Compactly supported analytic indices for Lie groupoids

Index theory has as a starting point the Atiyah-Singer index theorem [AS68]: Let $D$ be an elliptic operator over a compact smooth manifold $M$, in particular $D$ has finite dimensional Kernel and Cokernel and the Fredholm index can be defined as

The map $D\mapsto ind\,D$ is completely codified by a group morphism $K^{0}(T^{*}M)\longrightarrow\mathbb{Z},$ called the analytic index of $M$. That is, if $Ell(M)$ denotes the set of elliptic pseudodifferential operators over $M$, then the following diagram is commutative:

where $Ell(M)\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}K^{0}(T^{*}M)$ is the surjective map that associates the class of the principal symbol of the operator in $K^{0}(T^{*}M)$. This is a fundamental property because it allows to use the (cohomological) properties of $K-$theory and gives stability to the index. The topological formula given by Atiyah-Singer for the analytic index is the following:

This kind of formulas give very interesting invariants of the manifold [LM89, Mel93, Sha78].

We discuss now the Lie groupoids case. This concept is central in non commutative geometry. Groupoids generalize the concepts of spaces, groups and equivalence relations. In the late 70’s, mainly with the work of Alain Connes, it became clear that groupoids appeared naturally as substitutes of singular spaces [Con79, Mac87, Ren80, Pat99]. Furthermore, Connes showed that many groupoids and algebras associated to them appeared as ‘non commutative analogues‘ of smooth manifolds to which many tools of geometry such as K-theory and Characteristic classes could be applied [Con79, Con94]. Lie groupoids became a very natural place where to perform pseudodifferential calculus and index theory, [Con79, MP97, NWX99].

The study of the indices in the groupoid case is, as we will see, more delicate than the classical case. There are new phenomena appearing. If $\mathscr{G}$ is a Lie groupoid, a $\mathscr{G}$-pseudodifferential operator is a differentiable family (see [MP97, NWX99]) of operators. Let $P$ be such an operator, the index of $P$, $ind\,P$, is an element of $K_{0}(C_{c}^{\infty}(\mathscr{G}))$. There is a diagram similar to (1) for the Lie groupoid setting:

We would like, as above, that the map $D\mapsto ind\,D\in K_{0}(C_{c}^{\infty}(\mathscr{G}))$ factor through $K^{0}(A^{*}\mathscr{G})$: this group codifies the principal symbols of all the elliptic $\mathscr{G}$-operators and carries the topological invariants. Unfortunately, the map $D\mapsto ind\,D\in K_{0}(C_{c}^{\infty}(\mathscr{G}))$ does not factor in general through the principal symbol [Con94]. Nevertheless, if we consider the $K$-theory morphism $K_{0}(C_{c}^{\infty}(\mathscr{G}))\stackrel{{\scriptstyle j}}{{\longrightarrow}}K_{0}(C^{*}_{r}(\mathscr{G}))$ induced by the inclusion $C_{c}^{\infty}(\mathscr{G})\subset C^{*}_{r}(\mathscr{G})$, then the composition

do factor through the principal symbol by a morphism $ind_{a}$, called the analytic index of $\mathscr{G}$:

In fact, the $C^{*}$-analytic index $ind_{a}$ can be completely described using the Connes tangent groupoid, [Con94, MP97, MN05, DLN06, ANS06, HS87]. We briefly recall this fact.

The tangent groupoid associated to a Lie groupoid $\mathscr{G}\rightrightarrows\mathscr{G}^{(0)}$ is a Lie groupoid

that is compatible with $A\mathscr{G}$ and $\mathscr{G}$. Its $C^{*}$algebra is a continuous field of $C^{*}-$algebras over $[0,1]$ whose fiber at zero is $C_{r}^{*}(A\mathscr{G})\cong C_{0}(A^{*}\mathscr{G})$ and $C_{r}^{*}(\mathscr{G})$ elsewhere, it is a $C^{*}$-algebraic deformation in the sense of [Lan03]. In particular it gives a short exact sequence of $C^{*}-$algebras

Now, thanks to the good properties of the $K-$theory for $C^{*}$-algebras, Monthubert and Pierrot show in [MP97] that

where $C_{r}^{*}(\mathscr{G}^{T})\stackrel{{\scriptstyle ev_{1}}}{{\longrightarrow}}C_{r}^{*}(\mathscr{G})$ is the evaluation at $1$.

