# Renormalized Index Formulas for Elliptic Differential Operators on Boundary
Groupoids

Yu Qiao  School of Mathematics and Statistics, Shaanxi Normal University,
Xi’an, 710119, Shaanxi, China<EMAIL_ADDRESS>and  Bing Kwan So  School of
Mathematics, Jilin University, Changchun, 130023, Jilin, China
<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We consider the index problem of certain boundary groupoids of the form
$\mathcal{G}=M_{0}\times M_{0}\cup\mathbb{R}^{q}\times M_{1}\times M_{1}$.
Since it has been shown that for the case that $q\geq 3$ is odd,
$K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z}$, and moreover the $K$-theoretic
index coincides with the Fredholm index, we attempt in this paper to derive a
numerical formula for elliptic differential operators on $\mathcal{G}$. Our
approach is similar to that of renormalized trace of Moroianu and Nistor [41].
However, we find that when $q\geq 3$, the eta term vanishes, and hence the
$K$-theoretic and Fredholm indices of elliptic (respectively fully elliptic)
pseudo-differential operators on these groupoids are given only by the Atiyah-
Singer term. As for the $q=1$ case we find that the result depends on how the
singularity set $M_{1}$ lies in $M$.

Qiao was partially supported by the NSFC(11971282), Natural Science Basic
Research Program of Shaanxi (Program No. 2020JM-278), and the Fundamental
Research Funds for the Central Universities (GK202003006).

###### Contents

  1. 1 Introduction
    1. 1.1 Contents of the paper
  2. 2 Preliminaries
    1. 2.1 Lie groupoids, Lie algebroids, and groupoid $C^{*}$-algebras
    2. 2.2 Invariant submanifolds and composition series
    3. 2.3 Boundary groupoids and their composition series
    4. 2.4 The relation between Fredholm and $K_{0}(C^{*}(\mathcal{G}))$ indices
  3. 3 Renormalized Index formulas for boundary groupoids of the form $\mathcal{G}=(M_{0}\times M_{0})\cup(\mathbb{R}^{q}\times M_{1}\times M_{1}),$ with $q\geqslant 3$ odd
  4. 4 The case that $q=1$
    1. 4.1 Index formula for connected manifolds with connected boundaries
    2. 4.2 An Index formula for closed manifolds separated by interior hypersurfaces
  5. 5 Concluding remarks

##  1\. Introduction

The Atiyah-Singer index theorem is one of the most celebrated results of the
twentieth century mathematics. It states that given a compact manifold $M$:

  1. (i)

Given a closed manifold $M$, any elliptic differential operator $D$ acting
between sections of vector bundles over $M$ is invertible modulo compact
operators, hence $D$ is Fredholm;

  2. (ii)

The Fredholm index of $D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}$ is its
image under the connecting map in $K$-theory induced by the short exact
sequence

$0\to\mathcal{K}\to\mathfrak{U}\to\mathfrak{U}/\mathcal{K}\to 0;$

  3. (iii)

One has $K_{0}(\mathcal{K})\cong\mathbb{Z}$, and moreover, under such
isomorphism, an explicit formula for the index, which produces an integer, is
given:

$\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M}\operatorname{AS}(D).$

There has been many attempts to generalize the Atiyah-Singer index theorem.
One major direction is suggested by Connes in [20, Section 2]. Here one
considers any Lie groupoid $\mathcal{G}\rightrightarrows M$ with $M$ compact
(the classical case corresponds to the pair groupoid $\mathcal{G}=M\times M$).
On $\mathcal{G}$ one constructs the pseudo-differential calculus [1, 4, 8, 24,
25, 38, 44]. The sub-algebras of operators of order zero and $-\infty$ can be
completed to $C^{*}$-algebras $\mathfrak{U}(\mathcal{G})$ and
$C^{*}(\mathcal{G})$, respectively. Then one shows that any elliptic operator
is invertible modulo order $-\infty$ order operators. It follows that for any
elliptic operator of order zero, one can consider its analytic index in
$K_{0}(C^{*}(\mathcal{G}))$ through the short exact sequence

$0\to
C^{*}(\mathcal{G})\to\mathfrak{U}(\mathcal{G})\to\mathfrak{U}/C^{*}(\mathcal{G})\to
0.$

Equivalently, the analytic index can be defined by Lauter, Monthubert, and
Nistor in [27], or by Connes via the adiabatic groupoid [20, Chapter 2]. The
objective of the index theorem is to give an alternative description of such
index, preferably in “topological” terms.

Given a regular foliation $\mathcal{F}\subset TM$, one considers the holonomy
groupoid $\mathcal{G}$. Generalizing the Atiyah-Singer index theorem, one
considers an embedding $M\to\mathbb{R}^{N}$ for some sufficiently large $N$.
Then one can define a topological index using some Thom isomorphism and Morita
equivalence (see for instance [20, Chapter 2.9]).

The index theory for groupoids with leaves of different dimensions is less
known. The most well studied cases are mostly related to manifolds with
corners. The $K$-theory of the interior of the $b$-stretched product
$\mathcal{G}\rightrightarrows M$ is computed by Carrillo Rouse and Lescure in
[11], and Le Gall and Montubert in [30]. Recently Yamashita in [48] computed
the fibered boundary case corresponding to the $\Phi$ and edge calculus. In
the particular case of of manifolds with boundary, the $K$-groups are trivial.
Index theorems are again proved in [1, 12, 13, 23, 24, 39]. These results rely
heavily on the boundary defining function (in other words the groupoid
$\mathcal{G}$ under consideration must be the result of some blowups using the
boundary defining function). For example, Akrour and Carrillo Rouse [1], and
Monthubert and Nistor [39], used an auxiliary embedding of $M$ into a cube,
and the fact that given any transverse embedding of manifold with corners
$M_{1}\to M_{2}$, one can construct explicitly, using the boundary defining
functions, a groupoid homomorphism from the $b$-stretched product of $M_{1}$
to $b$-stretched product of $M_{2}$; while in the works of Carrillo Rouse,
Lescure, and Monthubert [12], Debord, Lescure, and Nistor [23], and Debord,
Lescure, and Rochon [24], various groupoids are constructed in place of the
tangent Lie algebroid.

On the other hand, one may also consider the Fredholm index of a Fredholm
operator, that is, an elliptic operator invertible modulo $C^{*}(M_{0}\times
M_{0})\cong\mathcal{K}$, the $C^{*}$-algebra of compact operators. Here,
recall [29] that an elliptic operator $D$ is Fredholm if $D$ is invertible
over all singular leaves (see [16, 17] for a discussion of the converse). By
using renormalized traces, or otherwise, one proves the Atiyah-Patodi-Singer
index theorem in the case of manifolds with boundary [2, 31, 36, 41], which is
of the form

$\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M}\operatorname{AS}(D)+\eta(D),$

where $\eta$ is an explicitly given non-local term, that is, in general, not
determined by the principal symbol of $D$. One also gets results of a similar
form for some Lie manifolds [10, 41] and the Bruhat sphere [46]. However, in
many cases the explicit from of the $\eta$ term is unknown.

The notion of boundary groupoids is introduced in [47], as a generalization of
manifolds with boundary. They also arise as the holonomy groupoid integrating
some singular foliations [3, 4, 6, 7, 22], or symplectic groupoid of some
Poison manifolds [32]. Roughly speaking, a boundary groupoid is a Lie groupoid
of the form:

$\mathcal{G}=\bigcup_{i}G_{i}\times M_{i}\times M_{i}.$

Here, it is important to note that the expression above does not uniquely
define a Lie groupoid. For example, the $b$-calculus, cusp calculus, and the
very small calculus all integrate to

$\mathcal{G}=M_{0}\times M_{0}\cup\mathbb{R}\times M_{1}\times M_{1},$

but each have different smooth structures and Lie algebroids. For the
integrability of Lie algebroids, one may consult the works of Crainic and
Fernandes [21], Debord [22], and Nistor [42]. In [47], the second author
constructed a calculus of pseudo-differential operators, using only intrinsic
quantities like Riemannian distance functions. The calculus includes the
finite rank parametrix of Fredholm operators, and could be served as a
framework for various problems (for example [43]). Thus, it appears that these
groupoids are representatives when one considers groupoids with leaves of
different dimensions.

In [14], Carrillo Rouse and the second author computed the $K$-groups of some
boundary groupoids, using composition series exact sequence [27]. Explicitly,
for a boundary groupoid of the form

$\mathcal{G}=M_{0}\times M_{0}\cup G\times M_{1}\times M_{1},$

where $G$ is exponential (i.e., a connected, simply-connected and solvable Lie
group), one has

$\displaystyle K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z},$ $\displaystyle\quad
K_{1}(C^{*}(\mathcal{G}))\cong\mathbb{Z},$ $\displaystyle\text{if $\dim G\geq
3$ odd};$ $\displaystyle K_{0}(C^{*}(\mathcal{G}))\cong\\{0\\},$
$\displaystyle\quad K_{1}(C^{*}(\mathcal{G}))\cong\mathbb{Z}\oplus\mathbb{Z},$
if $\dim G$ even.

Moreover, when $\dim G$ is odd, one has the commutative diagram

(1) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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where the left column is just the Fredholm index map. It follows that for a
fully elliptic operator, its index in $K_{0}(C^{*}(\mathcal{G}))$ equals to
its Fredholm index.

In this paper, we turn to the problem of the computation of the index. Suppose
we are given a fully elliptic operator $D$. Let $\varPhi$ be an inverse of $D$
up to finite rank operators in $C^{*}(\mathcal{G}_{0}))\cong\mathcal{K}$, then
its Fredholm index is given by

$\operatorname{ind}(D)=\int_{M_{0}}\operatorname{tr}((D\varPhi-1)|_{M_{0}})\mu_{M_{0}}-\int_{M_{0}}\operatorname{tr}((\varPhi
D-1)|_{M_{0}})\mu_{M_{0}}.$

Here, note that $\mu_{M_{0}}$ has infinite volume. In order to apply the
Calderon formula, we suppose further that $\mathcal{G}$ is of a special form
(see Example 2.17). Then we can extend the trace formula to a renormalized
trace

$\widehat{\operatorname{Tr}}([D,\varPhi]):=\frac{\partial^{2}}{\partial
z\partial\tau}\Big{|}_{\tau=z=0}\int_{\mathrm{M}_{0}}\operatorname{tr}(r^{z}[D,\varPhi]Q^{-\tau}|_{M_{0}})\mu_{M_{0}}.$

By a similar argument of Nistor and Moroianu [41], the renormalized trace
above can be written as the sum of an Atiyah-Singer term and eta term.
However, unlike [41], we find that when the codimension is odd and greater
than or equal to 3, the eta term vanishes, leaving the Aityah-Singer term (see
Theorem 3.7).

