# Interior Kasparov products for $\varrho$-classes on Riemannian foliated
bundles

Vito Felice Zenobi

###### Abstract

Let $\iota\colon\mathcal{F}_{0}\to\mathcal{F}_{1}$ be a suitably oriented
inclusion of foliations over a manifold $M$, then we extend the construction
of the lower shriek maps given by Hilsum and Skandalis to adiabatic
deformation groupoid C*-algebras: we construct an asymptotic morphism
$(\iota_{ad}^{[0,1)})_{!}\in
E_{n}\left(C^{*}(G_{ad}^{[0,1)}),C^{*}(H_{ad}^{[0,1)})\right)$, where $G$ and
$H$ are the monodromy groupoids associated with $\mathcal{F}_{0}$ and
$\mathcal{F}_{1}$ respectively. Furthermore, we prove an interior Kasparov
product formula for foliated $\varrho$-classes associated with longitudinal
metrics of positive scalar curvature in the case of Riemannian foliated
bundles.

Mathematical Subject Classification 2020: 53C12, 19K35, 22A22.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Definitions and notations
    2. 2.2 The deformation to the normal cone
    3. 2.3 Groupoid C*-algebras
  3. 3 Some KK-elements
    1. 3.1 Mapping cones and KK-classes
    2. 3.2 Bott bimodules
    3. 3.3 KK-elements associated with groupoid cocycles
    4. 3.4 The asymptotic morphism associated with a DNC
  4. 4 Longitudinal classes
  5. 5 Transverse classes
    1. 5.1 The isometric case
    2. 5.2 The almost isometric case
  6. 6 The equivariant setting
    1. 6.1 Semidirect products and imprimitivity bimodules
    2. 6.2 Equivariant KK-classes
  7. 7 Product formulas
    1. 7.1 Smooth fibrations
    2. 7.2 Foliated bundles
    3. 7.3 Functoriality of $\varrho$-classes through étale surjections
  8. 8 Open problems

##  1 Introduction

Let $M$ be a smooth compact manifold with fundamental group $\pi_{1}(M)$ and
let $\widetilde{M}$ be its universal covering. In [12] the authors introduced
the following long exact sequence of K-theory groups

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}(C^{*}(\pi_{1}(M)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{S}^{\pi_{1}(M)}_{*+1}(\widetilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\pi_{1}(M)}_{*+1}(\widetilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
(1.1)

known as the Higson-Roe exact sequence or the _analytic surgery exact
sequence_. The group on the right is the equivariant K-homology of
$\widetilde{M}$; on the left we have the K-theory of the group C*-algebra, the
receptacle of the index invariants; finally, the middle term is the so-called
_structure group_ whom the K-theoretic secondary invariants belong to.

These invariants, usually called $\varrho$-classes, were introduced in [12].
They are used to detect secondary geometric or topological structures, such as
topological/differential manifold structures or metrics with positive scalar
curvature, see for instance [22, 23, 30, 31, 28, 33]. All these papers treat
the subject within the methodologies of coarse geometry.

In [34], K-theoretic secondary invariants were adapted to the context of Lie
groupoids. Indeed, let $G\rightrightarrows M$ be a Lie groupoid over $M$, $AG$
its Lie algebroid and $G_{ad}^{[0,1)}$ its adiabatic deformation, whose
precise definition is not important for the moment. Then we can consider the
following long exact sequence of K-groups

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{K_{*}(C^{*}(G)\otimes
C_{0}(0,1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
151.94577pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
151.94577pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(G^{[0,1)}_{ad}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
238.92355pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
238.92355pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(AG))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
322.51248pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
322.51248pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ (1.2)

In [35] it is proved that, when $G$ is the fundamental groupoid of $M$, the
long exact sequence (1.2) is isomorphic to the Higson-Roe exact sequence (1.1)
and that the $\varrho$-classes defineded in [34] correspond to those defined
in [22].

This was a natural setting for a systematized and conceptual generalization to
foliations - both regular and singular ones, see also [24] \- of the
$\varrho$-invariants in K-theory. It is worth to mention that, if for long
time the coarse geometric methods were not fit to treat foliations, in the
recent works [3, 4] Benameur and Roy contributed to advance in this direction.

The specific motivation of this work is to improve the study of Kasparov
products for secondary classes. This topic, in the general setting of Lie
groupoids, was inspired by a seminal work of Siegel [25] where an exterior
product involving secondary classes was defined and then independently
generalized to the equivariant setting in [33] and [32]. It is given by a map
as follows

$\boxtimes\colon\mathrm{S}^{\pi_{1}(M)}_{i}(\widetilde{M})\times
K_{j}(N)\to\mathrm{S}^{\pi_{1}(M)\times\pi_{1}(N)}_{i+j}(\widetilde{M}\times\widetilde{N})$
(1.3)

and it gives a factorization of a $\varrho$-class $\varrho(M\times N)$ as the
product of the analogous $\varrho$-class $\varrho(M)$ on $M$ and the
fundamental class $[N]$. In [34, Section 3.6] a similar product was introduced
in the more general context of Lie groupoid

$\boxtimes\colon K_{i}(C^{*}(G^{[0,1)}_{ad}))\times K_{j}(C^{*}(AH))\to
K_{i+j}(C^{*}((G\times H)^{[0,1)}_{ad}))$ (1.4)

and similar product formulas are proved too. The main geometrical motivation
of this construction is to obtain stability results about secondary
structures, such as rigidity results for topological structures or for metrics
with positive scalar curvature.

In the coarse geometric setting the topic was deepened in other works such as
[8, 29]. An interesting advancement appeared in the recent paper [14], where
the authors consider the case of a manifold $M$ fibered over another manifold
$B$: they define an interior product between the longitudinal secondary class
on $M$ and the K-homology class of $B$ and prove a product formula for the
total secondary invariant of $M$.

In the present article we cope the task of putting the ad hoc construction
given in [14] into the more systematic setup offered by [13]. In this
classical paper, Hilsum and Skandalis provide the construction of a KK-element
associated with a K-oriented map of foliated manifolds and they proved the
functoriality (namely what, in this text, we are calling interior product
formulas) of their construction for the index classes associated with
longitudinal operators.

Let us give a quick overview of what is developed in the present paper. First,
let us fix a manifold $M$ and assume that there is a regular foliation
associated with an involutive sub-bundle $\mathcal{F}\subset TM$ over it. We
can then consider the monodromy groupoid
$G:=\mathrm{Mon}(M,\mathcal{F})\rightrightarrows M$ associated with this
foliation. As before, we can associate with $G$ a long exact sequence in
K-theory as in 1.2. Consider now for instance the inclusion of $AG$, the Lie
algebroid of $G$, into $TM$: this inclusion is an homomorphism of Lie
algebroids which integrates to an immersion of Lie groupoids $\iota\colon
G\looparrowright H$. Here $H$ denotes the fundamental groupoid
$\widetilde{M}\times_{\pi_{1}(M)}\widetilde{M}\rightrightarrows M$.

The focus of this paper is to construct a mapping of the exact sequence (1.2)
associated with $G$ to the one associated with $H$, isomorphic to the
classical Higson-Roe exact sequence, as follows

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31.38124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{K_{*}(C^{*}(G)\otimes
C_{0}(0,1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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79.97913pt\raise-30.29723pt\hbox{\hbox{\kern 0.0pt\raise
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153.8395pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{K_{*}(C^{*}(G^{[0,1)}_{ad}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
242.07976pt\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
185.32838pt\raise-20.71666pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 185.32838pt\raise-29.5pt\hbox{\hbox{\kern
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242.07976pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(AG))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
271.87422pt\raise-30.29723pt\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
326.29993pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}{\hbox{\kern-6.75pt\raise-41.43332pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise-41.43332pt\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
30.75pt\raise-41.43332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{K_{*}(C^{*}(H)\otimes
C_{0}(0,1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
153.20825pt\raise-41.43332pt\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
153.20825pt\raise-41.43332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(H^{[0,1)}_{ad}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
241.44852pt\raise-41.43332pt\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
241.44852pt\raise-41.43332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K_{*}(C^{*}(AH))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
326.29993pt\raise-41.43332pt\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
326.29993pt\raise-41.43332pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ (1.5)

where the vertical maps are given by asymptotic morphisms. This was already
done by Hilsum and Skandalis in [13] for the K-groups on the left and the
right hand side of (1.5). Here we use a mapping cone construction to
generalize their construction to the middle term of (1.2). Moreover, whereas
the lower shriek maps of Hilsum and Skandalis where defined only for ameanable
groupoids by means of KK-elements, our maps are defined for general groupoids:
this is only due to the fact that we can implement this mapping through
asymptotic morphisms in E-theory whereas, at the time when [13] was written,
E-theory did not exist yet.

The definition of these asymptotic morphisms is only a partial aim of this
paper. Indeed, the main result is given by Theorem 7.4: take a manifold $M$
which is the total space of a Riemannian foliated bundle associated with
$\mathcal{F}\subset TM$. Assume that there is a longitudinal metric
$g^{\mathcal{F}}$ with positive scalar curvature and let
$\varrho(g^{\mathcal{F}})\in
K_{*}(C^{*}(\mathrm{Mon}(M,\mathcal{F})_{ad}^{[0,1)}))$ be the corresponding
$\varrho$-class. Let us then consider the immersion of Lie groupoids
$\iota\colon\mathrm{Mon}(M,\mathcal{F})\looparrowright\widetilde{M}\times_{\pi_{1}(M)}\widetilde{M}$
obtained by integrating the inclusion of Lie algebroids
$\mathcal{F}\hookrightarrow TM$. Assume that the normal bundle admits a $Spin$
structure and let

$(\iota_{ad}^{[0,1)})_{!}\in
E_{n}\left(C^{*}(\mathrm{Mon}(M,\mathcal{F}))_{ad}^{[0,1)}),C^{*}(\widetilde{M}\times_{\pi_{1}(M)}\widetilde{M}_{ad}^{[0,1)})\right)$
(1.6)

be the asymptotic morphism associated with it. Let
$g^{M}:=g^{\mathcal{F}}\oplus g^{\mathcal{N}}$ be a suitable metric of $TM$
with positive scalar curvature. Then we can prove the following factorization
formula for the $\varrho$-class of $g^{M}$

$\varrho(g^{M})=\varrho(g^{\mathcal{F}})\otimes(\iota_{ad}^{[0,1)})_{!}\in
K_{*+n}\left(C^{*}(\widetilde{M}\times_{\pi_{1}(M)}\widetilde{M}_{ad}^{[0,1)})\right).$
(1.7)

While the construction of (1.6) can be given for the most general case of
suitably oriented immersion of foliations, proving the product formula (1.7)
presents more difficulties. The main issue is that we are dealing with
interior Kasparov product between elements in KK-groups of mapping cone
C*-algebras. In particular, we need to prove that the Kasparov product of a
degenerate Kasparov bimodule on the foliation and a transverse non-degenerate
Kasparov bimodule corresponds to a precise degenerate Kasparov bimodule on the
whole manifold. The key technical result to solve this problem is [26, Lemma
11], which allows to construct an operatorial homotopy through invertible
operators - namely through degenerate bimodules - between the longitudinal
operator and the total one, whenever one has a fine estimation from below of
their commutator. Fundamental to this end is the calculation of Bismut and
Cheeger in [5, Equation 4.46], see also [16, Lemma 17], for the simple
foliation given by a surjective submersion $\pi\colon M\to B$. Transposing all
that to the equivariant context and passing to quotients leads to prove
formula (1.7) in the case of a Riemannian foliated bundle. The general case
stays open.

As a by-product of the construction (1.6) we also obtain a generalization to
the context of general foliations of the main theorem in [10] about the
functoriality of $\varrho$-classes with respect to maps of principal bundle
over the same manifold $M$.

The paper is organized as follows: in Section 2 we introduce the elementary
definitions and notations about Lie groupoids and their C*-algebras; in
Section 3 the elementary KK-theoretic tools are recalled: the KK-theory of
mapping cone C*-algebras, the definition of the bimodules implementing the
Bott periodicity, the Thom isomorphisms in KK-theory defined in [13], the
asymptotic morphism associated with a deformation groupoid; in Section 4 we
recall the definitions of the primary invariants, the secondary invariants and
the fundamental classes as elements of the first, the second and the third
K-theory group in (1.2) respectively; Section 5 is dedicated to the
construction of the elements (1.6) both in the case of an isometric and an
almost isometric immersion of foliations; Section 6 is devoted to set the
previous constructions in the equivariant setting and to provide a precise
relation between equivariant objects with associated non-equivariant objects
over quotients; in Section 7 we give the proof of the main results of the
paper, namely the interior product formulas, and of the functoriality of
$\varrho$-classes for general surjective étale maps of foliation groupoids,
which generalizes [10, Theorem 1.1] and simplifies its proof; finally, in
Section 8 we list the problems which remain unsolved and we propose further
directions of research starting from this paper.

##  2 Preliminaries

###  2.1 Definitions and notations

######  Definition 2.1.

Let $G$ and $G^{(0)}$ be two sets. A groupoid structure on $G$ over $G^{(0)}$
is given by the following morphisms:

  * •

Two maps: $r,s:G\rightarrow G^{(0)}$, which are respectively the range and
source map.

  * •

A map $u:G^{(0)}\rightarrow G$ called the unit map that is a section for both
$s$ and $r$. We can identify $G^{(0)}$ with its image in $G$.

  * •

An involution: $i:G\rightarrow G$, $\gamma\mapsto\gamma^{-1}$ called the
inverse map. It satisfies: $s\circ i=r$.

  * •

A map $p:G^{(2)}\rightarrow G$,
$(\gamma_{1},\gamma_{2})\mapsto\gamma_{1}\cdot\gamma_{2}$ called the product,
where the set

$G^{(2)}:=\\{(\gamma_{1},\gamma_{2})\in G\times G\ |\
s(\gamma_{1})=r(\gamma_{2})\\}$

is the set of composable pair. Moreover for $(\gamma_{1},\gamma_{2})\in
G^{(2)}$ we have $r(\gamma_{1}\cdot\gamma_{2})=r(\gamma_{1})$ and
$s(\gamma_{1}\cdot\gamma_{2})=s(\gamma_{2})$.

