Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

We first obtain obstructions to $G$-invariant Riemannian metrics with positive scalar curvature, both in the $K$-theory of the maximal or reduced group $C^{*}$-algebra of $G$, and in terms of numerical topological invariants generalising the $\hat{A}$-genus. If $G$ is a connected Lie group, then these obstructions vanish under a mild assumption, as shown in [21]. In that case, we construct $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

Our most general obstruction result is the following.

See Theorem 2.1. In the case where $M$ is the universal cover of a manifold $X$ and $G$ is its fundamental group, this becomes Theorem A in [9].

Theorem 1.2 implies topological obstructions to $G$-invariant Riemannian metrics with positive scalar curvature. Let $g\in G$, and let $H^{g}\subset H$ be the fixed point set of $g$. Let $\mathcal{N}\to H^{g}$ be its normal bundle in $H$, and let $R^{\mathcal{N}}$ be the curvature of the Levi–Civita connection restricted to $\mathcal{N}$. The $g$-localised $\hat{A}$-genus of $H$ is

for a cutoff function $c^{g}$ on $X^{g}$. If $G$ acts freely, then $\hat{A}_{g}(H)=0$ if $g\neq e$, and $\hat{A}_{e}(H)$ is the $\hat{A}$-genus of $H/G$. In general, for example in the orbifold case, $\hat{A}_{g}(H)$ may be nonzero for different $g$.

See Theorem 2.4.

Theorem 2 in [21] is a generalisation of Atiyah and Hirzebruch’s vanishing result [3] to the noncompact case. It states that, if $G$ is connected, and not every stabiliser of its action on $H$ is maximal compact, then $\operatorname{index}_{G}(D^{H})=0$. This implies that $\hat{A}_{g}(H)=0$ as well. In view of Theorem 1.2 and Corollary 1.3, this makes it a natural question if $M$ admits a $G$-invariant Riemannian metric with positive scalar curvature if $G$ is a connected Lie group. The answer, given in the present paper, turns out to be yes under a certain equivariant bounded geometry assumption.

Suppose that $G$ is a connected Lie group, and let $K<G$ be a maximal compact subgroup. Abels’ slice theorem [1] implies that there is a diffeomorphism $M\cong G\times_{K}N$, for a $K$-invariant submanifold $N\subset M$. Consider the infinitesimal action map

mapping $(y,X)\in N\times\mathfrak{k}$ to $\left.\frac{d}{dt}\right|_{t=0}\exp(tX)y$. The action by $K$ on $N$ is said to have no shrinking orbits with respect to a $K$-invariant Riemannian metric on $N$, if the pointwise operator norm of $\varphi$ as a map from $\mathfrak{k}$ to a tangent space is uniformly positive outside a neighbourhood of the fixed point set $N^{K}$. We say that $N$ has $K$-bounded geometry if it has bounded geometry and no shrinking orbits.

In the compact case, the Atiyah–Hirzebruch vanishing theorem [3] implies that the obstruction $\hat{A}(N)$ to Riemannian metrics of positive scalar curvature vanishes if $K$ acts nontrivially on $N$. Then Lawson and Yau [27] constructed such metrics, under mild conditions. In a similar way, Theorem 1.4 complements the vanishing result in [21].

Two effective sources of index-theoretic obstructions to metrics of positive scalar curvature on noncompact manifolds are coarse index theory and Callias-type index theory. For some results involving coarse index theory, see for example [38] and the literature on the coarse Novikov conjecture, in particular [12] for the equivariant setting we are interested in here. We will use Callias-type index theory.

Not assuming that $M$ is $\operatorname{Spin}$ for now, and letting $G$ be any locally compact group, we consider a $G$-equivariant Dirac-type operator $D$ on a $G$-equivariant vector bundle $S\to M$. A Callias-type operator is of the form $D+\Phi$, for a $G$-equivariant endomorphism $\Phi$ of $S$ such that $D+\Phi$ is uniformly positive outside a cocompact set. Then this operator has an index $\operatorname{index}_{G}(D+\Phi)\in K_{*}(C^{*}(G))$, constructed in [15]. (See Theorem 4.2 in [17] for a realisation of this index in terms of coarse geometry.)

See Theorem 3.4. Versions of this result where $G$ is trivial were proved in [2, 5, 7, 8, 25]. Versions for operators on bundles of modules over operator algebras were proved in [6, 9]. Parts of our proof of Theorem 1.5 are based on a similar strategy as the proof of the index theorem in [9].

We deduce Theorem 2.1 from Theorem 1.5. This approach is an equivariant generalisation of the obstructions to metrics of positive scalar obtained in [2, 7, 9].