######  Theorem 1.1.

For any elliptic, differential operator $D\in\mathrm{D}^{1}(\mathcal{G},E)$,
the index of $D$ in $K_{0}(C^{*}(\mathcal{G}))$ equals

(2)
$\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M_{0}}r^{z-qN}\,a_{0}\,\mu_{M}\Big{|}_{z=0},$

where $a_{0}$ is the constant term in the $t\to 0$ asymptotic expansion of the
trace of heat kernel.

###  1.1. Contents of the paper

Section 2 recalls the background of necessary terminology we shall use in this
paper, such as Lie groupoids, Lie algebroids, pseudodifferential calculus on
Lie groupoids, boundary groupoids and their composition series. Throughout the
paper, we shall consider the simplest form of boundary groupoids

$\mathcal{G}=M_{0}\times M_{0}\cup\mathbb{R}^{q}\times M_{1}\times M_{1},$

for odd positive integer $q$, and explore index formulas for the cases
$q\geqslant 3$ odd and $q=1$ in Sections 3 and 4, respectively.

In Section 3, we assume that $q$ is an odd positive integer greater than $3$.
By the work of Carrillo Rouse and the second author [14], the Fredholm index
is in this case equivalent to the $K$-theoretic index. With this result in
mind, we turn to consider the index of elliptic operators. We assume that
$\dim M_{1}\geqslant 1$ and $q\geqslant 3$ odd, i.e., the singular structure
is similar to the very small calculus, a renormalized index along the lines of
[10, 41] can be defined, and we shall show in addition that the $\eta$ term
vanishes, leaving only the Atiyah-Singer term.

In Section 4, we assume that $q=1$. More prescisely, in Subsection 4.1, we
consider the case that $M_{1}$ is the connected boundary of $M:=M_{0}\cup
M_{1}$. Much about this case is known, and we reformulate an index formula in
the context of boundary groupoids. In Subection 4.2, we consider the case that
$M$ is a connected manifold without boundary partitioned by a codimensional 1
submanifold $M_{1}$, i.e., $M_{0}\setminus M_{1}$ has two components. The
corresponding Lie groupoid is of the form

$\mathcal{G}=(M_{0}^{\prime}\times
M_{0}^{\prime})\cup(M_{0}^{\prime\prime}\times
M_{0}^{\prime\prime})\cup(\mathbb{R}\times M_{1}\times M_{1}).$

(Strictly speaking, $\mathcal{G}$ is not a boundary groupoid.) We show that
$K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z}$, hence it is meaningful to consider
the index in $K_{0}(C^{*}(\mathcal{G}))$. We prove an index formula for this
case in Theorem 4.6,

In Section 5, we point out some future research directions.

Acknowledgments The authors are grateful to Victor Nistor for helpful
comments, enlightening discussions and email communications.

##  2\. Preliminaries

###  2.1. Lie groupoids, Lie algebroids, and groupoid $C^{*}$-algebras

We review first basic knowledge of Lie groupoids, Lie algebroids,
pseododifferential calculus on Lie groupoids, and groupoid $C^{*}$-algebras
[4, 27, 29, 33, 34, 40, 44, 45, 47].

######  Definition 2.1.

A Lie groupoid $\mathcal{G}\rightrightarrows M$ consists of the following
data:

  1. (i)

manifolds $M$, called the space of units, and $\mathcal{G}$;

  2. (ii)

a unit map (inclusion) $\mathbf{u}:M\rightarrow G$;

  3. (iii)

submersions $\mathbf{s},\mathbf{t}:\mathcal{G}\rightarrow M$, called the
source and target maps respectively, satisfying

$\mathbf{s}\circ\mathbf{u}=\operatorname{id}_{M}=\mathbf{t}\circ\mathbf{u};$

  4. (iv)

a multiplication map
$\mathbf{m}:\mathcal{G}^{(2)}:=\\{(g,h)\in\mathcal{G}\times\mathcal{G}\,:\,\mathbf{s}(g)=\mathbf{t}(h)\\}\rightarrow\mathcal{G}$,
$(g,h)\mapsto gh$, which is associative and satisfies

$\mathbf{s}(gh)=\mathbf{s}(g),\quad\mathbf{t}(gh)=\mathbf{t}(g),\quad
g(\mathbf{u}\circ\mathbf{s}(g))=g=(\mathbf{u}\circ\mathbf{t}(g))g;$

  5. (v)

an inverse diffeomorphism $\mathbf{i}:\mathcal{G}\rightarrow\mathcal{G}$,
$g\mapsto g=g^{-1}$, such that $\mathbf{s}(g^{-1})=\mathbf{t}(g)$,
$\mathbf{t}(g^{-1})=\mathbf{s}(g)$, and

$gg^{-1}=\mathbf{u}(\mathbf{t}(g)),\quad g^{-1}g=\mathbf{u}(\mathbf{s}(g)).$

######  Remark 2.2.

In general, the space $\mathcal{G}_{1}$ is not required to be Hausdorff.
However, since $\mathbf{s}$ is a submersion, it follows that each fiber
$\mathcal{G}_{x}:=\mathbf{s}^{-1}(x)$ is a smooth manifold, hence it is
Hausdorff. In this paper, all groupoids are assumed to be Hausdorff.

We sometimes use the following notations. Let $A,B$ be subsets of $M$. we
denote by $\mathcal{G}_{A}:=\mathbf{s}^{-1}(A)$,
$\mathcal{G}^{A}:=\mathbf{t}^{-1}(A)$, and
$\mathcal{G}_{A}^{B}:=\mathcal{G}_{A}\cap\mathcal{G}^{B}$. We call
$\mathcal{G}_{A}^{A}$ the reduction of $\mathcal{G}$ to $A$. In particular, if
$A=\\{x\\}$, then we write $\mathcal{G}_{x}=\mathcal{G}_{A}$ and
$\mathcal{G}^{x}=\mathcal{G}^{A}$ respectively. For any $x\in M$,
$\mathcal{G}_{x}^{x}=\mathbf{s}^{-1}(x)\cap\mathbf{t}^{-1}(x)$ is a group,
called the isotropy group at $x$.

To introduce the “Lie algebroid” of a Lie groupoid, let us recall the
definition of Lie algebroids.

######  Definition 2.3.

A _Lie algebroid_ $A$ over a manifold $M$ is a vector bundle $A$ over $M$,
together with a Lie algebra structure on the space $\Gamma(A)$ of the smooth
sections of $A$ and a bundle map $\nu:A\rightarrow TM$, extended to a map
between sections of theses bundles, such that

  1. (i)

$\nu([X,Y])=[\nu(X),\nu(Y)]$;

  2. (ii)

$[X,fY]=f[X,Y]+(\nu(X)f)Y$,

for all smooth sections $X$ and $Y$ of $A$ and any smooth function $f$ on $M$.
The map $\nu$ is called the _anchor_.

Give a Lie groupoid $\mathcal{G}$ with units $M$, we can associate a Lie
algebroid $A(\mathcal{G})$ to $\mathcal{G}$ as follows. (For more details, one
can read [33, 34].) The $\mathbf{s}$-vertical subbundle of $T\mathcal{G}$ for
$\mathbf{s}:\mathcal{G}\rightarrow M$ is denoted by
$T^{\mathbf{s}}(\mathcal{G})$ and called simply the $\mathbf{s}$-vertical
bundle for $\mathcal{G}$. It is an involutive distribution on $\mathcal{G}$
whose leaves are the components of the $\mathbf{s}$-fibers of $\mathcal{G}$.
(Here involutive distribution means that $T^{\mathbf{s}}(\mathcal{G})$ is
closed under the Lie bracket, i.e. if $X,Y\in\mathfrak{X}(\mathcal{G})$ are
sections of $T^{\mathbf{s}}(\mathcal{G})$, then the vector field $[X,Y]$ is
also a section of $T^{\mathbf{s}}(\mathcal{G})$.) Hence we obtain

$T^{\mathbf{s}}\mathcal{G}=\text{ker}\
\mathbf{s}_{*}=\displaystyle{\bigcup_{x\in M}T\mathcal{G}_{x}}\subset
T\mathcal{G}.$

The _Lie algebroid_ of $\mathcal{G}$, denoted by $A(\mathcal{G})$ (or simply
$A$ sometimes), is defined to be $T^{\mathbf{s}}(\mathcal{G})|_{M}$, the
restriction of the $\mathbf{s}$-vertical tangent bundle to the set of units
$M$. In this case, we say that $\mathcal{G}$ integrates $A(\mathcal{G})$.

Let $E\to M$ be a vector bundle. Recall [44] that an $m$th order pseudo-
differential operator on $\mathcal{G}$ is a right invariant, smooth family
$P=\\{P_{x}\\}_{x\in M}$, where each $P_{x}$ is an $m$th order classical
pseudo-differential operator on sections of
$\mathbf{t}^{*}\mathrm{E}\to\mathbf{s}^{-1}(x)$. We denote by
$\Psi^{m}(\mathcal{G},E)$ (resp. $\mathrm{D}^{m}(\mathcal{G},E)$) the algebra
of uniformly supported, order $m$ classical pseudo-differential operators
(resp. differential operators).

Suppose that $\mathcal{G}\rightrightarrows M$ is a Lie groupiod with $M$
compact. Then there is an unique (up to equivalence) Riemannian metric on its
Lie algebroid $A(\mathcal{G})\to M$, which induces a family of right invariant
Riemannian metrics on $\mathbf{s}^{-1}(x),x\in M$. We shall denote by
$\mu_{x}$ the Riemannian volume forms (which is a Haar system on
$\mathcal{G}$), and $d(\cdot,\cdot)$, the Riemannian distance function.

Recall [27] that one defines the strong norm for $P\in\Psi^{0}(\mathcal{G},E)$

$\|P\|:=\sup_{\rho}\|\rho(P)\|,$

where $\rho$ ranges over all bounded $*$-representations of
$\Psi^{0}(\mathcal{G},E)$ satisfying

$\|\rho(P)\|\leq\sup_{x\in
M}\Big{\\{}\int_{\mathbf{s}^{-1}(x)}|\kappa_{P}(g)|\,\mu_{x},\int_{\mathbf{s}^{-1}(x)}|\kappa_{P}(g^{-1})|\,\mu_{x}\Big{\\}},$

whenever $P\in\Psi^{-\dim M-1}(\mathcal{G},E)$ with (continuous) kernel
$\kappa_{P}$.