The following properties must be fulfilled:

  * •

The product is associative: for any $\gamma_{1},\ \gamma_{2},\ \gamma_{3}$ in
$G$ such that $s(\gamma_{1})=r(\gamma_{2})$ and $s(\gamma_{2})=r(\gamma_{3})$
the following equality holds

$(\gamma_{1}\cdot\gamma_{2})\cdot\gamma_{3}=\gamma_{1}\cdot(\gamma_{2}\cdot\gamma_{3})\
.$

  * •

For any $\gamma$ in $G$: $r(\gamma)\cdot\gamma=\gamma\cdot s(\gamma)=\gamma$
and $\gamma\cdot\gamma^{-1}=r(\gamma)$.

We denote a groupoid structure on $G$ over $G^{(0)}$ by $G\rightrightarrows
G^{(0)}$, where the arrows stand for the source and target maps.

We will adopt the following notations:

$G_{A}:=s^{-1}(A)\ ,\ G^{B}=r^{-1}(B)\ \mbox{ and }G_{A}^{B}=G_{A}\cap
G^{B}\,$

in particular if $x\in G^{(0)}$, the $s$-fiber (resp. $r$-fiber) of $G$ over
$x$ is $G_{x}=s^{-1}(x)$ (resp. $G^{x}=r^{-1}(x)$).

######  Definition 2.2.

We call $G$ a Lie groupoid when $G$ and $G^{(0)}$ are second-countable smooth
manifolds with $G^{(0)}$ Hausdorff, the structural homomorphisms are smooth.

Let us now consider a commutative diagram of Lie groupoids and vector bundles
as follows:

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{(0)}}$
(2.1)

Here we mean that $E\rightrightarrows E^{(0)}$ is a Lie groupoid (with source,
target, multiplication, identity, and inverse maps denoted by $\tilde{s}$,
$\tilde{r}$, $\tilde{m}$, $\tilde{u}$, and $\tilde{i}$), $G\rightrightarrows
G^{(0)}$ is a Lie groupoid (with source, target, multiplication, identity, and
inverse maps denoted by $s$, $t$, $m$, $u$, and $i$), $E\to G$ is a vector
bundle (with projection map and zero section denoted by $\tilde{q}$ and
$\tilde{0}$), $E^{(0)}\to G^{(0)}$ is a vector bundle bundle (with projection
map and zero section denoted by $q$ and $0$) and such that
$q\tilde{s}=s\tilde{q}$ and $q\tilde{t}=t\tilde{q}$.

######  Definition 2.3.

A $\mathcal{VB}$-groupoid is a commutative diagram of Lie groupoids and vector
bundles like (3.1) such that the following conditions hold:

  1. 1.

$(\tilde{s},s)$ and $(\tilde{t},t)$ are morphisms of vector bundles.

  2. 2.

$(\tilde{q},q)$ is a morphism of Lie groupoids.

  3. 3.

Interchange law:

$(\eta_{1}+\eta_{3})(\eta_{2}+\eta_{4})=\eta_{1}\eta_{2}+\eta_{3}\eta_{4}$

for any $(\eta_{1},\eta_{2}),(\eta_{3},\eta_{4})\in E^{(2)}$ such that
$\tilde{q}(\eta_{1})=\tilde{q}(\eta_{3})$ and
$\tilde{q}(\eta_{2})=\tilde{q}(\eta_{4})$.

######  Remark 2.4.

Recall that the usual definition includes the following technical condition.
Consider the vector bundle $s^{*}E^{(0)}=E^{(0)}\times_{s}G$ over $G$ and the
map $p\colon E\to s^{*}E^{(0)}$. The technical condition is that $p$ is
required to be a surjective submersion. But one can prove that this is implied
by the definition.

######  Definition 2.5.

Consider the sub-bundle $\ker p\subset E$. Then define the (right) core of $E$
as the vector bundle over $G^{(0)}$ given by $C:=u^{*}(\ker p)$.

For any pair $(\gamma_{1},\gamma_{2})\in G^{(2)}$, right-multiplication by
$\tilde{0}(\gamma_{2})$ produces a linear isomorphism from $(\ker
p)_{\gamma_{1}}$ to $(\ker p)_{\gamma_{1}\gamma_{2}}$. In particular, for any
$\gamma\in G$, right-multiplication by $\tilde{0}(\gamma)$ produces a linear
isomorphism from $C_{r(\gamma)}$ to $(\ker p)_{\gamma}$. Hence we have a
natural isomorphism of vector bundles over $G$ between $r^{*}C$ to $\ker p$
given by $(\gamma,c)\mapsto c\cdot\tilde{0}(\gamma)$.

So, for any section $h\colon s^{*}E^{(0)}\to E$, we have an isomorphism of
vector bundles

$s^{*}E^{(0)}\oplus r^{*}C\cong E\quad\mbox{given by}\quad(\gamma,e\oplus
c)\mapsto h(e)+c\cdot\tilde{0}(\gamma).$ (2.2)

######  Definition 2.6.

A _right-horizontal lift_ of the $\mathcal{VB}$-groupoid $E\rightrightarrows
E^{(0)}$ is given by a section $h\colon s^{*}E^{(0)}\to E$ of $p$ such that

$h(u(x),e)=\tilde{u}(e)\quad\forall x\in G^{(0)}\,\mbox{and}\,e\in
E^{(0)}_{x}.$

Observe that a right-horizontal lift $h$ is such that the associated
isomorphism (2.2) restricts to the natural splitting

$E_{u(x)}\cong C_{x}\oplus\tilde{u}(E^{(0)}_{x})\quad\forall x\in G^{(0)}.$

######  Remark 2.7.

It is important to point out that, by [9, Lemma 5.9], the space of right-
horizontal lifts of a $\mathcal{VB}$-groupoid is an affine space.

######  Remark 2.8.

Observe that the core bundle $C\to G^{(0)}$ is a $G$-bundle and the action of
$\gamma\in G$ which sends $C_{r(\gamma)}\to C_{s(\gamma)}$ is given by
$c\mapsto\tilde{0}(\gamma^{-1})\cdot c\cdot\tilde{0}(\gamma)$.

###  2.2 The deformation to the normal cone

Let $\iota\colon M_{0}\to M$ be an immersion of smooth manifolds, with normal
bundle $N_{\iota}$. As a set, the deformation to the normal cone is

$DNC^{\mathbb{R}}(M,M_{0}):=N_{\iota}\times\\{0\\}\sqcup
M\times\mathbb{R}^{*}.$ (2.3)

Notation: _we will consider restrictions of the $DNC$ to various subsets of
$\mathbb{R}$, in particular we will denote by $DNC$, without any decoration,
the restriction to $[0,1]$ and by $DNC^{[0,1)}$ the restriction to the
interval $[0,1)$ open at $1$._

In order to recall its smooth structure, we fix an exponential map, which is a
diffeomorphism $\theta$ from a neighbourhood $V^{\prime}$ of the zero section
$M_{0}$ in $N$ to a neighbourhood $V$ of $M_{0}$ in $M$. We may cover
$DNC(M,M_{0})$ with two open sets $M\times\mathbb{R}^{*}$, with the product
differentiable structure, and $W=N\times{0}\sqcup V\times\mathbb{R}^{*}$,
endowed with the differentiable structure for which the map

$\Psi\colon\\{(m,\xi,t)\in N\times\mathbb{R}\,|\,(m,t\xi)\in V^{\prime}\\}\to
W$ (2.4)

given by $(m,\xi,t)\mapsto(\theta(m,t\xi),t)$, for $t\neq 0$, and by
$(m,\xi,0)\mapsto(m,\xi,0)$, for $t=0$, is a diffeomorphism. One can verify
that the transition map on the overlap of these two charts is smooth, see for
instance [13, Section 3.1].

Let us consider a commutative square of the following type

$\textstyle{M_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{f_{0}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\prime}}$$\textstyle{M^{\prime}}$
(2.5)

where $\iota$ and $\iota^{\prime}$ are immersions, $f$ and $f_{0}$ are smooth
maps. Then we obtain the smooth map $DNC(f,f_{0})\colon DNC(M,M_{0})\to
DNC(M^{\prime},M_{0}^{\prime})$, defined by

$\begin{split}DNC(f,f_{0})(x,t)&=(f(x),t)\quad\mbox{for}\,t\neq 0,\\\
DNC(f,f_{0})(x,\xi,0)&=(f(x),df_{x}(\xi),0)\quad\mbox{for}\,t=0.\end{split}$

######  Remark 2.9.

Observe that, as for instance in [21, Proposition 1.1], $DNC(f,f_{0})$ is

  * •

a submersion if and only if both $f$ and $f_{0}$ are submersions,

  * •

an immersion if and only if $f$ is an immersion and
$T_{x}M_{0}=(df_{0})_{x}^{-1}(T_{f_{0}(x)}M^{\prime}_{0})$.

When there is no risk of confusion, we will denote $DNC(f,f_{0})$ just by
$DNC(f)$.

######  Definition 2.10.

Let $\iota\colon H\looparrowright G$ be a smooth immersion of Lie groupoids
and let us denote its restriction to the units by $\iota_{0}\colon
H^{(0)}\hookrightarrow G^{(0)}$.

  1. 1.

The normal bundle $N_{\iota}$ carries a Lie groupoid structure over
$N_{\iota_{0}}$. We denote this groupoid by
$\mathcal{N}_{\iota}\rightrightarrows N_{\iota_{0}}$. We will also use the
following notation $\mathcal{N}_{H}^{G}$ to denote $\mathcal{N}_{\iota}$.

  2. 2.

The smooth manifold $DNC(G,H)$ is naturally endowed with a structure of Lie
groupoid over $DNC(G^{(0)},H^{(0)})$. The source and range morphisms are given
by $DNC(s)$ and $DNC(r)$ respectively. Furthermore, the set of composable
arrows $DNC(G,H)^{(2)}$ identifies with $DNC(G^{(2)},H^{(2)})$ and the product
is given by $DNC(m^{G},m^{H})$.

  3. 3.

For the deformation to the normal cone associated with the unit morphism
$u\colon G^{(0)}\to G$ we will use the following notation
$G_{ad}:=DNC(G,G^{(0)})\rightrightarrows G^{(0)}\times[0,1]$ and we will call
it the _adiabatic groupoid_ associated with $G$. The same notation applies to
morphisms, namely $f_{ad}:=DNC(f)$. Finally, recall that in this particular
case $\mathcal{N}_{u}$ is just $\mathfrak{A}G$, the Lie algebroid of $G$.

######  Remark 2.11.

Notice that the projection $\mathcal{N}_{\iota}\to H$ is a groupoid morphism
and it follows that $\mathcal{N}_{\iota}$ is a $\mathcal{VB}$-groupoid over
$H$.

###  2.3 Groupoid C*-algebras

We can associate with a Lie groupoid $G$ the *-algebra
$C^{\infty}_{c}(G,\Omega^{\frac{1}{2}}(\ker ds\oplus\ker dr))$ of the
compactly supported sections of the half densities bundle associated with
$\ker ds\oplus\ker dr$, with:

  * •

the involution given by $f^{*}(\gamma)=\overline{f(\gamma^{-1})}$;

  * •

and the convolution product given by
$f*g(\gamma)=\int_{G_{s(\gamma)}}f(\gamma\eta^{-1})g(\eta)$.

For all $x\in G^{(0)}$ the algebra $C^{\infty}_{c}(G,\Omega^{\frac{1}{2}}(\ker
ds\oplus\ker dr))$ can be represented on
$L^{2}(G_{x},\Omega^{\frac{1}{2}}(G_{x}))$ by

$\lambda_{x}(f)\xi(\gamma)=\int_{G_{x}}f(\gamma\eta^{-1})\xi(\eta),$

where $f\in C^{\infty}_{c}(G,\Omega^{\frac{1}{2}}(\ker ds\oplus\ker dr))$ and
$\xi\in L^{2}(G_{x},\Omega^{\frac{1}{2}}(G_{x}))$.

######  Definition 2.12.

The reduced C*-algebra of a Lie groupoid G, denoted by $C^{*}_{r}(G)$, is the
completion of $C^{\infty}_{c}(G,\Omega^{\frac{1}{2}}(\ker ds\oplus\ker dr))$
with respect to the norm

$||f||_{r}=\sup_{x\in G^{(0)}}||\lambda_{x}(f)||.$

The full C*-algebra of $G$ is the completion of
$C^{\infty}_{c}(G,\Omega^{\frac{1}{2}}(\ker ds\oplus\ker dr))$ with respect to
all continuous representations.

######  Remark 2.13.

From now on, if $X$ is a $G$-invariant closed subset of $G^{(0)}$ we will call
$e_{X}\colon C^{\infty}_{c}(G)\to C^{\infty}_{c}(G_{|X})$ the restriction map
to $X$. That gives an exact sequence of full groupoid C*-algebras

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{C^{*}(G_{|G^{(0)}\setminus
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105.28467pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{*}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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161.58743pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
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224.52463pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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224.52463pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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but in general this is not true for the reduced ones: the reader can find
examples of this phenomenon in [11]. Let us precise that in what follows we
will mainly deal with the reduced groupoid C*-algebras, because there are more
details to check in the reduced situation. But everything we are going to
prove about the reduced C*-algebras works for the full C*-algebras too.

##  3 Some KK-elements

This section is dedicated to recall the technical constructions in KK-theory
which will be used all along this paper.

