Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Guo, Hao

doi:10.1017/s0004972718001338

Cited by 3 publications

(10 citation statements)

References 32 publications

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“…One of the key properties of Callias-type operators in the equivariant setting is that the their indices can be calculated by localising to a cocompact subset of the manifold [9,Theorem 3.4]. In the projective setting, a similar result holds.…”

Section: Localisation Of Projective Callias-type Indicesmentioning

confidence: 93%

“…(5.10) This is the projective analogue of the equivariant Callias-type index theorem [9,Theorem 3.4]. The proof is analogous to that in the untwisted setting, once we make the following modifications:…”

Section: Localisation Of Projective Callias-type Indicesmentioning

confidence: 94%

“…The indices of such operators can be defined either via Roe algebras, similar to what was done in subsection 3.3, or using Hilbert C * r (Γ, σ)-modules. We will take the latter approach in order to frame the discussion in parallel with that in [9] for the untwisted case.…”

Section: Projectively Invariant Callias-type Operatorsmentioning

confidence: 99%

“…Equivalently, this index can be formulated as in (5.3), using a bounded transform on a Hilbert module. As mentioned previously, we will adopt the latter in order to follow more closely the exposition of [9].…”

Section: Localisation Of Projective Callias-type Indicesmentioning

confidence: 99%

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Higher localised $\hat{A}$-genera for proper actions and applications

Guo¹,

Mathai²

2021

Preprint

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For a finitely generated discrete group Γ acting properly on a spin manifold M , we formulate new topological obstructions to Γ-invariant metrics of positive scalar curvature on M that take into account the cohomology of the classifying space BΓ for proper actions.In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher A-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of Γ-invariant positive scalar curvature on M . For classes arising from the subring of H * (BΓ, R) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted L 2 -Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of Mathai [15], which provided a partial answer to a conjecture of Gromov-Lawson on higher A-genera.If M is non-cocompact, we obtain obstructions to M being a partitioning hypersurface inside a non-cocompact Γ-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [10], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.2010 Mathematics Subject Classification. 46L80, 58B34, 53C20. Key words and phrases. Higher index, positive scalar curvature, fixed point theorem, coarse geometry, twisted Roe algebra, quantitative index.H.G. and V.M.

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Section: Localisation Of Projective Callias-type Indicesmentioning

confidence: 93%

Section: Localisation Of Projective Callias-type Indicesmentioning

confidence: 94%

Section: Projectively Invariant Callias-type Operatorsmentioning

confidence: 99%

Section: Localisation Of Projective Callias-type Indicesmentioning

confidence: 99%

See 2 more Smart Citations

Higher localised $\hat{A}$-genera for proper actions and applications

Guo¹,

Mathai²

2021

Preprint

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“…from [18], where C * r G is the reduced group C * -algebra of G. This index was defined in [18] in a more general setting, and applied to, for example, Callias-type operators and positive scalar curvature [19] and the quantisation commutes with reduction problem [20].…”

Section: Introductionmentioning

confidence: 99%

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

Hochs

Wang

2022

Math. Z.

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Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) in the K-theory of the reduced group $$C^*$$ C ∗ -algebra $$C^*_rG$$ C r ∗ G of G. This is a common generalisation of the Baum–Connes analytic assembly map and the (equivariant) Atiyah–Patodi–Singer index. In part I of this series, a numerical index $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) was defined for an element $$g \in G$$ g ∈ G , in terms of a parametrix of D and a trace associated to g. An Atiyah–Patodi–Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, $$\begin{aligned} \tau _g({{\,\mathrm{index}\,}}_G(D)) = {{\,\mathrm{index}\,}}_g(D), \end{aligned}$$ τ g ( index G ( D ) ) = index g ( D ) , for a trace $$\tau _g$$ τ g defined by the orbital integral over the conjugacy class of g. This implies that the index theorem from part I yields information about the K-theoretic index $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) . It also shows that $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) is a homotopy-invariant quantity.

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An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the $K$-theoretic index

Hochs¹,

Wang²,

Wang³

2020

Preprint

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Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index index G (D) in the K-theory of the reduced group C * -algebra C * r G of G. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index index g (D) was defined for an element g ∈ G, in terms of a parametrix of D and a trace associated to g. An Atiyah-Patodi-Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions,for a trace τ g defined by the orbital integral over the conjugacy class of g. This implies that the index theorem from part I yields information about the K-theoretic index index G (D). It also shows that index g (D) is a homotopy-invariant quantity.

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Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

Cited by 3 publications

References 32 publications

Higher localised $\hat{A}$-genera for proper actions and applications

Higher localised $\hat{A}$-genera for proper actions and applications

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the $K$-theoretic index

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