infinitesimal equivariant eta-invariant of S. Goette. We explicitly compute all the terms in this formula, which involve the equivariant Hirzebruch L-form and its transgression, for four-dimensional Kähler manifolds admitting a special Kähler-Ricci potential (SKR metrics), a class including many Kähler conformally Einstein manifolds, in the case where the boundary is given as the zero level set of this Killing potential. In the case of SKR metrics which are local Kähler products, these terms are zero, and we obtain a vanishing result for the infinitesimal equivariant eta invariant.

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Section: An Equivariant Collarmentioning

confidence: 99%

Equivariant APS index for Dirac operators of non-product type near the boundary

Braverman¹,

Maschler²

2019

Indiana Univ. Math. J.

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“…We similarly have that, for a tubular neighbourhood N pF q -DpνpF, M qq of F in M , τ pM q| N pF q -p˚τ pF q ' p˚νpF, M q for the restriction of p to F . Therefore, we may identify DpνpF, M qq Ă BDpξq with N pF q Ă Mˆt1u Ă MˆI and using standard equivariant gluing and smoothing techniques, see [18], obtain a tangentially stably complex T n -manifold whose boundary is pMˆt0uq \ M 1 . Thus M 1 also represents x in Ω U:T n m and its fixed point set has one less component of positive dimension, that is…”

Section: Equivariant Complex Bordismmentioning

confidence: 99%

Torus manifolds in equivariant complex bordism

Darby

2015

Topology and its Applications

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Abstract. We restrict geometric tangential equivariant complex T n -bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant K-theory characteristic numbers, that the information encoded in the oriented torus graph associated to a stably complex torus manifold completely describes its equivariant bordism class. We also consider the role of omnioriented quasitoric manifolds in this description.

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“…To begin, we need to 'fatten'L 0 , i.e. choose a B L -equivariant tubular neighbourhood Υ ofL 0 in S 3 [19]. Let ν ǫ S 1 = {(z 1 , z 2 ) ∈ C 2 : |z 2 | ≤ ǫ} ∩ S 3 for any 0 < ǫ < 1, considering it to be the total-space of bundle over S 1 via the projection map (z 1 , z 2 ) −→ z 1 ∈ S 1 .…”

Section: Example 511 a Hyperbolic Link L With B L ≃ D 3 (Dihedral Grmentioning

confidence: 99%

An operad for splicing

Budney

2012

Journal of Topology

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A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N → N where N is a manifold. The action of this operadis an extension of the action of the operad of (j + 1)-cubes on this space defined in [4]. Moreover the action of the splicing operad encodes a version of Larry Siebenmann's [1, 27] splicing construction for knots in S 3 in the j = 1, M = D 2 case, for which we denote the splicing operad SP 3,1 . The space of long knots in R 3 (denoted K 3,1 ) was shown to be a free algebra over the 2-cubes operad with free generating subspace P ⊂ K 3,1 , the subspace of long knots that are prime with respect to the connect-sum operation [4]. One of the main results of this paper is that K 3,1 is free with respect to the splicing operad SP 3,1 action, but the free generating space is the significantly smaller space of torus and hyperbolic knots T H ⊂ K 3,1 . Moreover, we show that SP 3,1 is a free product of two operads. The first free summand of SP 3,1 is a semi-direct product C 2 ⋊ O 2 operad which is not equivalent to the framed discs operad. The second free summand of SP 3,1 is a free Σ ≀ O 2 -operad, free on Σ ≀ O 2 -spaces which encode cabling and hyperbolic satellite operations, moreover the Σ ≀ O 2 -homotopy-type of these spaces is determined by finding adapted maximal symmetry positions for hyperbolic links in S 3 . This is an in-principle explicit description of the homotopytype of the space of knots in S 3 , and modulo the rather difficult problem of determining the symmetry groups of a class of hyperbolic links and their actions on the cusps, this is a closed form description of the homotopy-type.

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Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions

Cited by 22 publications

References 13 publications

Equivariant APS index for Dirac operators of non-product type near the boundary

Equivariant APS index for Dirac operators of non-product type near the boundary

Torus manifolds in equivariant complex bordism

An operad for splicing

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