###  3.1 Mapping cones and KK-classes

Degenerate bimodules. Recall that a Kasparov bimodule
$(\mathcal{H},F)\in\mathbb{E}(A,B)$ is said to be degenerate if

$[a,F]=a(F^{2}-1)=a(F-F^{*})=0,\quad\forall a\in A$

and that $\mathbb{D}(A,B)$ denotes the set of all degenerate Kasparov
$A,B$-bimodules. In this case $(\mathcal{H}\otimes C_{0}[0,1),F\otimes
1)\in\mathbb{E}(A,B[0,1])$ produces a canonical homotopy from
$(\mathcal{H},F)\in\mathbb{E}(A,B)$ to the zero bimodule $(0,0)$. Namely
$\mathbb{D}(A,B)$ is star-shaped. Thus, from now on, we will do a little abuse
by considering any degenerate bimodule as the zero bimodule. In particular, a
bimodule $(\overline{\mathcal{H}},\overline{F})\in\mathbb{E}(A,B[0,1])$ such
that $(\mathrm{ev}_{0})_{*}(\overline{\mathcal{H}},\overline{F})$ is
degenerate will define in a canonical way an element in
$\mathbb{E}(A,B[0,1))$.

Gluing bimodules [2, Section 2.3]. Let $\varphi_{0}\colon B_{0}\to C$ and
$\varphi_{1}\colon B_{1}\to C$ be two *-homomorphisms. We can then construct
the pull-back C*-algebra

$B_{0}\oplus_{C}B_{1}:=\\{(b_{0},b_{1})\in B_{0}\oplus
B_{1}|\varphi_{0}(b_{0})=\varphi_{1}(b_{1})\\}.$

Consider now two bimodules $(\mathcal{H},F)\in\mathbb{E}(A,B_{0})$ and
$(\mathcal{H}^{\prime},F^{\prime})\in\mathbb{E}(A,B_{1})$. Let us assume that
$\varphi_{0}$ and $\varphi_{1}$ are surjective *-homomorphism and recall that
this implies that, for instance, $\mathcal{H}\otimes_{\varphi_{0}}C$ is a
quotient of $\mathcal{H}$ (let us keep the notation $\varphi_{0}$ for the
quotient map).

######  Definition 3.1.

Let $w$ be a unitary equivalence between $(\varphi_{0})_{*}(\mathcal{H},F)$
and $(\varphi_{1})_{*}(\mathcal{H}^{\prime},F^{\prime})$. Then define the
Kasparov bimodule

$(\mathcal{H},F)\diamond^{w}_{\varphi_{0},\varphi_{1}}(\mathcal{H}^{\prime},F^{\prime})\in\mathbb{E}(A,B_{0}\oplus_{C}B_{1})$
(3.1)

to be given by the pair

  * •

$\mathcal{H}\diamond\mathcal{H}^{\prime}:=\\{(\eta,\xi)\in\mathcal{H}\times\mathcal{H}^{\prime}\,|\,\varphi_{0}(\xi)=w\varphi_{1}(\eta)\\}$,

  * •

$F\diamond F^{\prime}(\eta,\xi):=(F\eta,F^{\prime}\xi)$.

Mapping cones. Let $\varphi\colon A\to B$ be a *-homomorphism. Associated with
$\varphi$ we have the mapping cylinder C*-algebra

$\mathcal{Z}_{\varphi}=:\\{(a,f)\in A\oplus B[0,1]\,|\,\varphi(a)=f(0)\\}$

and the mapping cone C*-algebra

$\mathcal{C}_{\varphi}=:\\{(a,f)\in A\oplus B[0,1)\,|\,\varphi(a)=f(0)\\}$

which is an ideal in $\mathcal{Z}_{\varphi}$.

A typical element in $\mathbb{E}(D,\mathcal{C}_{\varphi})$ is given by a pair

$\left((\mathcal{H}_{A},G_{A}),(\overline{\mathcal{H}},\overline{G})\right),$
(3.2)

composed by a bimodule $(\mathcal{H}_{A},G_{A})\in\mathbb{E}(D,A)$ and a
bimodule $(\overline{\mathcal{H}},\overline{G})\in\mathbb{E}(D,B[0,1])$ which
is a homotopy between $\varphi_{*}(\mathcal{H}_{A},G_{A})$ and a degenerate
bimodule $(\mathcal{H}_{B},G_{B})\in\mathbb{D}(D,B)$. This fits with (3.1)
where $w$ is the identity of $B$.

######  Remark 3.2.

One could argue that a typical element in
$\mathbb{E}(D,\mathcal{C}_{\varphi})$ is defined by means of a homotopy
between $\varphi_{*}(\mathcal{H}_{A},G_{A})$ and the zero module, but by
gluing this homotopy with the canonical homotopy $(\mathcal{H}_{B}\otimes
C_{0}[0,1),G_{B}\otimes 1)$ we establish a canonical equivalence between these
definitions. We keep the degenerate bimodule at the end just because it allows
more flexibility with Kasparov products as we will see.

Let us consider two *-homomorphisms $\varphi_{i}\colon A_{i}\to B_{i}$, with
$i=0,1$, and two bimodules $(\mathcal{E}_{A},F_{A})\in\mathbb{E}(A_{0},A_{1})$
and $(\mathcal{E}_{B},F_{B})\in\mathbb{E}(B_{0},B_{1})$ such that there exists
a homotopy $(\overline{\mathcal{E}},\overline{F})$ between
$(\varphi_{1})_{*}(\mathcal{E}_{A},F_{A})$ and
$\varphi_{0}^{*}(\mathcal{E}_{B},F_{B})$ in $\mathbb{E}(A_{0},B_{1}[0,1])$.
Then this set of data produces a bimodule
$(\mathcal{E}_{\varphi},F_{\varphi})\in\mathbb{E}(\mathcal{C}_{\varphi_{0}},\mathcal{C}_{\varphi_{1}})$,
see [1, Lemma 5.2] for a detailed construction.

Let us see how we will intend in practice the Kasparov product of the element
$[(\mathcal{H}_{A_{0}},G_{A_{0}}),(\overline{\mathcal{H}},\overline{G})]\in
KK(D,\mathcal{C}_{\varphi_{0}})$ with $[\mathcal{E}_{\varphi},F_{\varphi}]\in
KK(\mathcal{C}_{\varphi_{0}},\mathcal{C}_{\varphi_{1}})$:

  * •

let $(\mathcal{H}_{A_{1}},G_{A_{1}})\in\mathbb{E}(D,A_{1})$ be a Kasparov
product of $(\mathcal{H}_{A_{0}},G_{A_{0}})$ and $(\mathcal{E}_{A},F_{A})$;

  * •

let
$(\overline{\mathcal{H}}^{\prime},\overline{G}^{\prime})\in\mathbb{E}(D,B_{1}[0,1])$
be a Kasparov product of $(\overline{\mathcal{H}},\overline{G})$ and
$(\overline{\mathcal{E}},\overline{F})$ such that the evaluation at $0$ of
$(\overline{\mathcal{H}}^{\prime},\overline{G}^{\prime})$ is equal to
$(\varphi_{1})_{*}(\mathcal{H}_{A_{1}},G_{A_{1}})$;

  * •

observe that $(\mathcal{H}_{B_{1}},G_{B_{1}})$, the evaluation at $1$ of
$(\overline{\mathcal{H}}^{\prime},\overline{G}^{\prime})$, is a Kasparov
product of the degenerate bimodule $(\mathcal{H}_{B},G_{B})$ and
$(\mathcal{E}_{B},F_{B})$. This is not necessarily degenerate, so the pair
$\left((\mathcal{H}_{A_{1}},G_{A_{1}}),(\overline{\mathcal{H}}^{\prime},\overline{G}^{\prime})\right)$
only defines an element in $\mathbb{E}(D,\mathcal{Z}_{\varphi_{1}})$;

  * •

but Lemma 3.3 below tells us that we have a canonical operatorial homotopy of
$(\mathcal{H}_{B_{1}},G_{B_{1}})$ to the degenerate bimodule
$(\mathcal{H}_{B_{0}}\otimes\mathcal{E}_{B},G_{B_{0}}\otimes 1)$ by means of
the operatorial homotopy $CS_{t}(G_{B_{1}},G_{B_{0}}\otimes 1)$ defined below
in (3.4). So the element in $\mathbb{E}(D,\mathcal{Z}_{\varphi_{1}})$, defined
in the previous point, lifts canonically to the element

$\left((\mathcal{H}_{A_{1}},G_{A_{1}}),(\overline{\mathcal{H}}^{\prime},\overline{G}^{\prime})\diamond^{\mathrm{id}}_{\mathrm{ev}_{1},\mathrm{ev}_{0}}(\mathcal{H}_{B_{1}},CS_{t}(G_{B_{1}},G_{B_{0}}\otimes
1))\right)\in\mathbb{E}(D,\mathcal{C}_{\varphi_{1}})$ (3.3)

which will be our canonical representative for a Kasparov product of this
kind.

######  Lemma 3.3.

[26, Lemma 11] Let $(\mathcal{E},F)$ and $(\mathcal{E},F^{\prime})$ be two
elements of $\mathbb{E}(A,B)$ such that $\forall a\in A$ we have that
$a[F,F^{\prime}]a^{*}-\lambda\geq 0$ modulo $\mathbb{K}(\mathcal{E})$, with
$\lambda>-2$. Then $(\mathcal{E},F)$ and $(\mathcal{E},F^{\prime})$ are
operatorially homotopic.

###### Proof.

Let us define the algebra

$\mathcal{A}:=\\{T\in\mathbb{B}(\mathcal{E})\,|\,[T,a]\in\mathbb{K}({\mathcal{E}})\,,\forall
a\in A\\}$

and its ideal

$\mathcal{J}:=\\{T\in\mathcal{A}\,|\,Ta\in\mathbb{K}({\mathcal{E}})\,,\forall
a\in A\\}.$

Then $[F,F^{\prime}]\in\mathcal{A}$ and it is positive modulo $\mathcal{J}$.
Write $[F,F^{\prime}]=P+K$, where $P\in\mathcal{A}$, $P\geq\lambda$ and
$K\in\mathcal{J}$. Note that as $F^{2}-1$ and $F^{\prime}-1$ belong to
$\mathcal{J}$ then $[F,P]$ and $[F^{\prime},P]$ belong to $\mathcal{J}$. Put

$CS_{t}(F,F^{\prime}):=(1+\sin(t)\cdot\cos(t)P)^{-\frac{1}{2}}(\cos(t)F+\sin(t)F^{\prime})\quad
t\in[0,\pi/2].$ (3.4)

Therefore, it needs a simple calculation to check that
$(\mathcal{E},CS_{t}(F,F^{\prime}))\in\mathbb{E}(A,B[0,1])$ and then it
realizes the desired operatorial homotopy. ∎

The well-known proof of the previous lemma is useful in order to prove the
following one, which will be key for our main result.

######  Lemma 3.4.

Let $F$ and $F^{\prime}$ be regular operators on $\mathcal{E}$ such that $F$
and $F^{\prime}$ are invertible and that $[F,F^{\prime}]-\lambda\geq 0$ in
$\mathbb{B}(\mathcal{E})$, with $\lambda>-2$. Then $F$ and $F^{\prime}$ are
homotopic through a path of invertible operators.

###### Proof.

Write in this case $[F,F^{\prime}]=P$. Then a simple calculation shows that
when $\lambda\in[0,-2)$ we have that

$(\cos(t)F+\sin(t)F^{\prime})^{2}=1+\sin(t)\cdot\cos(t)P\geq
1+\frac{\lambda}{2}>0,$

while when $\lambda$ is positive, we directly obtain that
$(\cos(t)F+\sin(t)F^{\prime})^{2}>0$. This implies that
$(\cos(t)F+\sin(t)F^{\prime})$ has a gap near zero in the spectrum and then
the same holds for $CS_{t}(F,F^{\prime})$. Thus
$(\mathcal{E},\mathrm{sgn}(CS_{t}(F,F^{\prime})))$ realizes the desired
homotopy through invertible operators. ∎

###  3.2 Bott bimodules

In this subsection we recall the constructions of Bott KK-elements given in
[17, Section 5]. Let $Cliff(\mathbb{R}^{n})$ be the Clifford algebra
associated with the euclidean metric on $\mathbb{R}^{n}$. Consider the
$\mathbb{R},C^{\mathbb{R}}_{0}(\mathbb{R}^{n})\otimes
Cliff(\mathbb{R}^{n})$-bimodule $C^{\mathbb{R}}_{0}(\mathbb{R}^{n})\otimes
Cliff(\mathbb{R}^{n})$. The bounded multiplier $F:=x\mapsto
x\cdot(1+\|x\|^{2})^{-1}$ is $GL_{n}(\mathbb{R})$-invariant and we have that
$F^{*}=F$ and that $1-F^{2}=(1+\|x\|^{2})^{-\frac{1}{2}}\in
C^{\mathbb{R}}_{0}(\mathbb{R}^{n})\otimes Cliff(\mathbb{R}^{n})$.

######  Definition 3.5.

Let us denote by $\mathcal{B}\in
KKO_{n}^{GL_{n}(\mathbb{R})}(\mathbb{R},C^{\mathbb{R}}_{0}(\mathbb{R}^{n})\otimes
Cliff(\mathbb{R}^{n}))$ the element given by

$[C^{\mathbb{R}}_{0}(\mathbb{R}^{n})\otimes Cliff(\mathbb{R}^{n}),F].$ (3.5)

Let $\mathfrak{j}\colon\mathcal{H}\to GL_{n}(\mathbb{R})$ be an homomorphism
of groups and let $S$ be an $\mathcal{H}$-equivariant finite dimensional
representation of $Cliff(\mathbb{R}^{n})$. This naturally gives an element of
$KK^{\mathcal{H}}(Cliff(\mathbb{R}^{n}),\mathbb{C})$ in the following way.