If $g\in G$, then under conditions, there is a subalgebra $A(G)\subset C^{*}(G)$ such that $K_{*}(A(G))=K_{*}(C^{*}(G))$, and there is a well-defined trace $\tau_{g}$ on $A(G)$ given by

where $Z$ is the centraliser of $g$. In various settings, including the three cases in Corollary 1.3, there are index formulas for the number $\tau_{g}(\operatorname{index}_{G}(D))$, see [24, 41, 40]. These index formulas imply that, in the setting of Theorem 1.2,

Hence Theorem 1.2 implies Corollary 1.3.

Theorem 1.4 is proved via a generalisation of Lawson and Yau’s arguments [27], together with a result from [17] that allows one to induce up metrics of positive scalar curvature from $N$ to $M=G\times_{K}N$.

Apart from using Theorem 1.5 to prove Theorem 2.1, we obtain some further applications, on the image (Corollary 7.1) and cobordism invariance (Corollary 7.2) of the analytic assembly map; on the Callias-type index of $\operatorname{Spin}^{c}$-Dirac operators (Corollary 7.3); on induction of Callias-type indices from compact groups to noncompact groups (Corollary 7.4); and on the $\operatorname{Spin}^{c}$-version [31] of the quantisation commutes with reduction problem [14, 28, 30, 39] for $\operatorname{Spin}^{c}$-Callias type operators (Corollary 7.5).

We state our obstruction and existence results in Section 2: Theorem 2.1, Corollary 2.4 and Theorem 2.9. In Section 3, we state the equivariant Callias-type index theorem, Theorem 3.4. Theorem 3.4 is proved in Sections 4 and 5. We then deduce Theorem 2.1 and Corollary 2.4 in Subsection 6.1. Theorem 2.9 is proved in Subsection 6.2. In Section 7, we obtain some further applications of the Callias-type index theorem, Corollaries 7.1–7.5.

HG was supported in part by funding from the National Science Foundation under grant no. 1564398. PH thanks Guoliang Yu and Texas A&M University for their hospitality during a research visit. VM was supported by funding from the Australian Research Council, through the Australian Laureate Fellowship FL170100020.

All manifolds, vector bundles, group actions and other maps between manifolds are implicitly assumed to be smooth. If a Hilbert space $H$ is mentioned, the inner product on that space will be denoted by $(\relbar,\relbar)_{H}$, and the corresponding norm by $\|\cdot\|_{H}$. Spaces of continuous sections are denoted by $\Gamma$; spaces of smooth sections by $\Gamma^{\infty}$. Subscripts $c$ denote compact supports.

If $G$ is a group, and $H$ is a subgroup acting on a set $S$, then we write $G\times_{H}S$ for the quotient of $G\times S$ by the $H$-action given by

for $h\in G$, $g\in G$ and $s\in S$. If $S$ is a manifold, $G$ is Lie group and $H$ is a closed subgroup, then this action is proper and free, and $G\times_{H}S$ is a manifold.

A continuous group action, and also the space acted on, is said to be cocompact if the quotient space is compact.

For a proper, cocompact action by $G$ on a manifold $N$, a $G$-equivariant elliptic differential operator $D$ on $N$ has an equivariant index $\operatorname{index}_{G}(D)\in K_{*}(C^{*}(G))$ defined by the analytic assembly map [4]. Here $C^{*}(G)$ is the maximal or reduced group $C^{*}$-algebra of $G$, and the index takes values in its even $K$-theory if $D$ is odd with respect to a grading, and in odd $K$-theory otherwise.

Our most general obstruction result is the following.

We will deduce this result from an equivariant index theorem for Callias-type operators, Theorem 3.4, which may be of independent interest and has some other applications as well.

Theorem 2.1 implies a set of topological obstructions to $G$-invariant positive scalar curvature metrics on $M$. Let $X$ be any Riemannian manifold on which $G$ acts properly, isometrically and cocompactly. Let $g\in G$, and let $X^{g}\subset X$ be its fixed point set. (Properness of the action implies that $X^{g}=\emptyset$ if $g$ is not contained in a compact subset of $G$.) Let $\mathcal{N}\to X^{g}$ be the normal bundle to $X^{g}$ in $X$. The connected components of $X^{g}$ are submanifolds of $X$ of possibly different dimensions, so the rank of $\mathcal{N}$ may jump between these components. In what follows, we implicitly apply all constructions to the connected components of $X^{g}$ and add the results together.

By a cutoff function we will mean a smooth function ${c}\colon M\rightarrow[0,1]$ such that $\operatorname{supp}({c})$ has compact intersection with each $G$-orbit, and for each $x\in M$ we have

We will also use cutoff functions for other group actions, which are defined analogously.