######  Definition 2.4.

The $C^{*}$-algebras $\mathfrak{U}(\mathcal{G})$ and $C^{*}(\mathcal{G})$ are
defined to be the completion of $\Psi^{0}(\mathcal{G},E)$ and
$\Psi^{-\infty}(\mathcal{G},E)$ respectively with respective to the strong
norm $\|\cdot\|$.

One also defines the reduced $C^{*}$-algebras $\mathfrak{U}_{r}(\mathcal{G})$
and $C^{*}_{r}(\mathcal{G})$ by completing $\Psi^{0}(\mathcal{G},E)$ and
$\Psi^{-\infty}(\mathcal{G},E)$) respectively with respect to the reduced norm

$\|P\|_{r}:=\sup_{x\in
M}\big{\\{}\|P_{x}\|_{L^{2}(\mathbf{s}^{-1}(x))}\big{\\}}.$

Recall that the strong and reduced norm coincides if $\mathcal{G}$ is
(metrically) amenable, which is the case for the groupoids we shall consider.

Suppose further that $\mathcal{G}$ has polynomial volume growth [47]. Then
convolution product is well defined between kernels with faster than
polynomial decay, and hence one may define:

######  Definition 2.5.

The sub-algebra of Schwartz kernels $S^{*}(\mathcal{G})\subset
C^{*}_{r}(\mathcal{G})$ is the set of all kernels $\psi\in C^{*}(\mathcal{G})$
satisfying:

  1. (i)

$\psi$ is continuous;

  2. (ii)

$\psi|_{\mathcal{G}_{i}}$ is smooth for all $i$;

  3. (iii)

For any collection of sections $V_{1},\cdots,V_{k}\in\Gamma^{\infty}(A)$,
regarded as first order differential operators (right invariant vector fields)
on $\mathcal{G}$ and $N=0,1,2,\cdots$,

$a\mapsto(V_{1}\cdots V_{i}\psi V_{i+1}\cdots
V_{k})(d(a,\mathbf{s}(a))+1)^{-N}$

is a bounded function.

The significance of $S^{*}(\mathcal{G})$ lies in (for instance [10], [28,
Section 6]):

######  Lemma 2.6.

The sub-algebras $S^{*}(\mathcal{G})\subset C^{*}_{r}(\mathcal{G})$ and
$\Psi^{0}(\mathcal{G},E)\oplus
S^{*}(\mathcal{G})\subset\mathfrak{U}_{r}(\mathcal{G})$ are dense, and are
invariant under holomorphic functional calculus.

###  2.2. Invariant submanifolds and composition series

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid.

######  Definition 2.7.

We say that an embedded sub-manifold $M^{\prime}\subset M$ is an invariant
submanifold if

$\mathbf{s}^{-1}(M^{\prime})=\mathbf{t}^{-1}(M^{\prime}).$

For any invariant submanifold $M^{\prime}$, we denote the sub-groupoid
$\mathbf{s}^{-1}(M^{\prime})=\mathbf{t}^{-1}(M^{\prime})$ by
$\mathcal{G}_{M^{\prime}}$ as before.

######  Definition 2.8.

Given a closed invariant submanifold $M^{\prime}$ of $\mathcal{G}$. For any
$P=\\{P_{x}\\}_{x\in M}\in\Psi^{m}(\mathcal{G},E)$, we define the restriction

$P|_{\mathcal{G}_{M^{\prime}}}:=\\{P_{x}\\}_{x\in\mathcal{G}_{M^{\prime}}}\in\Psi^{m}(\mathcal{G}_{M^{\prime}},E).$

Restriction extends to a map from $\mathfrak{U}(\mathcal{G})$ to
$\mathfrak{U}(\mathcal{G}_{M^{\prime}})$ and also from $C^{*}(\mathcal{G})$ to
$C^{*}(\mathcal{G}_{M^{\prime}})$.

Now suppose we are given a groupoid $\mathcal{G}\rightrightarrows M$ ($M$ not
necessarily compact), and closed invariant submanifolds

$M=\bar{M_{0}}\supset\bar{M_{1}}\supset\cdots\supset\bar{M_{r}}.$

For simplicity we shall denote
$\bar{\mathcal{G}}_{i}:=\mathcal{G}_{\bar{M_{i}}}$. Recall that $SA^{\prime}$
denotes the sphere sub-bundle of the dual of the Lie algebroid
$A(\mathcal{G})$ of $\mathcal{G}$.

######  Definition 2.9.

Let $\sigma:\Psi^{m}(\mathcal{G})\to C^{\infty}(SA^{\prime})$ denotes the
principal symbol map. For each $i=1,\cdots,r$, define the joint symbol maps

(3) $\mathbf{j}_{i}:\Psi^{m}(\mathcal{G})\to
C^{\infty}(SA^{\prime})\oplus\Psi^{m}(\bar{\mathcal{G}}_{i}),\quad\mathbf{j}_{i}(P):=(\sigma(P),P|_{\bar{\mathcal{G}}_{i}}).$

The map $\mathbf{j}_{i}$ extends to a homomorphism from
$\mathfrak{U}(\mathcal{G})$ to
$C_{0}(S\mathrm{A}^{*})\oplus\mathfrak{U}(\bar{\mathcal{G}}_{i})$.

We say that $P\in\Psi^{m}(\mathcal{G})$ is elliptic if $\sigma(P)$ is
invertible, and it is fully elliptic if $\mathbf{j}_{1}(P)$ is invertible.

######  Definition 2.10.

Denote by
$\mathcal{J}_{0}:=\overline{\Psi^{-1}(\mathcal{G})}\subset\mathfrak{U}(\mathcal{G})$,
and $\mathcal{J}_{i}\subset\mathfrak{U}(\mathcal{G}),i=1,\cdots,r$ the null
space of $\mathbf{j}_{r-i+1}$.

By construction, it is clear that

$\mathcal{J}_{0}\supset\mathcal{J}_{1}\supset\cdots\supset\mathcal{J}_{r}.$

Also, any uniformly supported kernels in
$\Psi^{-\infty}(\mathcal{G}_{\bar{M}_{i}\setminus\bar{M}_{j}},E|_{\bar{M}_{i}\setminus\bar{M}_{j}})$,
can be extended to a kernel in
$\Psi^{-\infty}(\mathcal{G}_{\bar{M}_{i}},E|_{\bar{M}_{i}})$ by zero. This
induces a $*$-algebra homomorphism from
$C^{*}(\mathcal{G}_{\bar{M}_{i}\setminus\bar{M}_{j}})$ to
$C^{*}(\mathcal{G}_{\bar{M}_{i}})$. We shall use the following key fact.

######  Lemma 2.11.

[27, Lemma 2 and Theorem 3] One has short exact sequences

(4) $\displaystyle 0\to\mathcal{J}_{i+1}\to\mathcal{J}_{i}\to
C^{*}(\mathcal{G}_{\bar{M}_{i}\setminus\bar{M}_{i+1}})\to 0,$ (5)
$\displaystyle 0\to C^{*}(\mathcal{G}_{\bar{M}_{i}\setminus\bar{M}_{j}})\to
C^{*}(\bar{\mathcal{G}}_{i})\to C^{*}(\bar{\mathcal{G}}_{j})\to
0,\quad\operatorname{\forall}j>i.$

###  2.3. Boundary groupoids and their composition series

######  Definition 2.12.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with $M$ compact. We
say that $\mathcal{G}$ is a boundary groupoid if:

  1. (i)

the singular foliation defined by the anchor map $\nu:A\to TM$ has finite
number of leaves $M_{0},M_{1},\cdots,M_{r}\subset M$ (which are invariant
submanifolds), such that $\dim M=\dim M_{0}>\dim M_{1}>\cdots>\dim M_{r}$;

  2. (ii)

For all $k=0,1,\cdots,r$, $\bar{M}_{k}:=M_{k}\cup\cdots\cup M_{r}$ are closed
submanifolds of $M$;

  3. (iii)

$\mathcal{G}_{0}:=\mathcal{G}_{M_{0}}$ is the pair groupoid, and
$\mathcal{G}_{k}:=\mathcal{G}_{M_{k}}\cong G_{k}\times M_{k}\times M_{k}$ for
some Lie group $G_{k}$;

  4. (iv)

For each $k$, there exists an unique sub-bundle $\bar{A}_{k}\subset
A|_{\bar{M}_{k}}$ such that $\bar{A}_{k}|_{M_{k}}=\ker(\nu|_{M_{k}})$
($=\mathfrak{g}_{k}\times M_{k}$).

Boundary groupoids are closely related to Fredholm groupoids and blowup
groupoids. Here we recall some basic properties of boundary groupoids.

By the same arguments as in [29], we can prove the following lemma.

######  Lemma 2.13.

(See [29, Lemma 7]) For any boundary groupoid of the form
$\mathcal{G}=(M_{0}\times M_{0})\cup(\mathbb{R}^{q}\times M_{1}\times M_{1})$,

$C^{*}(\mathcal{G})\cong C^{*}_{r}(\mathcal{G}).$

In other words, $\mathcal{G}$ is (metrically) amenable.

Moreover, since the additive group $\mathbb{R}^{q}$ is amenable, the groupoid
of the form $\mathbb{R}^{q}\times M_{1}\times M_{1}$ is (topologically)
amenable. By [18, Theorem 4.3], we have the following proposition.

######  Proposition 2.14.

The groupoid $\mathcal{G}=(M_{0}\times M_{0})\cup(\mathbb{R}^{q}\times
M_{1}\times M_{1})$ is a Fredholm groupoid.

######  Remark 2.15.

For the definition and basic properties of Fredholm groupoids, one may consult
[16, 17, 18]. Roughly speaking, Fredholm groupoids are those on which
Fredholmness of a pseudodifferential operator is completely characterized by
its ellipticity and invertibility at the boundary.

We are in position to give some examples of boundary groupoids.