######  Definition 3.6.

Let us denote by $\beta^{\mathfrak{j}}_{n}(S)\in
KKO^{\mathcal{H}}_{n}(\mathbb{R},C^{\mathbb{R}}_{0}(\mathbb{R}^{n}))$ the
element given by

$\mathfrak{j}^{*}(\mathcal{B})\otimes_{Cliff(\mathbb{R}^{n})}[S].$ (3.6)

When $\mathfrak{j}$ is the natural inclusion of a subgroup $\mathcal{H}$ or,
for instance, the natural map from $Spin$ or from the metalinear group $Ml$ to
$GL$, we will denote the Bott element associated with $S$ by
$\beta_{n}^{\mathcal{H}}(S)$.

######  Remark 3.7.

Notice that, after complexifying, one can construct analogous elements in
$KK^{\mathcal{H}}_{n}(\mathbb{C},C^{\mathbb{R}}_{0}(\mathbb{R}^{n}))$
associated with a complex representation of the complexified Clifford algebra
$\mathbb{C}liff(\mathbb{R}^{n}):=Cliff(\mathbb{R}^{n})\otimes\mathbb{C}$.

######  Exemple 3.8.

The most frequent geometrical examples of Bott elements are the following
ones:

  1. 1.

The oriented case: consider the representation of
$\mathbb{C}liff(\mathbb{R}^{n})$ given by the complexified exterior algebra
$\Lambda^{*}_{\mathbb{C}}\mathbb{R}^{n}$ of $\mathbb{R}^{n}$. Then we have the
class

$\beta^{SO_{n}}_{n}(\Lambda_{\mathbb{C}}\mathbb{R}^{n}):=\left[C_{0}(\mathbb{R}^{n},\Lambda^{*}_{\mathbb{C}}\mathbb{R}^{n}),x\mapsto\frac{\lambda_{x}+\lambda_{x}^{*}}{\sqrt{1+\|x\|^{2}}}\right]\in
KK_{n}^{SO_{n}}(\mathbb{C},C_{0}(\mathbb{R}^{n})),$

where $\lambda_{x}$ is the exterior multiplication by $x\in\mathbb{R}^{n}$.

  2. 2.

The $KO$-oriented case: consider the real spinor representation $\not{S}^{n}$
of $Cliff(\mathbb{R}^{n})$, then we obtain the class

$\not{\beta}_{n}:=\left[C^{\mathbb{R}}_{0}(\mathbb{R}^{n},\not{S}^{n}),x\mapsto\frac{cl(x)}{\sqrt{1+\|x\|^{2}}}\right]\in
KKO_{n}^{Spin_{n}}(\mathbb{R},C^{\mathbb{R}}_{0}(\mathbb{R}^{n})).$

  3. 3.

The $K$-oriented case: $\not{\beta}_{n}^{c}\in
KK_{n}^{Spin^{c}_{n}}(\mathbb{C},C_{0}(\mathbb{R}^{n}))$ is defined similarly
to the $KO$-oriented case.

###  3.3 KK-elements associated with groupoid cocycles

In this section we are going to quickly recall some constructions from [13,
Section II], concerning Thom elements in KK-theory.

Let $\mathcal{H}$ be a Lie group and let $i\colon G\to\mathcal{H}$ be a
cocycle on the Lie groupoid $G\rightrightarrows M$. More precisely it is given
by the following data:

  * •

an open cover $\\{U_{j}\\}_{j\in J}$ of $M$;

  * •

for all $j,k\in J$ an application $i_{j,k}\colon G^{j}_{k}\to\mathcal{H}$ such
that

$i_{j,h}(\gamma_{1}\gamma_{2})=i_{j,k}(\gamma_{1})i_{k,h}(\gamma_{2})$

for all $\gamma_{1}\in G^{j}_{k}$ and $\gamma_{2}\in G^{k}_{h}$. Here
$G^{j}_{k}$ denotes $G^{U_{j}}_{U_{k}}$.

It is equivalent to consider the principal $G$-equivariant fiber bundle
$P_{i}$ with structural group $\mathcal{H}$, constructed via the cocycle
$i_{0}$, which is the restriction of $i$ to $M$ (recall that in this situation
the actions of $\mathcal{H}$ and $G$ on $P_{i}$ commute).

Now, if $A$ is an $\mathcal{H}$-algebra, the associated bundle
$P_{i}\times_{\mathcal{H}}A$ is $G$-equivariant and we can construct the
crossed product C*-algebra
$A\rtimes_{i}G:=C^{*}(G;P_{i}\times_{\mathcal{H}}A)$. Notice that, if we
denote by $G^{P_{i}}$ the transformation groupoid $P_{i}\rtimes
G\rightrightarrows P_{i}$ associated with $i$, then $A\rtimes_{i}G$ is Morita
equivalent to $(A\otimes C^{*}(G^{P_{i}}))\rtimes\mathcal{H}$, let us denote
by $M_{A,i}$ the bimodule associated with this Morita equivalence.

######  Definition 3.9.

Let $i\colon G\to\mathcal{H}$ be a cocycle as before. Let $A$ and $B$ be
$\mathcal{H}$-algebras, then set the morphism of KK-groups

$i^{*}\colon KK^{\mathcal{H}}(A,B)\to KK(A\rtimes_{i}G,B\rtimes_{i}G)$

by

$x\mapsto M_{A,i}\otimes
j^{\mathcal{H}}(\mathrm{Id}_{C^{*}(G^{P_{i}})}\otimes_{\mathbb{C}}x)\otimes
M^{-1}_{B,i}$

Intuitively, $i^{*}$ assembles an equivariant bundle over $G$ whose typical
fiber is given by the Kasparov bimodule $x$.

Let $E\xrightarrow{\pi}G^{(0)}$ be a $G$-equivariant vector bundle, namely it
is associated with the restriction to $G^{(0)}$ of a cocycle $i\colon
G\to\mathcal{H}$, where $\mathcal{H}$ is the structural Lie group of $E$. Then
we can consider the action groupoid $G^{E}:=E\rtimes G\rightrightarrows E$. As
a set, $G^{E}$ is equal to the pull-back bundle $r^{*}E$ over $G$. We have the
following isomorphisms of C*-algebras

$C^{*}(G^{E})\cong C_{0}(E)\rtimes G\cong
C_{0}(P_{i}\times_{\mathcal{H}}\mathbb{R}^{n})\rtimes G\cong
C^{*}(G;P_{i}\times_{\mathcal{H}}C_{0}(\mathbb{R}^{n}))=:C_{0}(\mathbb{R}^{n})\rtimes_{i}G.$

######  Definition 3.10.

Let $E\to G$ be a vector bundle of rank $n$ associated with a cocycle $i\colon
G\to\mathcal{H}$. Let $\mathcal{S}$ be a representation of
$Cliff(\mathbb{R}^{n})$ which assembles to a $Cliff(E)$-module $S$ over $G$.
Then define the class $\beta(E,S)$ as the KK-element
$i^{*}(\beta^{\mathcal{H}}_{n}(\mathcal{S}))$ in the KK-groups
$KK_{n}(C^{*}(G),C^{*}(G^{E}))$, in the complex case, or in
$KKO_{n}(C^{*}_{\mathbb{R}}(G),C^{*}_{\mathbb{R}}(G^{E}))$, in the real case.

######  Remark 3.11.

The usual case where we want to apply this construction is the one of
$\mathcal{N}_{\iota}$, the normal bundle of an immersion $\iota\colon H\to G$
of Lie groupoids. It is a $\mathcal{VB}$-groupoid with objects $N_{\iota_{0}}$
and its C*-algebra $C^{*}(\mathcal{N}_{\iota})$ is not always identifiable in
the form $C^{*}(H^{E})$, for some $H$-bundle $E$ over $H^{(0)}$.

Examples where this happens are the following ones:

  1. 1.

if $N_{\iota_{0}}$ is trivial, then $C^{*}(\mathcal{N}_{\iota})$ is isomorphic
to $C^{*}(H^{C})$ via fiber-wise Fourier transform, where $C$ is the core of
$\mathcal{N}_{\iota}$ and, by Remark 2.8, it is an $H$-bundle.

  2. 2.

More generally, assume that $N_{\iota_{0}}$ is an $H$-bundle such that
$s^{*}N_{\iota_{0}}=r^{*}N_{\iota_{0}}$. Then, by (2.2), we have that
$C^{*}(\mathcal{N}_{\iota})\cong C^{*}(H^{N_{\iota_{0}}\oplus C})$. Observe
that this isomorphism uses the fiber-wise Fourier transform along $C$, but
thanks to Remark 2.7, the isomorphism induced in KK-theory does not depend on
the splitting used in (2.2).

_In this case we will shortly denote $\beta(\mathcal{N}_{\iota},S)$ just by
$\beta(\iota,S)$ and suppress $S$ from the notation when it is obvious from
the context._

###  3.4 The asymptotic morphism associated with a DNC

Let us fix an immersion of Lie groupoids $\iota\colon G\to H$ and consider the
associated deformation to the normal cone $DNC(G,H)\rightrightarrows
DNC(G^{(0)},H^{(0)})$. Its C*-algebra is a $C[0,1]$-algebra and it comes with
the following extension of $C[0,1]$-algebras

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(G)\otimes
C_{0}(0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(DNC(G,H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ev}_{0}}$$\textstyle{C^{*}(\mathcal{N}_{H}^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(3.7)

where $C[0,1]$ acts on $C^{*}(\mathcal{N}_{H}^{G})$ by evaluation at 0. Notice
that $C^{*}(G)\otimes C_{0}(0,1]$ is contractible and $\mathrm{ev}_{0}$
induces an invertible asymptotic morphism. Hence, as it is explained in [19,
Lemma A.2], we can give the following definition.

######  Definition 3.12.

The exact sequence (3.7) gives rise to an element in the E-theory group
$E(C^{*}(\mathcal{N}_{H}^{G}),C^{*}(G))$ which is explicitly obtained as

$\partial_{H}^{G}:=1_{C^{*}(\mathcal{N}_{H}^{G})}\otimes[\mathrm{ev}_{0}]^{-1}\otimes[\mathrm{ev}_{1}].$
(3.8)

_In some case, where the notation would be too heavy, we will denote it by
$\partial(\iota)$._

If, in addition, $\mathrm{ev}_{0}$ has a completely positive lifting of norm
1, then $\mathrm{ev}_{0}$ induces a KK-equivalence
$KK(C^{*}(\mathcal{N}_{H}^{G}),C^{*}(DNC(G,H)))\xrightarrow{[\mathrm{ev}_{0}]}KK(C^{*}(\mathcal{N}_{H}^{G}),C^{*}(\mathcal{N}_{H}^{G}))$
and $\partial_{H}^{G}$ is an element of
$KK(C^{*}(\mathcal{N}_{H}^{G}),C^{*}(G))$.

######  Remark 3.13.

Let us point out the obvious fact that one can see $\partial_{H}^{G}$ as the
boundary map associated with the short exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{C^{*}(G)\otimes
C_{0}(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
130.90964pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
130.90964pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{*}(DNC^{[0,1)}(G,H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
221.08907pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\mathrm{ev}_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 244.99289pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
244.99289pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{*}(\mathcal{N}_{H}^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
306.04318pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
306.04318pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
(3.9)

Observe that $C^{*}(DNC^{[0,1)}(G,H))$ is isomorphic to the mapping cone
C*-algebra of $\mathrm{ev}_{1}\colon C^{*}(DNC(G,H))\to C^{*}(G)$, then (3.9)
is equivalent in K-theory to the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(G)\otimes
C_{0}(0,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(DNC^{[0,1)}(G,H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(DNC(G,H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(3.10)

with associated boundary map given by the composition of $[\mathrm{ev}_{1}]$
and the suspension map.

Thus, an element in $\mathbb{E}(\mathbb{C},C^{*}(DNC^{[0,1)}(G,H)))$ is given
by a $\mathbb{C},C^{*}(DNC^{[0,1]}(G,H))$-bimodule $(\mathcal{E},F)$ and a
homotopy in $\mathbb{E}(\mathbb{C},C^{*}(G))$ of
$(\mathrm{ev}_{1})_{*}(\mathcal{E},F)$ to a degenerate bimodule.

##  4 Longitudinal classes

In this section we shall recall the construction of secondary invariants given
in [34]. The geometrical setting is the following one where, to keep the
presentation simple, we are going to present only the complex case, but
everything works analogously in the real case with KO-theory.

  * •

Let $G\rightrightarrows M$ be a Lie groupoid such that its Lie algebroid
$\mathfrak{A}G$ is of rank $n$ and let us assume that the anchor map is
injective (i.e. $\mathfrak{A}G$ is an involutive sub-bundle of the tangent
bundle $TM$).

  * •

Let $g$ be a metric on $\mathfrak{A}G$, by means of it we can define a
$G$-invariant metric on $\ker{ds}$ along the $s$-fibers of $G$.

  * •

Let $\mathrm{Cliff}_{g}\left(\mathfrak{A}G\right)$ be the Clifford algebra
bundle over $M$ associated with the metric $g$. Let us fix $E\to M$, an
hermitian bundle of $\mathrm{Cliff}_{g}(\mathfrak{A}G)$-modules and let
$cl(X)$ denote the Clifford multiplication by
$X\in\mathrm{Cliff}_{g}\left(\mathfrak{A}G\right)$.

  * •

We will denote by $\mathcal{E}(G)$ the $C^{*}(G)$-module obtained as the
completion of the module
$C^{\infty}_{c}(G,r^{*}E\otimes\Omega^{\frac{1}{2}}(G))$ with respect to the
usual $C^{*}(G)$-valued inner product.

  * •

Let $\nabla$ denote the fiberwise Levi-Civita connection associated with the
metric $g$.

Assume that $E$ is equipped with a metric $h$ and a compatible connection
$\nabla^{E}$ such that:

  * •

the Clifford multiplication is skew-symmetric, that is

$\langle cl(X)s_{1},s_{2}\rangle+\langle s_{1},cl(X)s_{2}\rangle=0$

for all $X\in C^{\infty}\left(M,\mathfrak{A}G\right)$ and $s_{1},s_{2}\in
C^{\infty}(M,E)$;

  * •

$\nabla^{E}$ is compatible with the Levi-Civita connection $\nabla$, namely

$\nabla^{E}_{X}(cl(Y)s)=cl(\nabla_{X}Y)s+cl(Y)\nabla^{E}_{X}(s)$

for all $X,Y\in C^{\infty}\left(M,\mathfrak{A}G\right)$ and $s\in
C^{\infty}(M,E)$.

