Hopf cyclic cohomology and Hodge theory for proper actions

Tang, Xiang; Yao, Yi-Jun; Zhang, Weiping

doi:10.4171/jncg/138

Cited by 10 publications

(8 citation statements)

References 13 publications

(23 reference statements)

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“…Essentially, arguments of the same spirit used in this appendix have appeared in [28]. We collect it here for the sake of convenience.…”

Section: Appendicesmentioning

confidence: 99%

Elliptic boundary value problem on non-compactG-manifolds

Wang

2017

Int. J. Math.

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In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property of the global elliptic boundary value problems of the Atiyah-Patodi-Singer type on a non-compact manifold is studied.Elliptic boundary value problem on non-compact G-manifolds 2 on which a locally compact group G acts symplectically. Suppose the group action is both proper and cocompact. Then we prove that an indices equality of type (1.2) still holds on M. 2 The main difficulty for such a generalization is to find proper definitions for both sides of (1.2) in the non-compact setting. On the left hand side, an index introduced by Hochs and Mathai [17] is a natural candidate. Therefore, the main effort of this paper centers around the right hand side of (1.2), i.e., establishing the well-definedness of the APS type index for non-compact manifolds. Admittedly, this is more or less of technical flavor.Indeed, let (M, g TM ) be an even dimensional non-compact Spin c Riemannian manifold with boundary. To carry out the usual argument about L 2 sections, one relies on a cut-off function, say f , as in [24]. Let S(TM) = S + (TM) ⊕ S − (TM) be the spinor bundle and E be the coefficient Hermitian vector bundle respectively. Suppose that Γ(M, S(TM) ⊗ E) G is the space of invariant smooth sections. An orthogonal projection operator P f is defined by f on the space of L 2 sections of S(TM) ⊗ E, whose range, denoted by H 0 f (M, S(TM) ⊗ E) G , is the closure of f Γ(M, S(TM) ⊗ E) G in the space of L 2 sections. In the same fashion, higher Sobolev spaces H i f (M, S(TM) ⊗ E) G associated to f are also defined.Roughly, two aspects should be checked when one tries to adapt the definition of the APS type index in order to take f into consideration. Firstly, it's well-known that the APS type index involves a kind of global boundary condition, the so-called APS boundary condition, whose definition relies on the spectral projection of the boundary operator. In this paper, with the help of f , certain spectral projection, denoted by P ≥0, f , plays a similar role on a non-compact manifold. Secondly, the Fredholm property of the Dirac operator with the APS boundary condition should be reestablished. For this purpose, we will apply the elementary functional analysis method as in [1] and [4]. More concretely, our strategy is as follows. A partial result, the Fredholmness of the boundary value problem with the product structure assumption, is proved in Subsection 4.1. Then the result is extended to the general case in Subsection 4.2, using the Rellich perturbation theorem about self-adjoint operators.Put it briefly, the major technical problem is the solution of the following boundary value problem on a non-compact manifold.

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“…Essentially, arguments of the same spirit used in this appendix have appeared in [28]. We collect it here for the sake of convenience.…”

Section: Appendicesmentioning

confidence: 99%

Elliptic boundary value problem on non-compactG-manifolds

Wang

2017

Int. J. Math.

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“…Corollary 2 is a special case of the following more general result which is part of Theorems 1.1 and 4.1 of [22]. The proofs are independent.…”

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confidence: 92%

“…For an action of a Lie group G on a manifold M we denote by H • dR,G (M ) the cohomology of the complex of G-invariant differential forms. Following [22], if the cohomology groups H p dR,G (M ) are finite dimensional then we define the Euler characteristic of the G-action on M by…”

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confidence: 99%

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The Euler Characteristic Of A Transitive Lie Algebroid

Waldron¹

2019

Preprint

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We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid A over a compact manifold M vanishes unless A = T M , and prove a general Künneth formula. As applications we give a short proof of a vanishing result for the Euler characteristic of a principal bundle calculated using invariant differential forms, and show that the cohomology of certain Lie algebroids are exterior algebras. The latter result can be seen as a generalization of Hopf's theorem regarding the cohomology of compact Lie groups.

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“…By Proposition 3.1 of [64] (adapted to a left Haar measure), P χ c is the orthogonal projection L 2 (E) → L 2 c (E) G . Now for each t ∈ [0, 1], define a linear map P χ t c on L 2 (E) ∋ µ by P χ t c µ = c(x) G (χ t ) −1 (g)(c(g −1 x) 2 ) dg G (χ t ) −1 (g)c(g −1 x)gµ(g −1 x) dg.…”

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confidence: 99%

Positive scalar curvature and Poincaré duality for proper actions

Guo

Mathai

Wang

2019

J. Noncommut. Geom.

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For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's results, and an analogue of Petrie's conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincaré duality between G-equivariant K-homology and K-theory, observing that Poincaré duality does not necessarily hold for general G.

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Hopf cyclic cohomology and Hodge theory for proper actions

Cited by 10 publications

References 13 publications

Elliptic boundary value problem on non-compactG-manifolds

Elliptic boundary value problem on non-compactG-manifolds

The Euler Characteristic Of A Transitive Lie Algebroid

Positive scalar curvature and Poincaré duality for proper actions

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