######  Example 2.16.

[14, Section 2.1]] Fix a smooth function $f:\mathbb{R}^{q}\to\mathbb{R}$ with
$f(0)=0$ and $f>0$ over $\mathbb{R}^{q}\setminus\\{0\\}$. Then there exists a
Lie algebroid structure over

$A_{f}:=T\mathbb{R}^{q}\to\mathbb{R}^{q}$

with anchor map

$(z,w)\mapsto(z,f(z)\cdot w),\quad\operatorname{\forall}(z,w)\in
T\mathbb{R}^{q}=\mathbb{R}^{q}\times\mathbb{R}^{q}$

and Lie bracket on sections

$[X,Y]_{A_{f}}:=f[X,Y]+(X\cdot f)Y-(Y\cdot f)X,$

whose image under the anchor map defines the foliation. This Lie algebroid is
almost injective (since the anchor is injective, in fact an isomorphism in
this case, over the open dense subset $\mathbb{R}^{q}\setminus\\{0\\}$), and
hence it is integrable by a Lie groupoid that is a quasi-graphoid (see [22,
Theorems 2 and 3]), which is of the form

$(\mathbb{R}^{q}\setminus\\{0\\})\times(\mathbb{R}^{q}\setminus\\{0\\})\cup\mathbb{R}^{q}.$

######  Example 2.17 (Renormalizable boundary groupoid).

Now let $M$ be a compact manifold without boundary, $M_{1}\subset M$ be a
closed submanifold of codimension $q\geq 2$. Fix a Riemannian metric $g_{M}$
on $M$. Let $r:M\to\mathbb{R}$ be the Riemannian distance from $M_{1}$. Then
$r^{2}$ is smooth near $M_{1}$. Fix an even number $N\geq 2$. Take coordinate
charts $(x_{\alpha},y_{\alpha})$ near $M_{1}$ such that

$M_{1}\cap U_{\alpha}=\\{x_{\alpha}^{1}=\cdots=x_{\alpha}^{n-q}=0\\}$

and

$r^{2}=(x_{\alpha}^{1})^{2}+\cdots+(x_{\alpha}^{q})^{2}.$

We consider the singular foliation spanned by

(6) $r^{N}\frac{\partial}{\partial x_{1}},\cdots,r^{N}\frac{\partial}{\partial
x_{q}},\frac{\partial}{\partial y_{1}},\cdots,\frac{\partial}{\partial y_{\dim
M_{1}}}$

on $U_{\alpha}$. These vector fields are locally free, and closed under Lie
bracket, hence by [3] they are the image of some Lie algebroid $A$. Indeed,
$A$ can be explicitly constructed similar to the last example.

In turn, $A$ can be integrated into a Lie groupoid $\mathcal{G}$. Locally,
over $U_{\alpha}$, $\mathcal{G}_{U_{\alpha}}$ is of the form

$\mathcal{G}_{U_{\alpha}}=(U^{\prime}_{\alpha}\times
U^{\prime}_{\alpha})\times\big{(}(U^{\prime\prime}_{\alpha}\times
U^{\prime\prime}_{\alpha})\cup\mathbb{R}^{q})\big{)},$

where $U^{\prime\prime}_{\alpha}\subset\mathbb{R}^{q}$ and
$(U^{\prime\prime}_{\alpha}\times
U^{\prime\prime}_{\alpha})\cup\mathbb{R}^{q}$ are defined in the previous
example. Then using [14, Section 2.2] or [42] one can “glue” these groupoids
together to form a groupoid over $M$.

Given a boundary groupoid, we consider the sequence of invariant submanifolds,

$M\supset\bar{M}_{1}\supset\cdots\supset\bar{M}_{r},$

where $\bar{M}_{i}$ is given to be (2) of Definition 2.12. The second short
exact sequence of Lemma 2.11 then induces the six-terms exact sequences:

(7)
$\begin{CD}K_{1}(C^{*}(\mathcal{G}_{M_{i}}))@>{}>{}>K_{1}(C^{*}(\bar{\mathcal{G}}_{i}))@>{}>{}>K_{1}(C^{*}(\bar{\mathcal{G}}_{i+1}))\\\
@A{}A{}A@V{}V{}V\\\
K_{0}(C^{*}(\bar{\mathcal{G}}_{i+1}))@<{}<{}<K_{0}(C^{*}(\bar{\mathcal{G}}_{i}))@<{}<{}<K_{0}(C^{*}(\mathcal{G}_{M_{i}}))\end{CD}$

If $r=1$, and $G_{1}$ is solvable, connected and simply connected (i.e.
exponential), then the system (7) will be greatly simplified. In this case, we
only have one six-terms exact sequence, and we have

$\displaystyle C^{*}(\mathcal{G}_{M_{0}})=C^{*}(M_{0}\times M_{0})\cong$
$\displaystyle C^{*}_{r}(M_{0}\times M_{0})\cong\mathcal{K},$ $\displaystyle
C^{*}(\bar{\mathcal{G}}_{0})=$ $\displaystyle C^{*}(\mathcal{G}),$
$\displaystyle C^{*}(\bar{\mathcal{G}}_{1})=C^{*}(\mathcal{G}_{1})=$
$\displaystyle C^{*}(G_{1}\times M_{1}\times M_{1})\cong\mathcal{K}\otimes
C^{*}(G_{1}),$

where the last isomorphism follows from [29, Proposition 2]. The $K$-groups of
$\mathcal{K}$ are well known —
$K_{0}(\mathcal{K})\cong\mathbb{Z},K_{0}(\mathcal{K})\cong\\{0\\}$. Hence, one
uses Connes’ Thom isomorphism [19] to compute the $K$ groups of
$\mathcal{K}\otimes C^{*}(G_{1})$. Then system (7) reads

(8)
$\displaystyle\begin{CD}\mathbb{Z}@>{}>{}>K_{0}(C^{*}(\mathcal{G}))@>{}>{}>\mathbb{Z}\\\
@A{}A{}A@V{}V{}V\\\ 0@<{}<{}<K_{1}(C^{*}(\mathcal{G}))@<{}<{}<0\end{CD}$ if
$\dim G_{1}$ is even, (9)
$\displaystyle\begin{CD}\mathbb{Z}@>{}>{}>K_{0}(C^{*}(\mathcal{G}))@>{}>{}>0\\\
@A{}A{}A@V{}V{}V\\\
\mathbb{Z}@<{}<{}<K_{1}(C^{*}(\mathcal{G}))@<{}<{}<0\end{CD}$ if $\dim G_{1}$
is odd.

In [14], Carrillo Rouse and the second author compute the $K$-theory of
boundary groupoids with $r=1$, namely:

$\displaystyle K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z},$ $\displaystyle\quad
K_{1}(C^{*}(\mathcal{G}))\cong\mathbb{Z},$ $\displaystyle\text{if $\dim G\geq
3$ odd};$ $\displaystyle K_{0}(C^{*}(\mathcal{G}))\cong\\{0\\},$
$\displaystyle\quad K_{1}(C^{*}(\mathcal{G}))\cong\mathbb{Z}\oplus\mathbb{Z},$
if $\dim G$ even.

###  2.4. The relation between Fredholm and $K_{0}(C^{*}(\mathcal{G}))$
indices

Suppose that $\mathcal{G}\rightrightarrows M$ is a boundary groupoid in the
sense of Definition 2.12. We turn to study the relationship between Fredholm
and $K_{0}(C^{*}(\mathcal{G}))$ index.

Recall [29] that an elliptic operator $P$ is Fredholm if
$P|_{\bar{\mathcal{G}}_{1}}$ is invertible, and its Fredholm index is its
image under the connecting map induced by the short exact sequence

$0\to C^{*}(M_{0}\times
M_{0})\cong\mathcal{K}\to\mathfrak{U}\to\mathfrak{U}/\mathcal{K}\to 0.$

Consider the morphism of short exact sequence

(10)
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>C^{*}(\mathcal{G}_{0})@>{}>{}>\mathfrak{U}@>{}>{}>\mathfrak{U}/C^{*}(\mathcal{G}_{0})@>{}>{}>0\\\
@V{}V{}V\Big{\|}@V{}V{}V\\\
0@>{}>{}>C^{*}(\mathcal{G})@>{}>{}>\mathfrak{U}@>{}>{}>\mathfrak{U}/C^{*}(\mathcal{G})@>{}>{}>0,\end{CD}$

where the left column is the incursion map of Lemma 2.11. The right column is
clearly defined by quotient out a bigger ideal, hence fits into the short
exact sequence

$0\to C^{*}(\mathcal{G})/C^{*}(\mathcal{G}_{0})\cong
C^{*}(\bar{\mathcal{G}}_{1})\to\mathfrak{U}/C^{*}(\mathcal{G}_{0})\to\mathfrak{U}/C^{*}(\mathcal{G})\to
0.$

By naturality of the index, one gets:

(11) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{1}(C^{*}(\bar{\mathcal{G}}_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
103.02643pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
103.02643pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{1}(\mathfrak{U}/C^{*}(\mathcal{G}_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
196.62228pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
128.10803pt\raise-20.31804pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\partial}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
137.82436pt\raise-29.49998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
196.62228pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{1}(\mathfrak{U}/C^{*}(\mathcal{G}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
230.0202pt\raise-20.31804pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\partial}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
230.0202pt\raise-29.49998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
287.41812pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
287.41812pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{0}(C^{*}(\bar{\mathcal{G}}_{1}))}$}}}}}}}{\hbox{\kern-3.0pt\raise-40.63608pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
50.01321pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
110.38753pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{0}(C^{*}(\mathcal{G}_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
202.59448pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
202.59448pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{0}(C^{*}(\mathcal{G})).}$}}}}}}}{\hbox{\kern
310.43134pt\raise-40.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

Thus, a class $[P]\in K_{1}(\mathfrak{U}/C^{*}(\mathcal{G}))$ is represented
by a Fredholm operator $[\tilde{P}]$, if and only if, its index in
$K_{0}(C^{*}(\bar{\mathcal{G}}_{1}))$ is zero, and in this case its index in
$K_{0}(C^{*}(\mathcal{G}))$ is the image of the Fredholm index of
$[\tilde{P}]$.

##  3\. Renormalized Index formulas for boundary groupoids of the form
$\mathcal{G}=(M_{0}\times M_{0})\cup(\mathbb{R}^{q}\times M_{1}\times M_{1}),$
with $q\geqslant 3$ odd

In this section, we shall derive index formulas for boundary groupoids of the
form

$\mathcal{G}=(M_{0}\times M_{0})\cup(\mathbb{R}^{q}\times M_{1}\times M_{1}),$

where $q\geqslant 3$ is an odd positive integer, as constructed in Example
2.17.

We first recall the $K$-theory computation in [14].

######  Theorem 3.1.

[14, Theorem 3.6] Let $\mathcal{G}=(M_{0}\times M_{0})\cup G_{1}\times
M_{1}\times M_{1}\rightrightarrows M$ be a boundary Lie groupoid with $G_{1}$
an exponential Lie group and $\dim G_{1}>1$ odd. Then

$K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z},\quad
K_{1}(C^{*}(\mathcal{G}))\cong\mathbb{Z}.$

######  Remark 3.2.

Putting the results of Theorem 3.1 into (9), we get the exact sequence

(12) $0\to K_{1}(C^{*}(\mathcal{G}))\cong\mathbb{Z}\to
K_{1}(C^{*}(G_{1})\otimes\mathcal{K})\to K_{0}(\mathcal{K})\cong\mathbb{Z}\to
K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z}\to 0,$

which implies the inclusion from $K_{0}(\mathcal{K})$ to
$K_{0}(C^{*}(\mathcal{G}))$ is an isomorphism. The commutative diagram (11)
now reads

(13)
$\begin{CD}K_{1}(\mathfrak{U}/C^{*}(\mathcal{G}_{M_{0}}))@>{}>{}>K_{1}(\mathfrak{U}/C^{*}(\mathcal{G}))\\\
@V{}V{\partial{}}V@V{}V{\partial{}}V\\\
K_{0}(C^{*}(\mathcal{G}_{M_{0}}))@>{\cong}>{}>K_{0}(C^{*}(\mathcal{G}))\end{CD}.$

It follows that unlike the manifold with boundary case, for elliptic Fredholm
operators, the $K$ theoretic index is the Fredholm index. Moreover, the
Fredholm index of such operator only depends on its principal symbol, as
opposed to full symbol.