######  Definition 4.1.

The generalized Dirac operator associated with this set of data is defined as

$D_{G}^{E}\colon s\mapsto\sum_{\alpha}c(e_{\alpha})\nabla^{E}_{\alpha}(s)$

for $s\in C_{c}^{\infty}(G,r^{*}E_{G}\otimes\Omega^{\frac{1}{2}}(G))$ and
$\\{e_{\alpha}\\}_{\alpha\in A}$ a local orthonormal frame. Here, by a little
abuse of notation, we still denote by $\nabla^{E}$ the pull-back of the
connection to $r^{*}E$.

Let $p\colon\mathfrak{A}^{*}G\to M$ be the bundle projection, then the symbol
of $D_{G}^{E}$ is given by the section $\sigma_{E}\in
C^{\infty}(\mathfrak{A}^{*}G,\mathrm{End}(p^{*}E))$, defined by
$\sigma_{E}\colon\xi\mapsto c(\xi)$.

######  Exemples 4.2.

The typical geometrical examples for $E$ are the following ones:

  1. 1.

if $\mathfrak{A}G$ is orientable, then for $E$ equal to
$\Lambda^{*}(\mathfrak{A}G)$, the exterior algebra of the Lie algebroid,
$D_{G}^{\Lambda^{*}}$ is equal to $D^{sign}_{G}$, the longitudinal Signature
operator on $G$;

  2. 2.

if $\mathfrak{A}G$ is $Spin$ or $Spin^{c}$, then for $E$ equal to
$\not{S}_{G}$, the spinor bundle associated with the $Spin$ or $Spin^{c}$
structure of the Lie algebroid, $D_{G}^{\not{S}}$ is equal to $\not{D}_{G}$,
the longitudinal $Spin$ or $Spin^{c}$ Dirac operator on $G$.

Now, let us see how we can define K-theory classes by means of these
operators. First recall that $D_{G}^{E}$ is a regular unbounded self-adjoint
operator on $\mathcal{E}(G)$, see [27]. Let $\psi$ be a continuous function on
the spectrum of $D_{G}^{E}$ (which is a subset of $\mathbb{R}$). We say that
it is a _normalizing function_ if it is odd (i.e.
$\psi(-s)=-\psi(s)\,\,\forall s\in\mathbb{R}$) and
$\lim_{s\to\pm\infty}\psi(s)=\pm 1$. It is a standard fact that the continuous
functional calculus of $D_{G}^{E}$ by means of $\psi$ gives a continuous
operator $\psi(D_{G}^{E})\in\mathbb{B}(\mathcal{E}(G))$, in particular it is
an elliptic 0-order pseudodifferential operator.

######  Definition 4.3.

Let us denote by $[D_{G}^{E}]$ the class in $KK_{n}(\mathbb{C},C^{*}(G))$
induced by the Kasparov bimodule $(\mathcal{E}(G),\psi(D_{G}^{E}))$.

######  Remark 4.4.

Let $\psi_{1}$ and $\psi_{2}$ two normalizing functions for $D_{G}^{E}$. Then,
for $t\in[0,1]$, $\psi_{t}:=t\cdot\psi_{1}+(1-t)\cdot\psi_{2}$ is a path of
normalizing functions. This implies that the class $[D_{G}^{E}]$ does not
depend on the choice of $\psi$.

Let $X$ be a closed $G$-invariant smooth submanifold of $M$. Then, as in
Remark 2.13, we have a restriction element $[\mathrm{ev}_{X}]\in
KK(C^{*}(G),C^{*}(G_{|X}))$. It is immediate to see that

$[D_{G}^{E}]\otimes_{C^{*}(G)}[\mathrm{ev}_{X}]=[D_{G_{|X}}^{E_{|X}}]\in
KK_{n}(\mathbb{C},C^{*}(G_{|X})).$ (4.1)

_From now on, let $E$ be implicitly understood_

Fundamental classes. If we see $\mathfrak{A}G$ as a Lie groupoid over $M$,
then the corresponding operator $D_{\mathfrak{A}G}$, constructed by using the
recipe of Definition 4.1, is nothing but the fiber-wise Fourier transform of
the Clifford multiplication $cl_{G}$. It then defines a class

$\left[\widehat{cl}_{G}\right]:=\left[\mathcal{E}(\mathfrak{A}G),\psi(\widehat{cl}_{G})\right]\in
KK_{n}(\mathbb{C},C^{*}(\mathfrak{A}G)).$ (4.2)

Let $G$ be a Lie groupoid over a closed smooth manifold $M$, such that its Lie
algebroid $\mathfrak{A}G$ is an orientable ($Spin$ or $Spin^{c}$) $M$-vector
bundle, where $M\rightrightarrows M$ denotes the trivial groupoid. Recall that
$u\colon M\to G$ denotes the unit map and that $\mathcal{N}_{u}$ is isomorphic
to $\mathfrak{A}G$. Then it is easy to check the following equality

$[pt]\otimes_{C(M)}\beta(u)=\left[\widehat{cl}_{G}\right]\in
KK_{n}(\mathbb{C},C^{*}(\mathfrak{A}G))$ (4.3)

where $[pt]\in KK(\mathbb{C},C(M))$ is the class induced by the collapsing map
$pt\colon M\to*$, which is proper if $M$ is compact. Recall that
$\mathfrak{A}G$ is the restriction of $G_{ad}\rightrightarrows M\times[0,1]$
to $M\times\\{0\\}$ and, as explained in Subsection 3.4, this restriction
induces a KK-equivalence. So, thanks to (4.1), it is clear that

$[D_{G_{ad}}]=\left[\widehat{cl}_{G}\right]\otimes_{C^{*}(\mathfrak{A}G)}[\mathrm{ev}_{0}]^{-1}\in
KK_{n}(\mathbb{C},C^{*}(G_{ad})).$ (4.4)

Primary invariants. Again by using (4.1), we obtain that

$[D_{G}]=[D_{G_{ad}}]\otimes_{C^{*}(G_{ad})}[\mathrm{ev}_{1}]\in
KK_{n}(\mathbb{C},C^{*}(G))$ (4.5)

which is primary invariant or the _index class_ of $D_{G}$.

Secondary invariants. Let us assume that there exists a bounded operator $A$
on $\mathcal{E}(G)$ such that $D_{G}+A$ is invertible. This implies that
$\mathrm{sgn}$, the sign function, is a continuous normalizing function on the
spectrum of $D_{G}+A$. It follows that
$(\mathcal{E}(G),\mathrm{sgn}(D_{G}+A))$ is a degenerate Kasparov bimodule.

######  Remark 4.5.

In this case it obviously follows that the index class (4.5) associated with
$D$ is the trivial element in $KK_{n}(\mathbb{C},C^{*}(G))$.

Now, we need the following ingredients: let $\psi$ be any continuous
normalizing function for $D_{G_{ad}}$; let
$\psi_{t}=t\cdot\mathrm{sgn}+(1-t)\cdot\psi$; finally, let
$\lambda\colon[0,1]\to[0,1]$ be the function which is given by $s\mapsto 2s$
for $s\in[0,1/2]$ and $s\mapsto 1$ for $s\in[1/2,1]$.

Consider the following two Kasparov bimodule

  * •

$\left(\mathcal{E}(G_{ad}),\psi(D_{G_{ad}})\right)\in\mathbb{E}_{n}(\mathbb{C},C^{*}(G_{ad}))$,

  * •

$\left(\mathcal{E}(G)\otimes
C_{0}[0,1],\psi_{t}(D_{G}+\lambda(t)A)\right)\in\mathbb{E}(\mathbb{C},C^{*}(G\times[0,1]))$,

and notice that the evaluation at 1 of the first one is equal to the
evaluation at 0 of the second one. Let us define, as in (3.2), the following
element in in $\mathbb{E}(\mathbb{C},C^{*}(G_{ad}^{[0,1)}))$ by the pair of
Kasparov bimodules

$\left(\left(\mathcal{E}(G_{ad}),\psi(D_{G_{ad}})\right),\left(\mathcal{E}(G)\otimes
C_{0}[0,1],\psi_{t}(D_{G}+\lambda(t)A)\right)\right).$ (4.6)

######  Definition 4.6.

[34] We will call the class of (4.6) the $\varrho$-class of the invertible
perturbation of $D_{G}$ associated with $A$ by

$\varrho(D_{G},A)\in KK_{n}(\mathbb{C},C^{*}(G_{ad}^{[0,1)})).$

If $D_{G}$ is already invertible, namely $A=0$, we will denote the associated
$\varrho$-class just by $\varrho(D_{G})$.

##  5 Transverse classes

###  5.1 The isometric case

Let $\iota\colon H\looparrowright G$ be an immersion of Lie groupoids. _Assume
that the normal groupoid $\mathcal{N}_{\iota}$ is the total space of a vector
bundle over $H$ of rank $n$ and it is associated with a cocycle $i\colon
H\to\mathcal{H}$, where $\mathcal{H}$ is the structural Lie group of
$\mathcal{N}_{\iota}$. Moreover suppose that $\mathcal{N}_{\iota}$ is the
pull-back through $r$ of an $H$-vector bundle over $H^{(0)}$ (as for instance
in Remark 3.11)._ Let $S\to G$ be a
$\mathrm{Cliff}(\mathcal{N}_{\iota})$-module.

######  Definition 5.1.

The lower shriek class associated with $\iota$ is given by the element

$\iota_{!}(S):=\beta(\iota,S)\otimes\partial(\iota)$ (5.1)

which belongs to $E_{n}(C^{*}(H),C^{*}(G))$. Here $\beta(\iota,S)$ is as in
Remark 3.11 and $\partial(\iota)$ as in Definition 3.12.

The immersion $\iota$, as in Remark 2.9, induces the following maps: the
immersion of adiabatic groupoids $\iota_{ad}\colon H_{ad}\looparrowright
G_{ad}$; by restriction to $[0,1)$, it also induces the immersion
$\iota_{ad}^{[0,1)}\colon H^{[0,1)}_{ad}\looparrowright G_{ad}^{[0,1)}$;
finally, by restriction to $\\{0\\}$, the immersion
$d\iota\colon\mathfrak{A}H\looparrowright\mathfrak{A}G$. These immersions
define in turn the lower shriek classes

$(\iota_{ad})_{!}\,,\quad(\iota_{ad}^{[0,1)})_{!}\quad\mbox{and}\quad
d\iota_{!}$

as elements of suitable E-theory groups.

######  Remark 5.2.

Notice that, by following the recipe of [1, Lemma 5.2] in the KK-theory of
mapping cone C*-algebras, is obtained by gluing $(\mathcal{E}_{ad},F_{ad})$,
which represents $(\iota_{ad})_{!}$, and $(\mathcal{E}\otimes
C_{0}([0,1)),F\otimes 1)$, where $(\mathcal{E},F)$ represents $\iota_{!}$.
Here it happens in E-theory.

It is then easy to show that the following diagram

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*}(C^{*}(H)\otimes
C_{0}(0,1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{!}\otimes
id_{C_{0}(0,1)}}$$\textstyle{K_{*}(C^{*}(H_{ad}^{[0,1)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{ad}^{[0,1)})_{!}}$$\textstyle{K_{*}(C^{*}(\mathfrak{A}H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\iota_{!}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*+n}(C^{*}(G)\otimes
C_{0}(0,1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*+n}(C^{*}(G_{ad}^{[0,1)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{*+n}(C^{*}(\mathfrak{A}G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
(5.2)

exists and commutes.

###  5.2 The almost isometric case

Even if the constructions in this section will not be used for the
applications, it is still important to present them in order to establish a
solid direction for a future generalization of the product formulas proved in
this article.

Let us consider two foliation $\mathcal{F}_{1}\subset\mathcal{F}_{2}$ of rank
$n_{1}$ and $n_{2}$, respectively, over $M$. Let us denote by $G_{1}$ and
$G_{2}$ the monodromy groupoids associated with $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$, respectively. The inclusion of the two foliations induces
an immersion $\iota\colon G_{1}\to G_{2}$ of Lie groupoids. Let us denote by
$d\iota$ the inclusion of Lie algebroids
$\mathcal{F}_{1}\hookrightarrow\mathcal{F}_{2}$ . Let us consider the normal
groupoid $\mathcal{N}_{\iota}$, it is isomorphic to
$G_{1}^{\mathcal{F}_{2}/\mathcal{F}_{1}}$. Recall that
$\mathcal{F}_{2}/\mathcal{F}_{1}$ is a $G_{1}$-vector bundle over $M$.

######  Definition 5.3.

[13, Remark 4.3] We say that $\mathcal{F}_{2}/\mathcal{F}_{1}$ is almost
isometric if there exists a $G_{1}$-invariant subbundle $E$, of rank $k$, and
a splitting $\mathcal{F}_{2}/\mathcal{F}_{1}\cong E\oplus E^{\prime}$ such
that both $E$ and $E^{\prime}$ are endowed with a $G_{1}$-equivariant
isometric structure. This means that the structural group of
$\mathcal{F}_{2}/\mathcal{F}_{1}$ reduces to the group

$\mathcal{H}:=\left\\{\begin{pmatrix}O_{k}&0\\\
M_{k,k^{\prime}}&O_{k^{\prime}}\end{pmatrix}\right\\}$ (5.3)

where $k$ and $k^{\prime}$ are the ranks of $E$ and $E^{\prime}$ respectively.
We say that it is almost isometric in the generalized sense if there exists a
sequence of $G_{1}$-invariant vector sub-bundles

$\mathcal{F}_{2}/\mathcal{F}_{1}=E_{i}\supseteq E_{i-1}\supseteq\dots\supseteq
E_{1}\supseteq E_{0}=\\{0\\}$

with $E_{j}/E_{j-1}$ endowed with a $G_{1}$-equivariant isometric structure
for $j=1,2,\dots,i$. Namely the structural group of
$\mathcal{F}_{2}/\mathcal{F}_{1}$ reduces to a group $\mathcal{H}$ of
triangular block matrices analogous to (5.3).