If we try to adapt the above arguments for the index $ind\,D\in K_{0}(C_{c}^{\infty}(\mathscr{G}))$, we find several difficulties: For instance, the map $K_{0}(C^{\infty}_{c}(A\mathscr{G}))\longrightarrow K_{0}(C^{*}_{r}(A\mathscr{G}))=K^{0}(A^{*}\mathscr{G})$ is not an isomorphism, i.e., $K_{0}(C^{\infty}_{c}(A\mathscr{G}))$ is not the group that we would like. On the other hand, the kernel of the evaluation map $C^{\infty}_{c}(\mathscr{G}^{T})\stackrel{{\scriptstyle ev_{0}}}{{\longrightarrow}}C^{\infty}_{c}(A\mathscr{G})$ is not easy to describe, and we cannot conclude its $K_{0}$ group vanishes since the $K$-theory for general topological algebras is not necessarily homotopy invariant. In general, the morphism $K_{0}(C_{c}^{\infty}(\mathscr{G}))\stackrel{{\scriptstyle j}}{{\rightarrow}}K_{0}(C^{*}_{r}(\mathscr{G}))$ is not an isomorphism. A very simple example of this situation is $\mathbb{R}\rightrightarrows\{*\}$ with the vectorial group structure ([Con94], p.142). When the groupoid is proper we do not perceive this nuance: it is an isomorphism. This is because $C_{c}^{\infty}(\mathscr{G})$ is stable under holomorphic calculus ([CMR07, Kar78]). In general, the $K$-theory groups as $K_{0}(C_{c}^{\infty}(\mathscr{G}))$ do not satisfy in general some fundamental properties as homotopy invariance and Bott periodicity.

Our first step is to consider a ”good quotient” of $K_{0}(C_{c}^{\infty}(\mathscr{G}))$ where we are going to be able to factorize $[ind\,D]\in K_{0}(C_{c}^{\infty}(\mathscr{G}))/\sim$ through the principal symbol, using the Connes tangent groupoid.

Let us note, for each $k\in\mathbb{N}$, $K_{0}^{h,k}(\mathscr{G})$ the quotient group of $K_{0}(C_{c}^{k}(\mathscr{G}))$ by the equivalence relation induced by $\tiny{K_{0}(C_{c}^{k}(\mathscr{G}\times[0,1]))\overset{e_{0}}{\underset{e_{1}}{\rightrightarrows}}K_{0}(C_{c}^{k}(\mathscr{G}))}$. Let $K_{0}^{F}(\mathscr{G})=\varprojlim_{k}K_{0}^{h,k}(\mathscr{G})$ be the projective limit relative to the inclusions $C_{c}^{k}(\mathscr{G})\subset C_{c}^{k-1}(\mathscr{G})$. The main result of this paper is the following.

The index of the above theorem is called “the compactly supported analytic index of $\mathscr{G}$”. The construction of these indices is also based on the Connes tangent groupoid, as in case of $C^{*}$-algebras. In fact, in [CR06] we constructed an algebra of smooth functions over the tangent groupoid that will allow in this paper to perform the compactly supported indices as ”deformations”. We recall the main result of [CR06].

The compactly supported index fits in a commutative diagram of the following type:

Let us recall what a groupoid is:

Along this paper we will only deal with Lie groupoids, that is, a groupoid in which $\mathscr{G}$ and $\mathscr{G}^{(0)}$ are smooth manifolds (possibly with boundary), and $s,r,m,u$ are smooth maps (with s and r submersions, see [Mac87, Pat99]). For $A,B$ subsets of $\mathscr{G}^{(0)}$ we use the notation $\mathscr{G}_{A}^{B}$ for the subset $\{\gamma\in\mathscr{G}:s(\gamma)\in A,\,r(\gamma)\in B\}$.

