Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature
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Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature Hao Guo1,2 Received: 24 April 2019 © Mathematica Josephina, Inc. 2019
Abstract We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G-equivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds. Keywords Positive scalar curvature · Equivariant index · Callias operator Mathematics Subject Classification Primary 58B34 · Secondary 58J20, 19K35, 19L47, 19K56, 47C15
1 Introduction It is well known that a Dirac operator D on a non-compact manifold M is not in general Fredholm, since the usual version of the Rellich lemma fails in this setting. Nevertheless, it is possible to modify D so as to make it Fredholm but still remain within the class of Dirac-type operators. One such modification is a Callias-type operator, which was initially studied by Callias [11] on M a Euclidean space, before being generalised to the setting of Riemannian manifolds by others [3,7,9,10]. A Calliastype operator may be written as B = D + , where is an endomorphism making B invertible at infinity. As in [10], one may form the order-0 bounded transform F of B, defined formally by
B
Hao Guo [email protected]; [email protected]
1
School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia
2
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
123
H. Guo
F := B(B 2 + f )−1/2 , where f is a compactly supported function. The formal computation F 2 − 1 = − f (B 2 + f )−1 shows that F is a Fredholm operator, since multiplication by the compactly supported function f defines a compact operator between Sobolev spaces H i → H j for all i > j. More generally, if M is a manifold with a proper, non-cocompact action of a Lie group G, a G-invariant Dirac operator on M need not be Fredholm in the sense of C ∗ -algebras. From the point of view of the equivariant index map in K K -theory, the lack of a general index for elements of the equivariant analytic K -homology K ∗G (M) can be traced back to the lack of a counterpart to the canonical projection in K K (C, C0 (M) G) defined by a compactly supported cut-off function when M/G is compact. In this paper we introduce G-equivariant analogues of the Sobolev spaces H i and the Rellich lemma. With these tools, the formal computation above can be made to work in the non-cocompact setting for any operator whose square is positive outside of a cocompact set. We establish that such operators have a G-equ
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