Next we compute an index formula for the groupoid constructed in Example 2.17.

Let $r:=d(\cdot,M_{1})$ be the Riemannian distance function on $M=M_{0}\cup
M_{1}$ with respect to some Riemannian metric. We assume that there are
coordinate charts $(U_{\alpha},\mathbf{x}_{\alpha})$ such that
$M_{1}\subset\cup_{\alpha}U_{\alpha}$, and on each $U_{\alpha}$ the image of
the anchor map is spanned by

$r^{N}\frac{\partial}{\partial x_{1}},\cdots,r^{N}\frac{\partial}{\partial
x_{q}},\frac{\partial}{\partial y_{1}},\cdots,\frac{\partial}{\partial y_{\dim
M_{1}}}.$

Note that these vector fields are smooth at $0$, which implies that $N$ is
necessarily even. In cylindrical coordinates the structural vector fields are
spanned by

$r^{N}\frac{\partial}{\partial
r},r^{N-1}\frac{\partial}{\partial\theta_{1}},\cdots,r^{N-1}\frac{\partial}{\partial\theta_{q-1}},\frac{\partial}{\partial
y_{1}},\cdots,\frac{\partial}{\partial y_{\dim M_{1}}}.$

The Riemannian density of the induced metric on $M_{0}$ is of the form

(14) $\mu_{M_{0}}:=r^{-qN}\mu_{M}\,(=r^{-Nq+(q-1)}\,dr\,d\theta\,dy)$

for some smooth density $\mu_{M}$ on $M$ restricted to $M_{0}$. Without loss
of generality we shall assume $\mu_{M}$ is the Lebesgue measure with respect
to each $\mathbf{x}_{\alpha}$.

The volume form (14) implies $M_{0}$ is of polynomial volume growth. By Lemma
2.6, the space of Schwartz kernels $S^{*}(\mathcal{G})$ and
$\Psi^{0}(\mathcal{G},E)\oplus S^{*}(\mathcal{G})$ are also closed under
holomorphic functional calculus in $\mathfrak{U}(\mathcal{G})$. In particular,
$\cup_{m}\Psi^{m}(\mathcal{G},E)\oplus S^{*}(\mathcal{G})$ satisfies all
axioms in [5, (i)-(vii),($\sigma$),($\psi$)]. It follows from [5, Theorem 7.2]
that for any uniformly supported elliptic, positive definite operator $Q$ of
order first order (for example $Q=(\varDelta+I)^{\frac{1}{2}}$, see [29]), the
complex powers $Q^{z}$ are well defined as pseudo-differential operators lying
in $\Psi^{-z}(\mathcal{G},E)\oplus\mathcal{S}^{*}(\mathcal{G})$.

For any $P\in\Psi^{m}(\mathcal{G},E)$, the operator $PQ^{-\tau}$ is given by a
sufficiently smooth kernel whenever the real part of $\tau$ is sufficiently
large. Hence it makes sense to consider

$\displaystyle Z(P;\tau,z)$ $\displaystyle:=$
$\displaystyle\int_{M_{0}}\operatorname{tr}(r^{z-qN}\,P\,Q^{-\tau})\,\mu_{M}$
$\displaystyle=$
$\displaystyle\int_{M_{0}}\operatorname{tr}(r^{-z}PQ^{-\tau}\big{)}\,\mu_{M_{0}},\quad\tau,z\in\mathbb{C},$

where we regard $r^{z-qN}$ as the operator defined by multiplying by a scalar
function. By [41, Lemma 1], $Z(P;\tau,z)$ is a meromorphic function of $\tau$
and $z$, with at most simple pole at $\tau=z=0$. Hence we make the following
definition.

######  Definition 3.3.

Given a holomorphic family $B(\tau,z)$ of pseudo-differential operators, its
renormalized trace is defined to be

$\displaystyle\widehat{\operatorname{Tr}}(B)$ $\displaystyle:=$
$\displaystyle\frac{\partial^{2}}{\partial\tau\partial
z}\Big{|}_{\tau=0,z=0}\int_{M_{0}}\tau
z\operatorname{tr}\big{(}r^{z-qN}B(\tau,z)Q^{-\tau}\big{|}_{M}\big{)}\,\mu_{M}$
$\displaystyle=$ $\displaystyle\frac{\partial^{2}}{\partial\tau\partial
z}\Big{|}_{\tau=0,z=0}\int_{M_{0}}\tau
z\operatorname{tr}\big{(}r^{z}B(\tau,z)Q^{-\tau}\big{|}_{M}\big{)}\,\mu_{M_{0}}.$

In view of Remark 3.2, to derive an index formula it suffices to consider the
Fredholm index. Suppose now that $D\in\mathrm{D}^{1}(\mathcal{G},E)$ is a
fully elliptic, first order differential operator. Using the results of [47,
Section 4], there exists a smooth pseudo-differential parametrix $\varPhi$ of
$D$ up to finite rank operators and vanishing up to infinite order at
$\mathcal{G}_{1}$. By the Calderon’s formula, we obtain the following
proposition.

######  Proposition 3.4.

With the notation as above, the Fredholm index of $D$ equals

(15)
$\partial([D])=\widehat{\operatorname{Tr}}([D,\varPhi])\,(=\int_{M_{0}}\operatorname{tr}([D,\varPhi])\,\mu_{M_{0}}).$

Our goal is to figure out $\widehat{\operatorname{Tr}}([D,\varPhi])$, which
will give us an index formula. Following the arguments of [41, Proposition 8],
we have by direct calculation

(16) $r^{z}[D,\varPhi]Q^{-\tau}=(r^{z}Dr^{-z}-D)r^{z}\varPhi
Q^{-\tau}+r^{z}\varPhi(Q^{-\tau}DQ^{\tau}-D)Q^{-\tau}+[D,r^{z}\varPhi
Q^{-\tau}].$

The right-hand side of Equation (16) has three terms. It is cleat that the
last term has vanishing trace (when the real part of $z$ and $\tau$ sufficient
large the renormalized trace is just the usual trace), hence it suffices to
work out the first two terms.

Let us consider the first term of Equation (16) which corresponds to the
$\eta$ term in [41, Equation (18)]. However, unlike the manifold with boundary
case, we now show the renormalized trace vanishes in our case.

######  Lemma 3.5.

The renormalized trace

(17) $\frac{\partial^{2}}{\partial\tau\partial z}\Big{|}_{\tau=0,z=0}\tau
z\int_{M_{0}}\operatorname{tr}(r^{-qN}(r^{z}Dr^{-z}-D)r^{z}\varPhi
Q^{-\tau})\,\mu_{M}=0.$

###### Proof.

The proof reduces to a local calculation by partition of unity argument. We
parameterize each $U_{\alpha}$ using cylindrical coordinates and let
$\sigma:U_{\alpha}\to U_{\alpha}$ (shrinking $U_{\alpha}$ if necessary) be the
cylindrical anti-pole map, i.e.,

$(r,\sigma(\theta),y):=(-(r,\theta),y).$

By assumption, the operator $D$ is the composition of one of the vector fields
in (6) and tensors. Taking local trivialization, one writes $D$ in cylindrical
coordinates:

$\displaystyle D=$ $\displaystyle
D_{0}(r,\theta,y)r^{N}\frac{\partial}{\partial
r}+D_{1}(r,\theta,y)r^{N-1}\frac{\partial}{\partial\theta_{1}}+\cdots+D_{q-1}(r,\theta,y)r^{N-1}\frac{\partial}{\partial\theta_{q-1}}$
$\displaystyle+D^{\prime}_{1}(r,\theta,y)\frac{\partial}{\partial
y_{1}}+\cdots+D^{\prime}_{\dim M_{1}}(r,\theta,y)\frac{\partial}{\partial
y_{\dim M_{1}}}+D^{\prime\prime}(r,\theta,y).$

Consider $(r^{z}Dr^{-z}-D)r^{z}$. One has

$\big{[}r^{N}\frac{\partial}{\partial r},r^{z}\big{]}=zr^{z+N-1},$

which implies

$\big{(}r^{z}(r^{N}\frac{\partial}{\partial
r})r^{-z}-r^{N}\frac{\partial}{\partial r}\big{)}r^{z}\varPhi
Q^{-\tau}=zr^{z+N-1}\varPhi Q^{-\tau};$

while other commutators vanish. It follows that

$(r^{z}Dr^{-z}-D)r^{z}\varPhi Q^{-\tau}=zr^{z-1}r^{N}D_{0}\varPhi Q^{-\tau}.$

In cylindrical coordinates, denoting the volume measure of the $q-1$
dimensional sphere by $d\theta$, we have

$\displaystyle\tau
z\,\int_{M_{0}}\operatorname{tr}(r^{-qN}(r^{z}Dr^{-z}-D)r^{z}\varPhi
Q^{-\tau})\,\mu_{M}$ $\displaystyle=$ $\displaystyle\tau
z^{2}\int_{M_{0}}r^{z-1+N-qN}\,\operatorname{tr}(D_{0}\varPhi
q^{-\tau})\,\mu_{M}$ $\displaystyle=$ $\displaystyle\tau
z^{2}\int_{M_{0}}r^{z-1+N-qN}\,\operatorname{tr}(D_{0}\varPhi
q^{-\tau})\,r^{q-1}\,dr\,d\theta\,dy$ $\displaystyle=$ $\displaystyle\tau
z\int_{M_{0}}z\,r^{z-1+N}\,r^{-qN+q-1}\,\operatorname{tr}(D_{0}\varPhi
q^{-\tau})\,dr\,d\theta\,dy.$

Hence the left-hand side in (17) becomes (by linearization)

(18) $\int z\,r^{z-1+N}r^{-qN+q-1}\operatorname{tr}(D_{0}\varPhi
Q^{-\tau})\,dr\,d\theta\,dy.$

Expand $D_{0}$ as a Taylor series

$D_{0}=A_{0}(\theta,y)+rA_{1}(\theta,y)+\cdots,$

for some matrices $A_{j}(\theta,y)$. Since $D_{0}r^{N}\frac{\partial}{\partial
r}$ is a smooth differential operator at $\\{r=0\\}$, it follows that

$A_{j}(\sigma(\theta),y)=-(-1)^{j}A_{j}(\theta,y),\quad j=0,1,2,\cdots.$

Similarly, for any $\tau$ with sufficiently large real part, $\varPhi
Q^{-\tau}$ is given by a sufficiently smooth kernel. It follows that
$\operatorname{tr}(D_{0}\varPhi Q^{-\tau})$ has Taylor expansion with respect
to $r$:

$\operatorname{tr}(D_{0}\varPhi
Q^{-\tau})=F_{0}(\theta,y)+F_{1}(\theta,y)r+\cdots.$

satisfying

(19) $F_{j}(\sigma(\theta),y)=-(-1)^{j}F_{j}(\theta,y),\quad j=0,1,2,\cdots.$

In (18), one replaces the trace factor $\operatorname{tr}(D_{0}\varPhi
Q^{-\tau})$ by its Taylor expansion, integrates with respect to $r$, and takes
the constant term to get:

(20) $\int F_{(N-1)(q-1)}(\theta,y)d\theta dy.$

Since $(q-1)(N-1)$ is even, it follows from parity (19) that the $\theta$
integral in (20) also vanishes for all $\tau$ with sufficiently positive real
part. Hence the lemma follows by meromorphic extension. ∎

It remains to consider the second term of Equation (16). This term has been
computed in [41, Proposition 12], and we shall briefly recall the result here.