We recall how to define an algebra structure in $C_{c}^{\infty}(\mathscr{G})$ using smooth Haar systems.

A Lie groupoid always posses a smooth Haar system. In fact, if we fix a smooth (positive) section of the 1-density bundle associated to the Lie algebroid we obtain a smooth Haar system in a canonical way. We suppose for the rest of the paper a given smooth Haar system given by 1-densities (for complete details see [Pat99]). We can now define a convolution product on $C_{c}^{\infty}(\mathscr{G})$: Let $f,g\in C_{c}^{\infty}(\mathscr{G})$, we set

This gives a well defined associative product.

As we mentioned in the introduction, we are going to consider some elements in the $K$-theory group $K_{0}(C_{c}^{\infty}(\mathscr{G}))$. We recall how these elements are usually defined (See [NWX99] for complete details): First we recall a few facts about $\mathscr{G}$-Pseudodifferential calculus:

A $\mathscr{G}$-$\it{Pseudodifferential}$ $\it{operator}$ is a family of pseudodifferential operators $\{P_{x}\}_{x\in\mathscr{G}^{(0)}}$ acting in $C^{\infty}_{c}(\mathscr{G}_{x})$ such that if $\gamma\in\mathscr{G}$ and

the induced operator, then we have the following compatibility condition

There is also a differentiability condition with respect to $x$ that can be found in [NWX99].

For $P$ a $\mathscr{G}$-Pseudodifferential elliptic operator there is a parametrix, $\it{i.e.}$, a $\mathscr{G}$-Pseudodifferential operator $Q$ such that $PQ-1$ and $QP-1$ belong to $C_{c}^{\infty}(\mathscr{G})$ (where we are identifying $C_{c}^{\infty}(\mathscr{G})$ with $\Psi^{-\infty}(\mathscr{G})$ as in [NWX99]). In other words, $P$ defines a quasi-isomorphism in $(\Psi^{+\infty},C_{c}^{\infty}(\mathscr{G}))$ and thus an element in $K_{0}(C_{c}^{\infty}(\mathscr{G}))$ that we call the index $ind(P)$. Similarly to the classical case, a $\mathscr{G}$-PDO operator has a principal symbol that defines an element in the $K$-theory group $K^{0}(A^{*}\mathscr{G})$.

The tangent groupoid is a particular case of a geometric construction that we describe here.

Let $M$ be a $C^{\infty}$ manifold and $X\subset M$ be a $C^{\infty}$ submanifold. We denote by $\mathscr{N}_{X}^{M}$ the normal bundle to $X$ in $M$, $\it{i.e.}$, $\mathscr{N}_{X}^{M}:=T_{X}M/TX$.

We define the following set

The purpose of this section is to recall how to define a $C^{\infty}$-structure in $\mathscr{D}_{X}^{M}$. This is more or less classical, for example it was extensively used in [HS87].

Let us first consider the case where $M=\mathbb{R}^{p}\times\mathbb{R}^{q}$ and $X=\mathbb{R}^{p}\times\{0\}$ (where we identify canonically $X=\mathbb{R}^{p}$). We denote by $q=n-p$ and by $\mathscr{D}_{p}^{n}$ for $\mathscr{D}_{\mathbb{R}^{p}}^{\mathbb{R}^{n}}$ as above. In this case we clearly have that $\mathscr{D}_{p}^{n}=\mathbb{R}^{p}\times\mathbb{R}^{q}\times\mathbb{R}$ (as a set). Consider the bijection $\psi:\mathbb{R}^{p}\times\mathbb{R}^{q}\times\mathbb{R}\rightarrow\mathscr{D}_{p}^{n}$ given by

which inverse is given explicitly by

We can consider the $C^{\infty}$-structure on $\mathscr{D}_{p}^{n}$ induced by this bijection.