Let $R^{\mathcal{N}}$ be the curvature of the Levi–Civita connection restricted to $\mathcal{N}$. Let $\hat{A}(X^{g})$ be the $\hat{A}$-class of $X^{g}$. Let $Z_{G}(g)<G$ be the centraliser of $g$. Let $c^{g}$ be a cutoff function for the action by $Z_{G}(g)$ on $X^{g}$.

If $g=e$, then

is the $L^{2}$-$\hat{A}$ genus of $X$ used in [41]. If $G$ is discrete and acts properly and freely on $X$, then $\hat{A}_{e}(X)=\hat{A}(X/G)$.

Theorem 2 in [21] is a generalisation of Atiyah and Hirzebruch’s vanishing theorem [3] to actions by noncompact groups. It states that if $G$ is a connected Lie group, and not all stabilisers of the action by $G$ on $H$ are maximal compact, then $\operatorname{index}_{G}(D^{H})=0$. So in this setting, the obstructions in Theorem 2.1 and Corollary 2.4 vanish. This makes it a natural question whether Riemannian metrics as in Theorem 2.1 exist if $G$ is a connected Lie group. A partial affirmative answer to that question is given in Subsection 2.2.

This also means that the natural place to look for examples and applications where Theorem 2.1 and Corollary 2.4 yield nontrivial obstructions is the setting where $G$ has infinitely many connected components. (The vanishing result generalises directly to the case where $G$ has finitely many connected components.) As noted in Remark 2.2, Theorem 2.1 implies Theorem A in [9], in the case where $G$ is the fundamental group of $M/G$ and $M$ is its universal cover. More generally, if $G$ is discrete, then $M/G$ is an orbifold. Then Theorem 2.1 and Corollary 2.4 lead to obstructions to orbifold metrics on $M/G$ with nonnegative scalar curvature, and positive scalar curvature near the sub-orbifold $N/G$. If $G$ acts freely, then $\hat{A}_{g}(H)$ is zero if $g\not=e$ (and $\hat{A}_{e}(H)=\hat{A}(H/G)$), but in the orbifold case the localised $\hat{A}$-genera for nontrivial elements $g$ are additional obstructions to positive scalar curvature.

In this subsection, we suppose that $G$ is a connected Lie group. As pointed out at the end of the previous subsection, for such groups the obstructions to $G$-invariant Riemannian metrics with positive scalar curvature in Theorem 2.1 and Corollary 2.4 vanish under a mild assumption on the action, so it is natural to investigate existence of such metrics.

If $G$ is connected, Abels’ global slice theorem [1] implies we have a diffeomorphism $M\cong G\times_{K}N$, for a $K$-invariant submanifold $N\subset M$. Our existence result, Theorem 2.9, supposes that such a slice $N$ has $K$-bounded geometry, a notion introduced in Definition 2.8.

Suppose a compact, connected Lie group $K$ acts isometrically on a complete Riemannian manifold $(N,g_{N})$. Let $b$ denote a bi-invariant Riemannian metric on $K$. For each $y\in N$ we have a linear map

defined by $\varphi_{y}(X):=\left.\frac{d}{dt}\right|_{t=0}\exp(tX)y$, for $X\in\mathfrak{k}$. Define a pointwise norm function

where $\|\,\cdot\|$ denotes the linear operator norm with respect to $b$ and $g_{N}$.

We remark that the condition of the action having no shrinking orbits is independent of $b$.

Now let us define the notion of $K$-bounded geometry, which is a strengthening of the standard notion of bounded geometry.

Our main existence result, proved in Subsection 6.2, is the following.

This result may be viewed as a strengthening of the vanishing of the obstructions to $G$-invariant metrics of positive scalar curvature in Theorem 2.1 and Corollary 2.4 in the case of connected Lie groups, by the result in [21], in the same way that Lawson and Yau’s [27] construction of metrics of positive scalar curvature strengthens the vanishing of the $\hat{A}$-genus as in Atiyah and Hirzebruch’s vanishing theorem [3] in the compact case. (See the diagram on page 233 of [27].)

From now on, $M$ will be a complete Riemannian manifold on which $G$ acts properly and isometrically. Let $S\to M$ denote a $\mathbb{Z}/2$-graded, $G$-equivariant Clifford module over $M$, and $D$ an odd-graded Dirac operator on $\Gamma^{\infty}(S)$, associated to a $G$-invariant Clifford connection on $S$ via the Clifford action by $TM$ on $S$.

Let $\Phi$ be an odd, $G$-equivariant, fibrewise Hermitian vector bundle endomorphism of $S$.