######  Lemma 3.6.

One has

$\widehat{\operatorname{Tr}}(\varPhi(Q^{-\tau}DQ^{\tau}-D)Q^{-\tau})=\int_{M_{0}}r^{z-qN}\,a_{0}\,\mu_{M}\Big{|}_{z=0}=\Big{(}\int_{M_{0}}r^{z}\,a_{0}\,\mu_{M_{0}}\Big{|}_{z=0}\Big{)},$

where $a_{0}$ is the constant term in the $t\to 0$ asymptotic expansion of the
trace of heat kernel

$\operatorname{tr}\big{(}(e^{-tDD^{*}}+e^{-tD^{*}D})\big{|}_{M}\big{)}.$

###### Proof.

We fix

$\displaystyle Q:=Q_{1}:=$
$\displaystyle(DD^{*}+\varPi_{\ker(D^{*})})^{\frac{1}{2}}$ $\displaystyle
Q_{2}:=$ $\displaystyle(D^{*}D+\varPi_{\ker(D)})^{\frac{1}{2}}$
$\displaystyle\varPhi:=$ $\displaystyle D^{*}Q_{1}^{-2}=Q_{2}^{-2}D^{*},$

where $\varPi_{\ker(D)}$ and $\varPi_{\ker(D^{*})}$ denote orthogonal
projections (in $L^{2}(M_{0})$) onto $\ker(D)$ and $\ker(D^{*})$,
respectively. Note that for any $\tau\in\mathbb{C}$,

$Q_{1}^{-\tau}D=DQ_{2}^{-\tau}.$

It follows that for any $\tau,z\in\mathbb{C}$:

$\displaystyle\varPhi(Q^{-\tau}DQ^{\tau}-D)Q^{-\tau}=$
$\displaystyle\>\varPhi(Q_{1}^{-\tau}D-DQ_{1}^{-\tau})$ $\displaystyle=$
$\displaystyle\>\varPhi D(Q_{2}^{-\tau}-Q_{1}^{-\tau})$
$\displaystyle\widehat{\operatorname{Tr}}(\varPhi(Q^{-\tau}DQ^{\tau}-D)Q^{-\tau})=$
$\displaystyle\widehat{\operatorname{Tr}}(Q_{2}^{-\tau}-Q_{1}^{-\tau}).$

Since $\tau^{-1}(Q_{2}^{-\tau}-Q_{1}^{-\tau})$ is a holomorphic family, by
[41, Lemma 1], the integral

$\int_{M_{0}}\operatorname{tr}(r^{z-qN}(Q_{2}^{-\tau}-Q_{1}^{-\tau}))\,\mu_{M}$

is holomorphic at $\tau=0$. Using the Mellin transform one gets for any $\tau$
with sufficiently large real part

$\operatorname{tr}(Q_{2}^{-\tau}-Q_{1}^{-\tau}|_{M})=\frac{1}{\Gamma(\tau/2)}\int_{0}^{\infty}t^{\frac{\tau}{2}-1}\operatorname{tr}\big{(}e^{-t(DD^{*}+\varPi_{\ker(D^{*})})}-e^{-t(D^{*}D+\varPi_{\ker(D^{*})})}|_{M}\big{)}dt.$

Since it is clear that

$\displaystyle e^{-t(DD^{*}+\varPi_{\ker(D^{*})})}=$ $\displaystyle
e^{-tDD^{*}}+(e^{t}-1)\varPi_{\ker(D^{*})}$ $\displaystyle
e^{-t(D^{*}D+\varPi_{\ker(D^{*})})}=$ $\displaystyle
e^{-tD^{*}D}+(e^{t}-1)\varPi_{\ker(D^{*})},$

the claim follows. ∎

By the discussion in Section 2.4, for any elliptic operator $D$ in
$\mathrm{D}^{1}(\mathcal{G},E)$, we can find a fully elliptic operator
$\widetilde{D}$ in $\mathrm{D}^{1}(\mathcal{G},E)$ such that
$[D]=[\widetilde{D}]$, i.e., in the same $K$-theory class. Summarizing the
calculations above, we obtain the following theorem.

######  Theorem 3.7.

For any elliptic, differential operator $D\in\mathrm{D}^{1}(\mathcal{G},E)$,
the index of $D$ in $K_{0}(C^{*}(\mathcal{G}))$ equals

(21)
$\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M_{0}}r^{z-qN}\,a_{0}\,\mu_{M}\Big{|}_{z=0},$

where $a_{0}$ is the constant term in the $t\to 0$ asymptotic expansion of the
trace of heat kernel.

Recall that Bohlen and Schrohe in [10] proved that for a Dirac type operator
$D$ on a Lie manifold (and by [9] general fully elliptic operators reduce to
the Dirac case), its index is

$\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M}\operatorname{AS}(D)+\eta(D),$

where

$\eta(D)=\frac{1}{2}\int_{0}^{\infty}{}^{\nu}\operatorname{Tr}_{s}([D,e^{-tD^{2}}])dt$

and the ${}^{\nu}\operatorname{Tr}_{s}$ is the Hadamard (i.e. cutoff)
regularized super-trace.

Comparing with Equation (17), we obtain the following non-trivial result.

######  Corollary 3.8.

With the notation as above, we have

$\int_{0}^{\infty}{}^{\nu}\operatorname{Tr}_{s}([D,e^{-tD^{2}}])dt=0.$

##  4\. The case that $q=1$

We turn to consider boundary groupoids with one-dimensional isotropy
subgroups.

###  4.1. Index formula for connected manifolds with connected boundaries

In this subsection, we assume that $M=M_{0}\cup M_{1}$ is a connected manifold
with connected boundary, where we denote by $M_{0}$ and $M_{1}$ the interior
and boundary of $M$, respectively. Then the groupoid that we consider in this
subsection is of the form

$\mathcal{G}=(M_{0}\times M_{0})\cup(\mathbb{R}\times M_{1}\times M_{1}),$

where $M_{1}$ is the connected boundary of $M$.

With a modified argument in [14, Remark 3.7], we conclude the following
$K$-theory result.

######  Lemma 4.1.

Under the same assumption as above, one has

$K_{0}(C^{*}(\mathcal{G}))\cong\\{0\\},\quad
K_{1}(C^{*}(\mathcal{G}))\cong\\{0\\},$

that is, the $K$-theoretic index is trivial.

######  Remark 4.2.

We have the exact sequence [37]

$0\to
K_{1}(C^{*}(\mathcal{G}))\to\mathbb{Z}\xrightarrow{\partial}\mathbb{Z}\to
K_{0}(C^{*}(\mathcal{G})).$

It follows that the connecting map $\partial$ is an isomorphism. More
concretely, we have

(22) $\partial(1)=1.$

Suppose that $M_{1}=r^{-1}(0)$ for some defining function $r\in
C^{\infty}(M)$, and furthermore that the structural vector fields near $M_{1}$
are spanned by

$r^{N}\frac{\partial}{\partial r},\frac{\partial}{\partial
y_{1}},\cdots,\frac{\partial}{\partial y_{\dim M_{1}}},$

for some integer $N$, then formulas computing the Fredholm index of an
elliptic operator $D$ that is invertible on $\mathcal{G}_{1}$ is well known —
one of which is by adapting the arguments in Section 3. We just briefly recall
the result here.

######  Proposition 4.3.

[41, Proposition 12, Proposition 13] For any first order elliptic differential
operator $D$ on the groupoid $\mathcal{G}=(M_{0}\times
M_{0})\cup(\mathbb{R}\times M_{1}\times M_{1})$ defined as above, that is
invertible on $\mathbb{R}\times M_{1}\times M_{1}$, the Fredholm index of
$D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}$ equals

(23)
$\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M_{0}}a_{0}\,\mu_{M}+\eta(D),$

where $a_{0}$ is the constant term in $t\to 0$ asymptotic expansion of the
trace of heat kernel

$\operatorname{tr}\big{(}(e^{-tDD^{*}}+e^{-tD^{*}D})\big{|}_{M}\big{)},$

and

$\eta(D):=\frac{\partial^{2}}{\partial\tau\partial
z}\Big{|}_{\tau=0,z=0}\int_{M_{0}}\operatorname{tr}\big{(}r^{z-N}(D-r^{-z}Dr^{z})(DD^{*}+\operatorname{id})^{-\frac{1}{2}-\tau}\big{)}\mu_{M}$

depends only on $D|_{\mathbb{R}\times M_{1}\times M_{1}}$.

###  4.2. An Index formula for closed manifolds separated by interior
hypersurfaces

In this subsection, let $M$ be a compact manifold without boundary, $M_{1}$ a
closed sub-manifold of co-dimension one. We suppose further that

$M=M^{\prime}_{0}\cup M_{1}\cup M^{\prime\prime}_{0},$

where $M^{\prime}_{0}$ and $M^{\prime\prime}_{0}$ are open and connected
submanifolds, and their closures $\bar{M}^{\prime}_{0}$ and
$\bar{M}^{\prime\prime}_{0}$ are manifolds with the same boundary $M_{1}$. In
particular, our assumption implies the normal bundle of $M_{1}$ is trivial.