We pass now to the general case. A local chart $(\mathscr{U},\phi)$ in $M$ is said to be a $X$-slice if

• 1)

  $\phi:\mathscr{U}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}U\subset\mathbb{R}^{p}\times\mathbb{R}^{q}$

• 2)

  If $\mathscr{U}\cap X=\mathscr{V}$, $\mathscr{V}=\phi^{-1}(U\cap\mathbb{R}^{p}\times\{0\})$ (we note $V=U\cap\mathbb{R}^{p}\times\{0\}$)

With this notation, $\mathscr{D}_{V}^{U}\subset\mathscr{D}_{p}^{n}$ as an open subset. We may define a function

in the following way: For $x\in\mathscr{V}$ we have $\phi(x)\in\mathbb{R}^{p}\times\{0\}$. If we write $\phi(x)=(\phi_{1}(x),0)$, then

is a diffeomorphism. We set $\tilde{\phi}(v,\xi,0)=(\phi_{1}(v),d_{N}\phi_{v}(\xi),0)$ and $\tilde{\phi}(u,t)=(\phi(u),t)$ for $t\neq 0$. Here $d_{N}\phi_{v}:N_{v}\rightarrow\mathbb{R}^{q}$ is the normal component of the derivate $d\phi_{v}$ for $v\in\mathscr{V}$. It is clear that $\tilde{\phi}$ is also a bijection (in particular it induces a $C^{\infty}$ structure on $\mathscr{D}_{\mathscr{V}}^{\mathscr{U}}$). Now, let us consider an atlas $\{(\mathscr{U}_{\alpha},\phi_{\alpha})\}_{\alpha\in\Delta}$ of $M$ consisting of $X-$slices. Then the collection $\{(\mathscr{D}_{\mathscr{V}_{\alpha}}^{\mathscr{U}_{\alpha}},\tilde{\phi_{\alpha})}\}_{\alpha\in\Delta}$ is a $C^{\infty}$-atlas of $\mathscr{D}_{X}^{M}$ (proposition 3.1 in [CR06]).

One important feature about this construction is that it is in some sense functorial. More explicitly, let $(M,X)$ and $(M^{\prime},X^{\prime})$ be $C^{\infty}$-pairs as above and let $F:(M,X)\rightarrow(M^{\prime},X^{\prime})$ be a pair morphism, i.e., a $C^{\infty}$ map $F:M\rightarrow M^{\prime}$, with $F(X)\subset X^{\prime}$. We define $\mathscr{D}(F):\mathscr{D}_{X}^{M}\rightarrow\mathscr{D}_{X^{\prime}}^{M^{\prime}}$ by the following formulas:

$\mathscr{D}(F)(x,\xi,0)=(F(x),d_{N}F_{x}(\xi),0)$ and

$\mathscr{D}(F)(m,t)=(F(m),t)$ for $t\neq 0$, where $d_{N}F_{x}$ is by definition the map

induced by $T_{x}M\stackrel{{\scriptstyle dF_{x}}}{{\longrightarrow}}T_{F(x)}M^{\prime}$.

Then $\mathscr{D}(F):\mathscr{D}_{X}^{M}\rightarrow\mathscr{D}_{X^{\prime}}^{M^{\prime}}$ is $C^{\infty}$-map (proposition 3.4 in [CR06]).

In this section we will recall how to construct the deformation algebra mentioned at the introduction. For complete details, we refer the reader to [CR06].

The Schwartz algebra for the Tangent groupoid will be a particular case of a construction associated to any deformation to the normal cone.

Now, the spaces $\mathscr{S}_{c}(\Omega_{V}^{U})$ are invariant under diffeomorphisms. More precisely: Let $F:U\rightarrow U^{\prime}$ be a $C^{\infty}$-diffeomorphism such that $F(V)=V^{\prime}$; let $\tilde{F}:\Omega_{V}^{U}\rightarrow\Omega_{V^{\prime}}^{U^{\prime}}$ the induced map. Then, for every $g\in\mathscr{S}_{c}(\Omega_{V^{\prime}}^{U^{\prime}})$, we have that $\tilde{g}:=g\circ\tilde{F}\in\mathscr{S}_{c}(\Omega_{V}^{U})$ (proposition 4.2 in [CR06]).

This compatibility result allows to give the following definition.