Let us assume that
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{2}/\mathcal{F}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
induces an almost isometric structure on the $G_{1}$-bundle
$\mathcal{F}_{2}/\mathcal{F}_{1}$, namely it is associated with a cocycle
$i\colon G_{1}\to\mathcal{H}$ as in (5.3). Moreover, let us assume that
$\mathcal{F}_{2}/\mathcal{F}_{1}$ is $Spin$, hence we have a hermitian
$G_{1}$-vector bundle $S$ of dimension $2^{\lfloor(n_{2}-n_{1})/2\rfloor}$ and
a $G_{1}$-equivariant homomorphism $cl\colon E^{*}\oplus(E^{\prime})^{*}\to
End(S)$ such that $cl(\xi)=cl(\xi)^{*}$, $cl(\xi)^{2}=\|\xi\|^{2}$ for all
$\xi\in E^{*}\oplus(E^{\prime})^{*}$.

Let $p\colon(\mathcal{F}_{2}/\mathcal{F}_{1})^{*}\to(E^{\prime})^{*}$ be the
transposed of the inclusion
$E^{\prime}\hookrightarrow\mathcal{F}_{2}/\mathcal{F}_{1}$. Let $s\colon
E\to\mathcal{F}_{2}/\mathcal{F}_{1}$ any section and let
$q\colon(\mathcal{F}_{2}/\mathcal{F}_{1})^{*}\to E^{*}$ be its transposed.
Finally let $1/2<\rho<1$.

Let us put
$b_{q}(x,\xi):=((1+\|p(\xi)\|^{2}+\|q(\xi)\|^{2})q(\xi),p(\xi))\in(E^{*}\oplus(E^{\prime})^{*})_{x}$
for $(x,\xi)\in(\mathcal{F}_{2}/\mathcal{F}_{1})^{*}_{x}$ and let us define a
bounded multiplier of $C_{0}((\mathcal{F}_{2}/\mathcal{F}_{1})^{*};\pi^{*}S)$
as follows

$a_{q}\colon(x,\xi)\mapsto(1+\|b_{q}(x,\xi)\|^{2})^{-1/2}cl(b_{q}(x,\xi))\in
End(S_{x}).$ (5.4)

Let us fix a point $x\in M$: we have a vector space
$V:=(\mathcal{F}_{2}/\mathcal{F}_{1})^{*}_{x}$ and a representation $E:=S_{x}$
of $Cliff(V)$. We naturally obtain an element $[C_{0}(V,E),a_{q}(x,\cdot)]$ in
$KK^{\mathcal{H}}_{k}(\mathbb{C},C_{0}(V))$, where $\mathcal{H}$ is as in
(5.3).Thanks to [13, Lemma 4.1], if we apply to this KK-element the
construction of Definition 3.10, we obtain the element

$\beta^{a.i.}(\iota)\in KK_{k}(C^{*}(G_{1}),C^{*}(\mathcal{N}_{\iota})).$
(5.5)

######  Definition 5.4.

The lower shriek class associated with an immersion of foliations
$\iota\colon\mathcal{F}_{1}\to\mathcal{F}_{2}$ with almost isometric normal
bundle is given by the element

$\iota_{!}^{a.i.}:=\beta^{a.i.}(\iota)\otimes\partial(\iota)\in
KK_{k}(C^{*}(G_{1}),C^{*}(G_{2})).$ (5.6)

It is obvious from the construction that if the normal bundle of $\iota$ has
an isometric structure $\iota_{!}$ and $\iota^{a.i.}_{!}$ coincide. As in the
previous section, it is now easy to define the lower shriek class associated
with $\iota_{ad}^{[0,1)}$ and $d\iota$ and obtain diagram (5.2) also in the
almost isometric case.

######  Remark 5.5.

It is immediate from the definition that this is an equivalent description of
the class constructed in [13, Definition 4.2].

##  6 The equivariant setting

###  6.1 Semidirect products and imprimitivity bimodules

Let us first recall some abstract definitions from [18, Section 3.2].

######  Definition 6.1.

Let $X$ be a proper, $\sigma$-compact $\Gamma$-space. Denote by $\\{v_{i}\\}$
some countable approximate unit in $C_{0}(X/\Gamma)$. For any $\Gamma-
C_{0}(X)$-algebra $B$, define $B^{\Gamma}$ as the subalgebra of
$\mathbb{B}(B)$ consisting of those $\Gamma$-invariant elements
$b\in\mathbb{B}(B)$ for which $f\cdot b\in B$, $\forall f\in C_{0}(X)$ , and
$\lim\|v_{i}b-b\|=0$ (where elements $v_{i}$ are considered as functions on
$X$ via the projection $X\to X/\Gamma$). Clearly, $B^{\Gamma}$ is a
$C_{0}(X/\Gamma)$-algebra. Now, for any Hilbert $\Gamma-B$-module
$\mathcal{E}$, one can define a Hilbert $B^{\Gamma}$-module
$\mathcal{E}^{\Gamma}$ as follows. The space of operators
$\mathbb{B}(B,\mathcal{E})$ is a Hilbert $\mathbb{B}(B)$\- module. There is an
inclusion $\mathcal{E}\hookrightarrow\mathbb{B}(B,\mathcal{E})$ given by
$e\mapsto\tilde{e}$, where $\tilde{e}(b)=e\cdot b$, $\forall b\in B$. We shall
define $\mathcal{E}^{\Gamma}$ as the subspace of $\mathbb{B}(B,\mathcal{E})$
consisting of those $\Gamma$-invariant elements
$\tilde{e}\in\mathbb{B}(B,\mathcal{E})$ for which
$f\cdot\tilde{e}\in\mathcal{E}$, $\forall f\in C_{0}(X)$, and
$\lim\|v_{i}\tilde{e}-\tilde{e}\|=0$.

Let $\Gamma$ be a group and let $G\rightrightarrows G^{(0)}$ be a Lie
groupoid. Consider an homomorphism $\omega\colon\Gamma\to\mathrm{Aut}(G)$.
This induces an action $G\times\Gamma\to G$ given by
$(g,\gamma)\mapsto\omega_{\gamma^{-1}}(g)$.

######  Definition 6.2.

We can define a groupoid structure on $G\times\Gamma$ over $G^{(0)}$, which we
will denote by $G\rtimes_{\omega}\Gamma$, in the following way:

  * •

the source and range maps are given by
$s(g,\gamma):=\omega_{\gamma^{-1}}(s(g))$ and $r(g,\gamma):=r(g)$;

  * •

the inverse is given by
$(g,\gamma)^{-1}:=(\omega_{\gamma}(g^{-1}),\gamma^{-1})$;

  * •

the product is defined as follows:
$(g,\gamma)\cdot(g^{\prime},\gamma^{\prime}):=(g\cdot\omega_{\gamma}(g^{\prime}),\gamma\gamma^{\prime})$.

######  Proposition 6.3.

Let $\Gamma$ act on $G$ in a free and properly continuous way. Then
$G\rtimes\Gamma\rightrightarrows G^{(0)}$ is Morita equivalent to
$G/\Gamma\rightrightarrows G^{(0)}/\Gamma$.

###### Proof.

Let $q\colon G\to G/\Gamma$ denote the quotient map. Consider the following
groupoid

$L:=G\rtimes\Gamma\sqcup\overline{G}\sqcup\overline{G}^{-1}\sqcup
G/\Gamma\rightrightarrows G^{(0)}\sqcup G^{(0)}/\Gamma$

where as sets $G=\overline{G}=\overline{G}^{-1}$ and

  * •

$s_{L}$ is equal to: the corresponding source maps on $G\rtimes\Gamma$ and
$G/\Gamma$, $q\circ s_{G}$ on $\overline{G}$ and $s_{G}$ on
$\overline{G}^{-1}$;

  * •

$r_{L}$ is equal to: the corresponding range maps on $G\rtimes\Gamma$ and
$G/\Gamma$, $r_{G}$ on $\overline{G}$ and $q\circ r_{G}$ on
$\overline{G}^{-1}$;

  * •

$i_{L}$ is equal to: the corresponding inverse maps on $G\rtimes\Gamma$ and
$G/\Gamma$, sends $g\in\overline{G}$ to $g^{-1}\in\overline{G}^{-1}$ and
similarly for elements in $\overline{G}^{-1}$;

  * •

the composition given by:

    * –

the corresponding one on $G\rtimes\Gamma$ and $G/\Gamma$;

    * –

$(g,\gamma)\cdot g^{\prime}:=(g\cdot\omega_{\gamma}(g^{\prime}),\gamma)$ for
$(g,\gamma)\in G\rtimes\Gamma$ and $g^{\prime}\in\overline{G}$;

    * –

$g^{\prime}\cdot(g,\gamma):=(g^{\prime}\cdot g,\gamma)$ for
$g^{\prime}\in\overline{G}^{-1}$ and $(g,\gamma)\in G\rtimes\Gamma$; $g\cdot
q(g^{\prime}):=g\cdot\omega_{\gamma^{\prime}}(g^{\prime})$ for
$g\in\overline{G}$ and $q(g^{\prime})\in G/\Gamma$, where $\gamma^{\prime}$ is
uniquely determined so that $s(g)=r(\omega_{\gamma}(g^{\prime}))$;

    * –

similarly $q(g)\cdot g^{\prime}=\omega_{\gamma}(g)\cdot g^{\prime}$ for
$q(g)\in G/\Gamma$ and $g^{\prime}\in\overline{G}$;

    * –

for $g\in\overline{G}$ and $g^{\prime}\in\overline{G}^{-1}$ with
$s(g)=\omega_{\gamma^{\prime}}(r(g^{\prime}))$, $g\cdot
g^{\prime}=g\cdot\omega_{\gamma^{\prime}}(g^{\prime})$;

    * –

finally for $g\in\overline{G}^{-1}$ and $g^{\prime}\in\overline{G}$, we set
$g\cdot g^{\prime}=q(g\cdot g^{\prime})$.

By [7, Definition 2] the result follows. ∎

The proof of this classical result is useful here to recall the explicit
construction of a link algebra and then the imprimitivity bimodules associated
with the Morita equivalence between $C^{*}(G\rtimes\Gamma)$ and
$C^{*}(G/\Gamma)$. Indeed, $C^{*}(L)$ does the role of the link algebra and
the imprimitivity bimodule is given by the completion of
$C^{\infty}_{c}(\overline{G})$ inside $C^{*}(L)$.

######  Definition 6.4.

Let us denote the imprimitivity
$C^{*}(G\rtimes\Gamma),C^{*}(G/\Gamma)$-bimodule associated with the previous
Morita equivalence by $\mathcal{M}_{\Gamma}^{G}$.

Let $E\to G^{(0)}$ be a $\Gamma$-equivariant vector bundle. Denote by
$\mathcal{E}_{c}$ the $C_{c}^{\infty}(G)$-module
$C^{\infty}_{c}(G,r^{*}E\otimes\Omega^{\frac{1}{2}})$ and by $\mathcal{E}$ its
C*-completion with respect to the usual $C^{*}(G)$-inner product

$\langle\xi,\xi^{\prime}\rangle(\gamma):=\int_{G_{s(\gamma)}}\langle\overline{\xi(\gamma\eta^{-1})}\xi^{\prime}(\eta)\rangle_{E},$
(6.1)

moreover recall that the right action of $C^{*}(G)$ on $\mathcal{E}$ is given
by

$\xi\cdot f(\gamma):=\int_{G_{s(\gamma)}}\xi(\gamma\eta^{-1})f(\eta).$ (6.2)

As in [18, Definition 3.8], associated with the action of $\Gamma$ on
$\mathcal{E}$, we can construct the $C^{*}(G)\rtimes\Gamma$-module
$\mathcal{E}\rtimes\Gamma$ as the C*-completion of
$C_{c}\left(\Gamma,C^{\infty}_{c}(G,r^{*}E\otimes\Omega^{\frac{1}{2}})\right)$.

Another way to obtain a $C^{*}(G)\rtimes\Gamma$-module from $\mathcal{E}_{c}$
is to endow it with a $C^{*}(G)\rtimes\Gamma$-valued inner product
$\langle\cdot,\cdot\rangle_{\rtimes}$ defined by
$\langle\xi,\eta\rangle_{\rtimes}(\gamma):=\langle\xi,\omega_{\gamma}^{*}(\eta)\rangle$.
Let us denote its C*-completion by $\mathcal{E}^{\rtimes}$.

Furthermore, observe that $\mathcal{E}^{\Gamma}$, the $C^{*}(G/\Gamma)$-module
of $\Gamma$-invariant elements of $\mathcal{E}$, is isomorphic to the
$C^{*}(G/\Gamma)$-completion of
$C^{\infty}_{c}(G/\Gamma,r^{*}\bar{E}\otimes\Omega^{\frac{1}{2}})$, where
$\bar{E}$ is the quotient of $E$ by the action of $\Gamma$.

Finally, notice that both $\mathcal{E}^{\Gamma}$ and $\mathcal{E}^{\rtimes}$
are left $C(G^{(0)}/\Gamma)$-modules, that $\mathcal{E}$ is a left
$C(G^{(0)})$-module and then that $\mathcal{E}\rtimes\Gamma$ is a left
$C_{0}(G^{(0)})\rtimes\Gamma$-module.

######  Lemma 6.5.

There exists an isomorphism between $\mathcal{E}^{\rtimes}$ and
$\mathcal{E}^{\Gamma}\otimes_{C^{*}(G/\Gamma)}(\mathcal{M}_{\Gamma}^{G})^{-1}$
as $C^{*}(G)\rtimes\Gamma$-modules.