We consider the subspace of vector fields tangential to $M_{1}$. The
corresponding Lie groupoid is

$\mathcal{G}=(M^{\prime}_{0}\times
M^{\prime}_{0})\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})\cup(\mathbb{R}\times M_{1}\times M_{1}),$

which contains

$\mathcal{G}^{\prime}:=(M^{\prime}_{0}\times
M^{\prime}_{0})\cup(\mathbb{R}\times M_{1}\times
M_{1}),\quad\mathcal{G}^{\prime\prime}=(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})\cup(\mathbb{R}\times M_{1}\times M_{1})$

as invariant subspaces.

######  Example 4.4.

Let $\mathbb{R}$ act on itself by $(x,t)\rightarrow x\cdot e^{t}$. There are
three orbits $(-\infty,0)$, $\\{0\\}$, and $(0,\infty)$. Then set
$M^{\prime}=(-\infty,0)$, $M^{\prime\prime}=(0,\infty)$, and $M_{1}=\\{0\\}$.
The action groupoid
$\mathcal{G}=\mathbb{R}\rtimes\mathbb{R}\rightrightarrows\mathbb{R}$ satisfies
the above conditions.

We first compute $K$-theory of the groupoid $C^{*}$-algebra of $\mathcal{G}$
defined as above.

######  Theorem 4.5.

Under the same assumption as above, one has

$K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z}.$

###### Proof.

For the composition series of $\mathcal{G}$, we have

$C^{*}((M^{\prime}_{0}\times M^{\prime}_{0})\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))=C^{*}(M^{\prime}_{0}\times M^{\prime}_{0})\oplus
C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})=\mathcal{K}\oplus\mathcal{K},$

where each copy of $\mathcal{K}$ corresponds to one component. Therefore
Diagram (9) becomes the exact sequence:

(24) $K_{1}(C^{*}(\mathbb{R}\times M_{1}\times
M_{1}))\cong\mathbb{Z}\xrightarrow{\partial{}}\mathbb{Z}\oplus\mathbb{Z}\xrightarrow{\iota_{*}}K_{0}(C^{*}(\mathcal{G}))\to
0.$

We compute the connecting map $\partial$ using the morphism of exact sequences

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>\begin{array}[]{c}C^{*}(M^{\prime}_{0}\times
M^{\prime}_{0})\\\ \oplus C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})\end{array}@>{}>{}>C^{*}(\mathcal{G})@>{}>{}>C^{*}(\mathcal{G})/(\mathcal{K}\oplus\mathcal{K})@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\
0@>{}>{}>\mathcal{K}\oplus\mathcal{K}@>{}>{}>\begin{array}[]{c}C^{*}(\mathcal{G}^{\prime})\\\
\oplus
C^{*}(\mathcal{G}^{\prime\prime})\end{array}@>{}>{}>\begin{array}[]{c}C^{*}(\mathcal{G}^{\prime})/\mathcal{K}\\\
\oplus
C^{*}(\mathcal{G}^{\prime\prime})/\mathcal{K}\end{array}@>{}>{}>0,\end{CD}$

where the middle column map is defined by restriction to
$\mathcal{G}^{\prime}$ and $\mathcal{G}^{\prime}$. Thus, one gets the
commutative diagram:

$\begin{CD}K_{1}(C^{*}(\mathbb{R}\times M_{1}\times
M_{1}))@>{\partial{}}>{}>K_{0}(C^{*}(M^{\prime}_{0}\times
M^{\prime}_{0}))\oplus K_{0}(C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))\\\ @V{}V{}V\Big{\|}\\\
K_{1}(C^{*}(\mathcal{G}^{\prime})/\mathcal{K})\oplus
<EMAIL_ADDRESS>
Here, the bottom row is just two copies of the connection map in (22). On the
other hand, we have

$C^{*}(\mathbb{R}\times M_{1}\times
M_{1})=C^{*}(\mathcal{G}^{\prime})/C^{*}(M^{\prime}_{0}\times
M^{\prime}_{0})=C^{*}(\mathcal{G}^{\prime\prime})/C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}),$

and the left column is just two copies of the identity map. It follows that

$\partial(1)=1\oplus 1.$

Hence $K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z}$ and
$\iota_{*}:\mathbb{Z}\oplus\mathbb{Z}\to\boldsymbol{\mathbb{K}}_{0}(C^{*}(\mathcal{G}))$
in (24) is realized as

(25) $\iota_{*}(1\oplus 0)=1,\iota_{*}(0\oplus 1)=-1.\qed$

We then turn to exploit the index of an elliptic pseudo-differential operators
on $\mathcal{G}$.

First, given any elliptic pseudo-differential operator $P$ that is invertible
on $\mathcal{G}_{1}$, we consider the index of $P$ in
$K_{0}(C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\cup(M^{\prime\prime}_{0}\times M^{\prime\prime}_{0})))$. One
has the homomorphism of short exact sequences:

$\begin{CD}0\to\begin{array}[]{c}C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\\\ \quad\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))\end{array}@>{}>{}>\mathfrak{U}(\mathcal{G})@>{}>{}>\mathfrak{U}(\mathcal{G})\Big{/}\begin{array}[]{c}C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\\\ \quad\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))\end{array}\to 0\\\ \Big{\|}@V{}V{}V@V{}V{}V\\\
0\to\begin{array}[]{c}C^{*}(M^{\prime}_{0}\times M^{\prime}_{0})\\\ \oplus
C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})\end{array}@>{}>{}>\begin{array}[]{c}\mathfrak{U}(\mathcal{G}^{\prime})\\\
\oplus\mathfrak{U}(\mathcal{G}^{\prime\prime})\end{array}@>{}>{}>\begin{array}[]{c}\mathfrak{U}(\mathcal{G}^{\prime})/C^{*}(M^{\prime}_{0}\times
M^{\prime}_{0})\\\
\oplus\mathfrak{U}(\mathcal{G}^{\prime\prime})/C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})\end{array}\to 0,\end{CD}$

where the middle map is given by (the extension of) restricting the pseudo-
differential operator on $\mathcal{G}$ to the invariant subspaces
$\mathcal{G}^{\prime}$ and $\mathcal{G}^{\prime\prime}$, and it is clear that
the restriction map induces a map for the right column.

By the naturality of the index map, one has

(26)
$\begin{CD}K_{1}\Big{(}\mathfrak{U}(\mathcal{G})\Big{/}\begin{array}[]{c}C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\\\ \quad\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))\end{array}\Big{)}@>{\partial}>{}>K_{0}\Big{(}\begin{array}[]{c}C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\\\ \quad\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))\end{array}\Big{)}\\\ @V{}V{}V\Big{\|}\\\
\begin{array}[]{c}K_{1}(\mathfrak{U}(\mathcal{G}^{\prime})/C^{*}(M^{\prime}_{0}\times
M^{\prime}_{0}))\\\ \oplus
K_{1}(\mathfrak{U}(\mathcal{G}^{\prime\prime})/C^{*}(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}))\end{array}@>{\partial\oplus\partial}>{}>K(\mathcal{K}\oplus\mathcal{K})\cong\mathbb{Z}\oplus\mathbb{Z},\end{CD}$

where ${\partial}\oplus{\partial}$ is just the Fredholm index maps for each
copy.

To conclude, the index of $P$ in $K_{0}(C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\cup(M^{\prime\prime}_{0}\times M^{\prime\prime}_{0})))$ is the
direct sum of the Fredholm indices of $P|_{\mathcal{G}^{\prime}}$ and
$P|_{\mathcal{G}^{\prime\prime}}$, and the Fredholm index of $P$ is clearly
the arithmetic sum of the two indices in $\mathbb{Z}$.

We are in position to derive an index formula (in $K_{0}(C^{*}\mathcal{G})$)
for first order elliptic differential operators on $\mathcal{G}$.

######  Theorem 4.6.

Let $D$ be a first order elliptic differential operator in
$\mathrm{D}^{1}(\mathcal{G},E)+S^{*}(\mathcal{G})$. The following index
formula holds

$\partial([D(DD)^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M^{\prime}_{0}}a^{\prime}_{0}-\int_{M^{\prime\prime}_{0}}a^{\prime\prime}_{0}\in
K_{0}(C^{*}(\mathcal{G}))\cong\mathbb{Z},$

where $a_{0}^{\prime}$ and $a_{0}^{\prime\prime}$ are defined in the same way
as in Proposition 4.3 .

###### Proof.

We first suppose that $D$ is a first order (pseudo-)elliptic differential
operator that is invertible on $\mathbb{R}\times M_{1}\times M_{1}$, and
compare its index in $K_{0}(C^{*}(\mathcal{G}))$ and
$K_{0}(C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\cup(M^{\prime\prime}_{0}\times M^{\prime\prime}_{0})))$ using
Equation (11). One gets

(27) $\begin{CD}K_{1}(\mathfrak{U}/C^{*}((M^{\prime}_{0}\times
M^{\prime}_{0})\cup(M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0})))@>{}>{}>K_{0}(\mathcal{K}\oplus\mathcal{K})=\mathbb{Z}\oplus\mathbb{Z}\\\
@V{}V{}V@V{}V{\iota_{*}}V\\\
K_{1}(\mathfrak{U}/C^{*}(\mathcal{G}))@>{\partial{}}>{}>K_{0}(C^{*}(\mathcal{G})),\end{CD}$

where the right column is just (25). It follows from Proposition 4.3 that

(28)
$\partial([D(DD)^{*}+\operatorname{id})^{-\frac{1}{2}}])=\int_{M^{\prime}_{0}}a^{\prime}_{0}-\int_{M^{\prime\prime}_{0}}a^{\prime\prime}_{0}\in
K_{0}(\mathcal{C}^{*}(\mathcal{G}))\cong\mathbb{Z},$

where $a^{\prime}_{0}$ and $a^{\prime\prime}_{0}$ are respectively the
constant term in $t\to 0$ asymptotic expansion of the traces of the heat
kernels

$e^{-tDD^{*}}|_{M^{\prime}_{0}\times
M^{\prime}_{0}}+e^{-tD^{*}D}|_{M^{\prime}_{0}\times M^{\prime}_{0}}\,\text{
and }\,e^{-tDD^{*}}|_{M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}}+e^{-tD^{*}D}|_{M^{\prime\prime}_{0}\times
M^{\prime\prime}_{0}}.$

Next, let us consider general case when $D$ belongs to
$\mathrm{D}^{1}(\mathcal{G},E)+S^{*}(\mathcal{G})$ and is elliptic. By the
discussion in Section 2.4, there exists a Fredholm operator $\widetilde{D}$ of
the form

$\widetilde{D}=D+R,\,\,R\in S^{*}(\mathcal{G})$

such that

$\partial([\widetilde{D}(\widetilde{D}\widetilde{D}^{*}+\operatorname{id})^{-\frac{1}{2}}])=\partial([D(DD^{*}+\operatorname{id})^{-\frac{1}{2}}])\in
K_{0}(C^{*}(\mathcal{G})).$

Since $\widetilde{D}$ and $D$ have the same principal symbol, the asymptotic
expansions as $t\to 0$ of the traces of their heat kernels have the same
constant. So we are done. ∎

######  Remark 4.7.

The above theorem shows that in this case the $K$-theory index is the
difference of Fredholm indices.