In the rest of the paper, we will use the following notation. Let $\mu$ denote the modular function on $G$. Let $C^{*}(G)$ be either the reduced or maximal group $C^{*}$-algebra of $G$. Let $G$ act on sections of the bundle $S$ by

for $s$ a section, $g\in G$ and $x\in M$.

Equip the space $\Gamma_{c}(S)$ with a right $C_{c}(G)$-action defined by

and a $C_{c}(G)$-valued inner product defined by

for $s_{1},s_{2}\in\Gamma^{\infty}_{c}(S)$, $b\in C_{c}(G)$, $g\in G$, and $x\in M$. Let $\mathcal{E}$ be the Hilbert $C^{*}(G)$-module completion of $\Gamma^{\infty}_{c}(S)$ with respect to this structure.

The definition of the equivariant index of $G$-Callias-type operators is based on the following result, Theorem 4.19 in [15].

For details about the definition of the operator $F$, we refer to Definition 4.11 in [15].

In the setting of the previous subsection, we now suppose that $S=S_{0}\oplus S_{0}$ for an ungraded, $G$-equivariant Clifford module $S_{0}$ over $M$, where the first copy of $S_{0}$ is the even part of $S$, and the second copy is the odd part. Suppose that

for a Dirac operator $D_{0}$ on $S_{0}$, and that

for a Hermitian endomorphism $\Phi_{0}$ of $S_{0}$. (The conditions on $D\Phi+\Phi D$ then become conditions on $[D_{0},\Phi_{0}]$.)

Let $Z$ be as in Definition 3.1. Let $M_{-}\subset M$ be a $G$-invariant, cocompact subset containing $Z$ in its interior, such that $N\coloneqq\partial M_{-}$ is a smooth submanifold of $M$. Let $M_{+}$ be the closure of the complement of $M_{-}$, so that $N=M_{-}\cap M_{+}$ and $M=M_{-}\cup M_{+}$. In this and similar settings, we write

By (3.1), the restriction of $\Phi_{0}$ to $N$ is fibrewise invertible. Let $S^{N}_{+}\to N$ and $S^{N}_{-}\to N$ be its positive and negative eigenbundles. (These are vector bundles, even though eigenbundles for single eigenvalues may not be.) Clifford multiplication by the unit normal vector field $\hat{n}$ to $N$ pointing into $M_{+}$, times $-i$, defines $G$-invariant gradings on $S^{N}_{\pm}$.

Let $\nabla^{S_{0}}$ be the Clifford connection on $S_{0}$ used to define $D_{0}$. By restriction and projection, it defines connections $\nabla^{S^{N}_{\pm}}$ on $S^{N}_{\pm}$. The Clifford action by $TM|_{N}$ on $S_{0}|_{N}$ preserves $S^{N}_{\pm}$ by the first condition in Definition 3.1; see also Remark 1.2 in [2]. Hence the connections $\nabla^{S^{N}_{\pm}}$ define Dirac operators $D^{S^{N}_{\pm}}$ on $\Gamma^{\infty}(S^{N}_{\pm})$. These operators are odd-graded. Because $N$ is cocompact, $D^{S^{N}_{+}}$ has an equivariant index

defined by the analytic assembly map [4].

Versions of this result where $G$ is trivial were proved in [2, 5, 7, 8, 25]. Versions for operators on bundles of modules over operator algebras are proved in [6, 9].

There are various index theorems for the the image of the right hand side of (3.7) under traces [24, 41, 40] or pairings with higher cyclic cocycles [23, 33, 34]. Via these results, Theorem 3.4 yields topological expressions for the corresponding images of the left hand side of (3.7). The results in [24, 41, 40] will be used to deduce Corollary 2.4 from Theorem 2.1.

We start by recalling the definition of Sobolev Hilbert $C^{*}(G)$-modules from [15]. Let $M$, $G$, $S$ and $D$ be as in Subsection 3.1.

The module $\mathcal{E}$ defined above Theorem 3.2 equals $\mathcal{E}^{0}$. The following version of the Rellich lemma holds for Sobolev modules (Theorem 3.12 in [15]).

We will state and prove a homotopy invariance result, Proposition 4.9, for the index in Definition 3.3, that will be of use later. A hypothesis in this result is that a certain vector bundle endomorphism defines adjointable operators on the Sobolev modules $\mathcal{E}^{0}$ and $\mathcal{E}^{1}$. In order to check this condition in some geometric situations relevant to us, we will need Propositions 4.3 and 4.4 below.

The proof uses the next lemma. To state it, let $H^{1}(S)$ be the completion of $\Gamma_{c}^{\infty}(S)$ with respect to the inner product