###### Proof.

Consider the morphism
$\alpha\colon\mathcal{E}^{\Gamma}_{c}\otimes_{C_{c}^{\infty}(G/\Gamma)}C_{c}^{\infty}(G)\to\mathcal{E}_{c}$
given by $\alpha(\tilde{\eta}\otimes\xi)=\tilde{\eta}(\xi)$. Following the
reasoning in [18, Lemma 3.2] \- notice that here we consider an isomorphism of
$C^{*}(G)\rtimes\Gamma$-modules whereas there it is considered a morphism of
$C^{*}(G)$-modules - it is easy to prove that this is an isometric map with
dense range. ∎

######  Remark 6.6.

Observe that from the proof of [18, Theorem 3.14], we have the following
isomorphisms of $C_{0}(G^{(0)}),C^{*}(G)\rtimes\Gamma$-bimodules

$\mathcal{E}^{\rtimes}\cong\mathcal{E}^{\Gamma}\otimes_{C^{*}(G/\Gamma)}(\mathcal{M}_{\Gamma}^{G})^{-1}\cong(\mathcal{M}_{\Gamma}^{G^{(0)}})^{-1}\otimes_{C_{0}(G^{(0)})\rtimes\Gamma}\mathcal{E}\rtimes\Gamma$
(6.3)

###  6.2 Equivariant KK-classes

Consider the following situation: let $\Gamma$ be a discrete group acting on
the Lie groupoid $G$ in a free and properly discontinuous way; let $i\colon
G\to\mathcal{H}$ be a $\Gamma$-invariant cocycle, namely it descends to a
cocycle $l\colon G/\Gamma\to\mathcal{H}$. Then we have the following result.

######  Proposition 6.7.

The cocycle $i$ defines a map $i\colon KK^{\mathcal{H}}(A,B)\to
KK^{\Gamma}(A\rtimes_{i}G,B\rtimes_{i}G)$ and the following equality holds

$\mathcal{M}_{A,\Gamma}^{-1}\otimes
j^{\Gamma}(i^{*}(x))\otimes\mathcal{M}_{B,\Gamma}=l^{*}(x)$ (6.4)

where $x\in KK^{\mathcal{H}}(A,B)$ and $\mathcal{M}_{A,\Gamma}$ is an
imprimitivity bimodule for the algebras $(A\rtimes_{i}G)\rtimes\Gamma$ and
$A\rtimes_{l}(G/\Gamma)$ and similarly for $\mathcal{M}_{B,\Gamma}$.

###### Proof.

The first statement follows from the fact that the KK-element
$\mathrm{Id}_{C^{*}(G^{P_{i}})}$ actually lies in
$KK^{\Gamma}(C^{*}(G^{P_{i}}),C^{*}(G^{P_{i}}))$ and (6.4) follows from the
following calculations:

$\begin{split}&\mathcal{M}_{A,\Gamma}^{-1}\otimes
j^{\Gamma}(i^{*}(x))\otimes\mathcal{M}_{B,\Gamma}=\\\
=&\mathcal{M}_{A,\Gamma}^{-1}\otimes j^{\Gamma}(\mathcal{M}_{A,i}\otimes
j^{\mathcal{H}}(\mathrm{Id}_{C^{*}(G^{P_{i}})}\otimes
x)\otimes\mathcal{M}^{-1}_{B,i})\otimes\mathcal{M}_{B,\Gamma}=\\\
=&\mathcal{M}_{A,\Gamma}^{-1}\otimes j^{\Gamma}(\mathcal{M}_{A,i})\otimes
j^{\Gamma}\circ j^{\mathcal{H}}(\mathrm{Id}_{C^{*}(G^{P_{i}})}\otimes
x)\otimes j^{\Gamma}(\mathcal{M}^{-1}_{B,i})\otimes\mathcal{M}_{B,\Gamma}=\\\
=&\mathcal{M}_{A,\Gamma}^{-1}\otimes j^{\Gamma}(\mathcal{M}_{A,i})\otimes
j^{\mathcal{H}}(\mathcal{M}_{\Gamma}^{i,l})\otimes
j^{\mathcal{H}}(\mathcal{M}_{\Gamma}^{i,l})^{-1}\otimes j^{\mathcal{H}}\circ
j^{\Gamma}(\mathrm{Id}_{C^{*}(G^{P_{i}})}\otimes x)\otimes\\\ &\otimes
j^{\mathcal{H}}(\mathcal{M}_{\Gamma}^{i,l})\otimes
j^{\mathcal{H}}(\mathcal{M}_{\Gamma}^{i,l})^{-1}\otimes
j^{\Gamma}(\mathcal{M}^{-1}_{B,i})\otimes\mathcal{M}_{B,\Gamma}=\\\
=&\mathcal{M}_{A,l}^{-1}\otimes
j^{\mathcal{H}}\left((\mathcal{M}_{\Gamma}^{i,l})^{-1}\otimes
j^{\Gamma}(\mathrm{Id}_{C^{*}(G^{P_{i}})}\otimes
x)\otimes\mathcal{M}_{\Gamma}^{i,l}\right)\otimes\mathcal{M}_{B,l}=\\\
=&\mathcal{M}_{A,l}^{-1}\otimes
j^{\mathcal{H}}\left(\left((\mathcal{M}_{\Gamma}^{i,l})^{-1}\otimes
j^{\Gamma}(\mathrm{Id}_{C^{*}(G^{P_{i}})})\otimes\mathcal{M}_{\Gamma}^{i,l}\right)\otimes
x\right)\otimes\mathcal{M}_{B,l}=\\\ =&\mathcal{M}_{A,l}^{-1}\otimes
j^{\mathcal{H}}\left(C^{*}((G/\Gamma)^{P_{l}})\otimes
x\right)\otimes\mathcal{M}_{B,l}=l^{*}(x)\end{split}$ (6.5)

where we used the distibuitivity of the descent maps and the fact that
$j^{\Gamma}$ and $j^{\mathcal{H}}$ commute since the two actions commute;
moreover $\mathcal{M}_{\Gamma}^{i,l}$ is the imprimitivity bimodule for the
C*-algebras $C^{*}(G^{P_{i}}\rtimes\Gamma)$ and $C^{*}((G/\Gamma)^{P_{l}})$ as
in Definition 6.4; finally we repeatedly used the fact that a crossed product
bimodule $\mathcal{E}\rtimes\Gamma$ is conjugated the to bimodule of invariant
elements $\mathcal{E}^{\Gamma}$ by means of interior tensor product with
suitable imprimitivity bimodules, see the proof of [18, Theorem 3.14] for the
details about this fact. ∎

######  Proposition 6.8.

Let $\Gamma$ be a discrete group acting in a free and properly discontinuous
way on both $G$ and $H$. Let $\iota\colon G\looparrowright H$ be a
$\Gamma$-equivariant immersion. Then the element $\partial^{G}_{H}$ in (3.8)
belongs to $E^{\Gamma}(C^{*}(\mathcal{N}_{H}^{G}),C^{*}(G))$ and

$(\mathcal{M}_{\Gamma}^{\mathcal{N}})^{-1}\otimes
j^{\Gamma}(\partial^{G}_{H})\otimes\mathcal{M}_{\Gamma}^{G}=\partial^{G/\Gamma}_{H/\Gamma}.$
(6.6)

###### Proof.

The proof is immediate, since the element in (3.8) is the composition of a
morphism and the inverse of a morphism. ∎

From now on let us assume that the action of $\Gamma$ on $G^{(0)}$ is
cocompact, so that the map $pt\colon G^{(0)}/\Gamma\to\ast$ is proper. Fix a
$\Gamma$-invariant metric $\overline{g}$ on $AG$, which descends to a metric
$g$ on $AG/\Gamma$. Thanks to Proposition 6.7 and 6.8 one can easily see that
the following equalities hold:

$[\hat{cl}_{G/\Gamma}]=[pt]\otimes(\mathcal{M}_{\Gamma}^{G^{(0)}})^{-1}\otimes
j^{\Gamma}(\beta(u))\otimes\mathcal{M}^{AG}_{\Gamma},$ (6.7)

where $\beta(u)$ belongs to $KK^{\Gamma}_{n}(C_{0}(G^{(0)}),C^{*}(AG))$;

$[D_{G/\Gamma}]=[pt]\otimes(\mathcal{M}_{\Gamma}^{M})^{-1}\otimes
j^{\Gamma}([D_{G}])\otimes\mathcal{M}^{G}_{\Gamma},$ (6.8)

where $[D_{G}]$ is the class of the Dirac operator in
$KK^{\Gamma}_{n}(C_{0}(G^{(0)}),C^{*}(G))$.

######  Proposition 6.9.

Let $\Gamma$ be a discrete group acting in a free and properly discontinuous
way on both $G$ and $H$. Assume that $\overline{\iota}\colon G\to H$ is
$\Gamma$-equivariant and that the normal bundle of $\bar{\iota}$ is associated
with a $\Gamma$-invariant cocycle. Then $(\bar{\iota}_{ad}^{[0,1)})_{!}$ gives
an element in $E^{\Gamma}_{n}(C^{*}(H_{ad}^{[0,1)}),C^{*}(G_{ad}^{[0,1)}))$.
Moreover, let $\iota\colon G/\Gamma\to H/\Gamma$ be the immersion induced by
$\iota$ between the quotients, then

$(\mathcal{M}_{\Gamma}^{H_{ad}})^{-1}\otimes
j^{\Gamma}\left((\overline{\iota}_{ad}^{[0,1)})_{!}\right)\otimes(\mathcal{M}_{\Gamma}^{G_{ad}})=(\iota_{ad}^{[0,1)})_{!}.$
(6.9)

###### Proof.

It follows immediately from Proposition 6.7 and Proposition 6.8. ∎

Now, let us consider a Dirac type operator $D_{G/\Gamma}$ acting on a
$C^{*}(G/\Gamma)$-module $\mathcal{E}^{\Gamma}$, given by the completion of
$\overline{C^{\infty}_{c}(G/\Gamma,r^{*}\overline{E}\otimes\Omega^{\frac{1}{2}})}$,
and let us assume that $D_{G/\Gamma}$ is invertible. The operator
$D_{G/\Gamma}\otimes 1$ on
$\mathcal{E}^{\Gamma}\otimes(\mathcal{M}_{\Gamma}^{G})^{-1}$ defines, by means
of the isomorphism $\alpha$ in Lemma 6.5 an operator on
$\mathcal{E}^{\rtimes}$ which we can identify in the following way. Take
$\eta\in\overline{C^{\infty}_{c}(G/\Gamma,r^{*}\overline{E}\otimes\Omega^{\frac{1}{2}})}$
and $\xi\in C^{\infty}_{c}(G,\Omega^{\frac{1}{2}})$, then we have

$\begin{split}&\alpha\circ D_{G/\Gamma}\otimes
1\circ\alpha^{-1}(\tilde{\eta}(\xi))=\\\ =&\alpha\circ D_{G/\Gamma}\otimes
1(\tilde{\eta}\otimes\xi)=\\\
=&\alpha(D_{G/\Gamma}(\tilde{\eta})\otimes\xi)=D_{G/\Gamma}(\tilde{\eta})(\xi).\end{split}$
(6.10)

The last term, following Definition 6.1, corresponds to the
$\Gamma$-equivariant lift $D_{G}^{\Gamma}$ of $D_{G/\Gamma}$ acting on the
element $\eta\cdot\xi$ defined as in (6.2), which is an element in
$C^{\infty}_{c}(G,r^{*}E\otimes\Omega^{\frac{1}{2}})$ seen as a
$C^{*}(G)\rtimes\Gamma$-module. Here $E$ is the pull-back of $\overline{E}$
through the quotient map $G^{(0)}\to G^{(0)}/\Gamma$. Namely $D^{\Gamma}_{G}$
defines an operator on the Hilbert module $\mathcal{E}^{\rtimes}$. Finally
observe that $D^{\Gamma}_{G}$ is invertible.

######  Proposition 6.10.

We have that
$\varrho(D_{G/\Gamma})\otimes(\mathcal{M}^{G}_{\Gamma})^{-1}=\varrho(D^{\Gamma}_{G})\in
KK(\mathbb{C},C^{*}(G)\rtimes\Gamma)$.

###### Proof.

Let the subscript $ad$ indicate the Hilbert module associated with the pull-
back of the vector bundle $E$ on the adiabatic deformation groupoid. Then, let
us recall that $\varrho(D_{G/\Gamma})$ is represented, as in (3.2), by

$\left[\left(pt^{*}\mathcal{E}^{\Gamma}_{ad},\psi(D_{G/\Gamma_{ad}})\right),\left(pt^{*}\mathcal{E}^{\Gamma}\otimes
C_{0}[0,1],\psi_{t}(D_{G/\Gamma})\right)\right]\in
KK(\mathbb{C},C^{*}(G/\Gamma_{ad}^{[0,1)}))$ (6.11)

as in (4.6). Analogously $\varrho(D^{\Gamma}_{G})$ is represented by

$\left[\left(pt^{*}\mathcal{E}^{\rtimes}_{ad},\psi(D_{G_{ad}})^{\Gamma}\right),\left(pt^{*}\mathcal{E}^{\rtimes}\otimes
C_{0}[0,1],\psi_{t}(D^{\Gamma}_{G})\right)\right]\in
KK(\mathbb{C},C^{*}(G_{ad}^{[0,1)})\rtimes\Gamma).$ (6.12)

Now, consider
$\varrho(D_{G/\Gamma})\otimes_{C^{*}(G/\Gamma)}(\mathcal{M}^{G}_{\Gamma})^{-1}$
and let us represent it by following the recipe given in (3.3). Thus, thanks
to Lemma 6.5, we obtain the mapping cone class given, as in (3.2), by the pair

$\begin{split}&\left(pt^{*}\mathcal{E}^{\rtimes}_{ad},\psi(D_{G/\Gamma_{ad}})\otimes
1\right)\,\mbox{and}\\\ \left(pt^{*}\mathcal{E}^{\rtimes}\otimes
C_{0}[0,1],\psi_{t}(D_{G/\Gamma})\otimes
1\right)&\diamond(pt^{*}\mathcal{E}^{\rtimes}\otimes
C_{0}[0,1],CS_{t}(\mathrm{sgn}(D_{G/\Gamma})\otimes
1,\mathrm{sgn}(D_{G/\Gamma}\otimes 1))\end{split}$ (6.13)

which, thanks to (6.10) and the fact that $\mathrm{sgn}(D_{G/\Gamma})\otimes
1=\mathrm{sgn}(D_{G/\Gamma}\otimes 1)$, is equal to (6.12). ∎

##  7 Product formulas

###  7.1 Smooth fibrations

Let $(M,g^{M})$ and $(B,g^{B})$ be two smooth compact connected Riemannian
manifolds and let $\pi\colon M\to B$ be a surjective Riemannian submersion.
This means that the tangent bundle $TM$ splits as $\ker d\pi\oplus\pi^{*}TB$
and that the metric $g^{M}$ can be expressed as the sum
$g^{M/B}\oplus\pi^{*}g^{B}$, where $g^{M/B}$ is a metric for $\ker d\pi$.