##  5\. Concluding remarks

In this paper we have derived index formulas for some boundary groupoid
$C^{*}$-algebras. It is obvious that we are not considering the most general
cases. It seems clear that more examples can be computed by combining the
above results with combinatorial techniques.

It seems that for the groupoids constructed in Example 2.17, one can define
classifying space as in [14] (with $M=\mathbb{S}^{N}\times\mathbb{S}^{q}$).
Then an embedding $M\to\mathbb{S}^{N}\times\mathbb{S}^{q}$ induces a groupoid
homomorphism and the arguments of [39] can be used to derive an index formula.
That should work for all $q\geq 2$.

However, constructing such groupoid homomorphisms may not be feasible for
general boundary groupoids with $r=1$. When $M_{1}$ is a point and
$G_{1}=\mathbb{R}^{q}$, one can adapt the argument index formula [15] for
‘asymptotically Abelian’ manifolds, and then try to generalize this approach
by using excision.

It would also be interesting to consider manifolds with fibered boundary. In
other words, one considers groupoids which are a union of fibered products:

$\mathcal{G}=\bigcup_{i}G_{i}\times(M_{i}\times_{B_{i}}M_{i}).$

This includes the examples considered in [24, 35] and many others. It should
be particularly interesting to generalize the Dirac operator and index pairing
approach [1, 9, 26, 48] to this case.

Also it is not clear how a renormalized integral can be defined for boundary
groupoids with $r\geq 2$. Last but not the least the $\eta$-term of the even
dimensional case is still mysterious. We expect some interesting new
invariants arising from the relevant index formulas.

## References

  * [1] I. Akrour and P. Carrillo Rouse.  Longitudinal b-operators, blups and index theorems.  arXiv:1711.11197v3, 2018.
  * [2] P. Albin.  A renormalized index theorem for some complete asymptotically regular metrics: the Gauss-Bonnet theorem.  Adv. Math., 213(1):1–52, 2007.
  * [3] B. Ammann, R. Lauter, and V. Nistor.  On the geometry of Riemannian manifolds with a Lie structure at infinity.  Int. J. Math. Math. Sci., (1-4):161–193, 2004.
  * [4] B. Ammann, R. Lauter, and V. Nistor.  Pseudodifferential operators on manifolds with a Lie structure at infinity.  Ann. of Math. (2), 165(3):717–747, 2007.
  * [5] B. Ammann, R. Lauter, V. Nistor, and A. Vasy.  Complex powers and non-compact manifolds.  Comm. Partial Differential Equations, 29(5-6):671–705, 2004.
  * [6] I. Androulidakis and G. Skandalis.  The analytic index of elliptic pseudodifferential operators on a singular foliation.  J. K-Theory, 8(3):363–385, 2011.
  * [7] I. Androulidakis and G. Skandalis.  Pseudodifferential calculus on a singular foliation.  J. Noncommut. Geom., 5(1):125–152, 2011.
  * [8] K. Bohlen.  Groupoids and singular manifolds.  arXiv:1601.04166, 2016.
  * [9] K. Bohlen and J.-M. Lescure.  A geometric approach of $K$-homology for Lie manifolds.  arXiv:/1904.04069v1, 2019.
  * [10] K. Bohlen and E. Schrohe.  Getzler rescaling via adiabatic deformation and a renormalized index formula.  J. Math. Pures Appl. (9), 120:220–252, 2018.
  * [11] P. Carrillo Rouse and J.-M. Lescure.  Geometric obstructions for Fredholm boundary conditions for manifolds with corners.  Ann. K-Theory, 3(3):523–563, 2018.
  * [12] P. Carrillo Rouse, J.-M. Lescure, and B. Monthubert.  A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary.  J. Topol. Anal., 6(1):27–74, 2014.
  * [13] P. Carrillo-Rouse and B. Monthubert.  An index theorem for manifolds with boundary.  C. R. Math. Acad. Sci. Paris, 347(23-24):1393–1398, 2009.
  * [14] P. Carrillo Rouse and B. K. So.  K-theory and index theory for some boundary groupoids.  Results Math., 75(4):Paper No. 172, 20pp, 2020.
  * [15] C. Carvalho and V. Nistor.  An index formula for perturbed Dirac operators on Lie manifolds.  J. Geom. Anal., 24(4):1808–1843, 2014.
  * [16] C. Carvalho, V. Nistor, and Y. Qiao.  Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras.  Electron. Res. Announc. Math. Sci., 24:68–77, 2017.
  * [17] C. Carvalho, V. Nistor, and Y. Qiao.  Fredholm conditions on non-compact manifolds: theory and examples.  In Operator theory, operator algebras, and matrix theory, volume 267 of Oper. Theory Adv. Appl., pages 79–122. Birkhäuser/Springer, Cham, 2018.
  * [18] R. Côme.  The Fredholm property for groupoids is a local property.  Results Math., 74(4):Paper No. 160, 33, 2019.
  * [19] A. Connes.  An analogue of the Thom isomorphism for crossed products of a $C^{\ast}$-algebra by an action of ${\bf R}$.  Adv. in Math., 39(1):31–55, 1981.
  * [20] A. Connes.  Noncommutative geometry.  Academic Press, Inc., San Diego, CA, 1994.
  * [21] M. Crainic and R. L. Fernandes.  Integrability of Lie brackets.  Ann. of Math. (2), 157(2):575–620, 2003.
  * [22] C. Debord.  Holonomy groupoids of singular foliations.  J. Differential Geom., 58(3):467–500, 2001.
  * [23] C. Debord, J.-M. Lescure, and V. Nistor.  Groupoids and an index theorem for conical pseudo-manifolds.  J. Reine Angew. Math., 628:1–35, 2009.
  * [24] C. Debord, J.-M. Lescure, and F. Rochon.  Pseudodifferential operators on manifolds with fibred corners.  Ann. Inst. Fourier (Grenoble), 65(4):1799–1880, 2015.
  * [25] C. Debord and G. Skandalis.  Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus.  arXiv:1705.09588, 2017.
  * [26] C. Debord and G. Skandalis.  Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps.  J. Geom. Phys., 129:255–268, 2018.
  * [27] R. Lauter, B. Monthubert, and V. Nistor.  Pseudodifferential analysis on continuous family groupoids.  Doc. Math., 5:625–655, 2000.
  * [28] R. Lauter, B. Monthubert, and V. Nistor.  Spectral invariance for certain algebras of pseudodifferential operators.  J. Inst. Math. Jussieu, 4(3):405–442, 2005.
  * [29] R. Lauter and V. Nistor.  Analysis of geometric operators on open manifolds: a groupoid approach.  In Quantization of singular symplectic quotients, volume 198 of Progr. Math., pages 181–229. Birkhäuser, Basel, 2001.
  * [30] P.-Y. Le Gall and B. Monthubert.  $K$-theory of the indicial algebra of a manifold with corners.  $K$-Theory, 23(2):105–113, 2001.
  * [31] P. Loya.  The index of $b$-pseudodifferential operators on manifolds with corners.  Ann. Global Anal. Geom., 27(2):101–133, 2005.
  * [32] J.-H. Lu.  A note on Poisson homogeneous spaces.  In Poisson geometry in mathematics and physics, volume 450 of Contemp. Math., pages 173–198. Amer. Math. Soc., Providence, RI, 2008.
  * [33] K. Mackenzie.  Lie groupoids and Lie algebroids in differential geometry, volume 124 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 1987.
  * [34] K. Mackenzie.  General theory of Lie groupoids and Lie algebroids, volume 213 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2005.
  * [35] R. Mazzeo.  Elliptic theory of differential edge operators. I.  Comm. Partial Differential Equations, 16(10):1615–1664, 1991.
  * [36] R. Melrose.  The Atiyah-Patodi-Singer index theorem, volume 4 of Research Notes in Mathematics.  A K Peters, Ltd., Wellesley, MA, 1993.
  * [37] R. Melrose and V. Nistor.  $K$-theory of $C^{*}$-algebras of $b$-pseudodifferential operators.  Geom. Funct. Anal., 8(1):88–122, 1998.
  * [38] B Monthubert.  Pseudodifferential calculus on manifolds with corners and groupoids.  Proc. Amer. Math. Soc., 127(10):2871–2881, 1999.
  * [39] B. Monthubert and V. Nistor.  A topological index theorem for manifolds with corners.  Compos. Math., 148(2):640–668, 2012.
  * [40] B. Monthubert and F. Pierrot.  Indice analytique et groupoïdes de Lie.  C. R. Acad. Sci. Paris Sér. I Math., 325(2):193–198, 1997.
  * [41] S. Moroianu and V. Nistor.  Index and homology of pseudodifferential operators on manifolds with boundary.  In Perspectives in operator algebras and mathematical physics, volume 8 of Theta Ser. Adv. Math., pages 123–148. Theta, Bucharest, 2008\.
  * [42] V. Nistor.  Groupoids and the integration of Lie algebroids.  J. Math. Soc. Japan, 52(4):847–868, 2000.
  * [43] V. Nistor.  Pseudodifferential operators on non-compact manifolds and analysis on polyhedral domains.  In Spectral geometry of manifolds with boundary and decomposition of manifolds, volume 366 of Contemp. Math., pages 307–328. Amer. Math. Soc., Providence, RI, 2005.
  * [44] V. Nistor, A. Weinstein, and P. Xu.  Pseudodifferential operators on differential groupoids.  Pacific J. Math., 189(1):117–152, 1999.
  * [45] J. Renault.  A groupoid approach to $C^{\ast}$-algebras, volume 793 of Lecture Notes in Mathematics.  Springer, Berlin, 1980.
  * [46] B. K. So.  Pseudo-differential operators, heat calculus and index theory of groupoids satisfying the Lauter-Nistor condition.  PhD thesis, The University of Warwick, 2010.
  * [47] B.K. So.  On the full calculus of pseudo-differential operators on boundary groupoids with polynomial growth.  Adv. Math., 237:1–32, 2013.
  * [48] M. Yamashita.  A topological approach to indices of geometric operators on manifolds with fibered boundaries.  arXiv:1902.03767v1, 2019.