Let $\nabla^{\alpha}$ be the Levi-Civita connection for $g^{\alpha}$, with
$\alpha=M,B$ or $M/B$. We thus obtain two metric connections $\nabla^{M}$ and
$\nabla^{\oplus}:=\nabla^{M/B}\oplus\pi^{*}\nabla^{B}$ on $TM$ whose
difference is calculated in terms of the tensor $\omega\in
C^{\infty}(M;T^{*}M\otimes\bigwedge^{2}T^{*}M)$ defined by

$\begin{split}\omega(X)(Y,Z):=&S(X,Z,Y)-S(X,Y,Z)+\\\
+&\frac{1}{2}\left(\Omega(X,Z,Y)-\Omega(X,Y,Z)+\Omega(Y,Z,X)\right)\end{split}$
(7.1)

for $X,Y,Z\in C^{\infty}(M;TM)$. Here, set $P$ the projection $TM\to\ker
d\pi$, then

  * •

$S\in C^{\infty}(M;T^{*}M^{\otimes 3})$ is the _second fundamental form_
defined by

$S(X,Y,Z):=g^{M}\left(\nabla^{M/B}_{(1-P)Z}(PX)-[(1-P)Z,PX],PY\right),$ (7.2)

  * •

we will also need the trace of $S$, which gives a 1-form $k\in
C^{\infty}(M;T^{*}M)$, the _mean curvature_ , defined as

$k(X):=\sum_{i}S(e_{i},e_{i},X),$ (7.3)

  * •

$\Omega\in C^{\infty}(M;\bigwedge^{2}T^{*}M\otimes T^{*}M)$ is the _curvature
of the fibration $\pi$_ defined by

$\Omega(X,Y,Z):=-g^{M}\left([(1-P)X,(1-P)Y],PZ\right),$ (7.4)

for $X,Y,Z\in C^{\infty}(M;TM)$.

######  Lemma 7.1.

The connections $\nabla^{M}$ and $\nabla^{\oplus}$ are related by the
following formula

$g^{M}\left(\nabla^{M}_{X}Y,Z\right)=g^{M}\left(\nabla^{\oplus}_{X}Y,Z\right)+\omega(X)(Y,Z).$
(7.5)

In [5] and [15] the following factorization result is stated for the
$Spin^{(c)}$ Dirac operator, but the proof, which essentially depends only on
(7.5), works verbatim for generalized Dirac operator too. Consider the
following objects:

  * •

the Dirac operator $D^{B}$ acting on the sections of a bundle of
$\mathrm{Cliff}(TB,g^{B})$-modules $E^{B}\to B$;

  * •

the Dirac operator $D^{M/B}$ acting on the sections of a bundle of
$\mathrm{Cliff}(\ker d\pi,g^{M/B})$-modules $E^{M/B}\to M$.

Let us consider the following metric depending on a parameter $\varepsilon$

$g^{M}_{\varepsilon}=g^{M/B}\oplus\varepsilon^{-1}\pi^{*}g_{B},$ (7.6)

then the Dirac-type operator $D_{\varepsilon}^{M}$ acts on the sections of the
bundle of $\mathrm{Cliff}(TM,g_{\varepsilon}^{M})$-modules
$E^{M}:=E^{M/B}\otimes\pi^{*}E^{B}$ and can be written, as in [5, Eq. (4.26)],
in the following way

$D^{M}_{\varepsilon}=D^{M/B}\otimes
1+\varepsilon^{\frac{1}{2}}\sum_{i}c(f_{i})\widetilde{\nabla}^{B}_{f_{i}}-\frac{\varepsilon}{4}\sum_{i<j}c(f_{i}f_{j})c(\Omega(f_{i},f_{j}))$
(7.7)

where $\widetilde{\nabla}^{B}_{X}=(\pi^{*}\nabla^{B})_{X}-\frac{1}{2}k(X)$ for
$X\in C^{\infty}(M;TM)$ and $\\{f_{i}\\}$ is a local orthonormal frame of
$TM$.

In [15] it is meticulously proved that the unbounded Kasparov
$C(M)$-$\mathbb{C}$-bimodule $(L^{2}(M;E^{M}),D^{M}_{\varepsilon})$ is a
bounded perturbation of the unbounded Kasparov product of the unbounded
$C(M)$-$C(B)$-bimodule $(L^{2}(M;E^{M/B}),D^{M/B})$ and the unbounded
$C(B)$-$\mathbb{C}$-bimodule $(L^{2}(B;E^{B}),D^{B}_{\varepsilon})$, where
$D^{B}_{\varepsilon}$ is the Dirac operator associated with the metric
$\varepsilon^{-1}g_{B}$ and where the bounded defect is given by the last term
$-\frac{\varepsilon}{4}\sum_{i<j}c(f_{i}f_{j})c(\Omega(f_{i},f_{j}))$ in
(7.7).

Now, notice that the Lie groupoid $M\times_{\pi}M\rightrightarrows M$ is
Morita equivalent to $B\rightrightarrows B$ and that $M\times
M\rightrightarrows M$ is Morita equivalent to the trivial groupoid
$pt\rightrightarrows pt$. Let us denote by $\mathcal{M}_{\pi}$ and
$\mathcal{M}_{pt}$ the imprimitivity bimodules associated with these Morita
equivalences. More precisely: $\mathcal{M}_{\pi}$ is the closure of
$C_{c}^{\infty}(M)$ equipped with the $C(B)$-valued inner product $\langle
f,g\rangle(b):=\int_{\pi^{-1}(b)}\overline{f(x)}g(x)$ with the obvious left
action of $C_{c}^{\infty}(M\times_{\pi}M)$; $\mathcal{M}_{pt}$ is just
$L^{2}(M)$ with the obvious left action of $C^{*}(M\times M)$. Now, it is an
easy observation that the previous factorization corresponds, by means of
these Morita equivalences, to the factorization at the unbounded level of the
following composition of inclusions of Lie groupoids over $M$

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{\pi}}$$\scriptstyle{u}$$\textstyle{M\times_{\pi}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{M\times
M}$

namely that $u_{!}=u^{\pi}_{!}\otimes\iota_{!}$, where $u^{\pi}$ is the unit
map of $M\times_{\pi}M$ and $u$ is the unit map of $M\times M$. Thanks to
(4.3), (4.4), (4.5), by precomposing with the KK-class $[pt]$ associated with
the map $M\to pt$, this just gives us the unbounded factorization of the
primary class $[D_{M\times M}]=[D_{M\times_{\pi}M}]\otimes\iota_{!}$.

In particular, with the notation of Definition 4.3 and $\simeq$ standing for
unitarily equivalent, observe that:

  * •

$\left(\mathcal{E}(M\times M,E^{M}),D^{E^{M}}_{M\times
M}\right)\otimes\mathcal{M}_{pt}\simeq(L^{2}(M;E^{M}),D^{M})\in\mathbb{E}(\mathbb{C},\mathbb{C})$;

  * •

$\left(\mathcal{E}(M\times_{\pi}M,E^{M/B}),D^{E^{M/B}}_{M\times_{\pi}M}\right)\otimes\mathcal{M}_{\pi}\simeq(L^{2}(M\to
B;E^{M/B}),D^{M/B})\in\mathbb{E}(\mathbb{C},C(B))$;

  * •

then, up to the last bounded term in (7.7),

$\begin{split}&\left(\mathcal{E}(M\times M,E^{M}),D^{E^{M}}_{M\times
M}\right)\simeq\\\
&\simeq\left(L^{2}(M;E^{M}),D^{M}\right)\otimes\mathcal{M}_{pt}^{-1}\simeq\\\
&\simeq\left(L^{2}(M\to
B;E^{M/B}),D^{M/B}\right)\otimes\left(L^{2}(B;E^{B}),D^{B}\right)\otimes\mathcal{M}_{pt}^{-1}\simeq\\\
&\simeq\left(\left(\mathcal{E}(M\times_{\pi}M,E^{M/B}),D^{E^{M/B}}_{M\times_{\pi}M}\right)\otimes\mathcal{M}_{\pi}\otimes(L^{2}(B;E^{B}),D^{B})\right)\otimes\mathcal{M}_{pt}^{-1}.\end{split}$
(7.8)

  * •

so that

$D^{E^{M}}_{M\times M,\varepsilon}=D^{E^{M/B}}_{M\times_{\pi}M}\otimes
1\otimes 1+\varepsilon^{\frac{1}{2}}(1\otimes_{\nabla}D^{B})\otimes
1-\varepsilon A$ (7.9)

where $1\otimes_{\nabla}D^{B}$ is a $D^{B}$-connection on
$\left(\mathcal{E}(M\times_{\pi}M,E^{M/B})\otimes\mathcal{M}_{\pi}\right)\otimes
L^{2}(B;E^{B})$, unitarily equivalent to the second term in (7.7), and $A$ is
a zero order term corresponding to the last term in (7.7).

The reason we did this detailed translation to the Lie groupoid setting of the
factorization (7.7) is that we need a fine estimate of the commutator of
$D^{E^{M/B}}_{M\times_{\pi}M}\otimes 1\otimes 1$ and $D^{E^{M}}_{M\times
M,\varepsilon}$ in order to apply Lemma 3.4 in the following theorem.

######  Theorem 7.2.

Let $\pi\colon(M,g^{M})\to(B,g^{B})$ be a Riemannian submersion between $Spin$
manifolds. Let $g^{M/B}$ be a metric of $\ker d\pi$ with positive scalar
curvature and let $\varrho(D_{M\times_{\pi}M})\in
KK_{n}(\mathbb{C},C^{*}(M\times_{\pi}M_{ad}^{[0,1)}))$ be the associated
$\varrho$-class. Let $\varepsilon_{0}$ be such that $g^{M}_{\varepsilon_{0}}$
has positive scalar curvature too. Then we have that

$\varrho(D_{M\times_{\pi}M})\otimes\iota_{!}^{ad}=\varrho(D^{\varepsilon_{0}}_{M\times
M})\in KK_{n+k}(\mathbb{C},C^{*}(M\times M_{ad}^{[0,1)}))$ (7.10)

where $D$ denotes the $Spin$ Dirac operator, $\iota$ is the inclusion of Lie
groupoids $M\times_{\pi}M\hookrightarrow M\times M$, $n=rk(\ker d\pi)$ and
$k=\dim B$.

###### Proof.

Let us denote by $H$ the Lie groupoid $M\times_{\pi}M$ and by $G$ the Lie
groupoid $M\times M$. Recall from (4.6) that

$\varrho(D_{H})=\left[\left(\mathcal{E}(H_{ad}),\psi(D_{H_{ad}})\right)\left(\mathcal{E}(H)\otimes
C_{0}[0,1],\psi_{t}(D_{H})\right)\right]\in
KK_{n}(\mathbb{C},C^{*}(H_{ad}^{[0,1)})).$

Moreover, as in (3.3), we have that
$\varrho(D_{M\times_{\pi}M})\otimes\iota_{!}^{ad}\in
KK_{n+k}(\mathbb{C},C^{*}(G_{ad}^{[0,1)}))$ is represented by

$\left[\left(\mathcal{E}(G_{ad}),\psi(D_{G_{ad}})\right),\left(\mathcal{E}(G)\otimes
C_{0}[0,1],CS_{t}(\psi(D_{G}),\mathrm{sgn}(D_{H})\otimes 1)\right)\right].$
(7.11)

Observe that the class of $\iota_{!}$ does not depend on $\varepsilon$ which
parametrizes the family of metrics $\varepsilon^{-1}g_{B}$ one can use to
construct it. Then the family of bimodules

$\left[\left(\mathcal{E}(G_{ad}),\psi(D^{\varepsilon}_{G_{ad}})\right)\left(\mathcal{E}(G)\otimes
C_{0}[0,1],CS_{t}(\psi(D^{\varepsilon}_{G}),\mathrm{sgn}(D_{H})\otimes
1)\right)\right]$ (7.12)

defines the same class in $KK_{n+k}(\mathbb{C},C^{*}(G_{ad}^{[0,1)}))$ for any
$\varepsilon$. Observe now that [5, Equation (4.46)], see also [16, Lemma 17]
for more details, tells us that there exists an $\varepsilon_{0}$ and
$\lambda>-2$ such that the commutator $[D_{G}^{\varepsilon_{0}},D_{H}\otimes
1]-\lambda\geq 0$ and that $D_{G}^{\varepsilon_{0}}$ is invertible. Thanks to
Lemma 3.4

$\left[\left(\mathcal{E}(G_{ad}),\psi(D^{\varepsilon_{0}}_{G_{ad}})\right),\left(\mathcal{E}(G)\otimes
C_{0}[0,1],CS_{t}(\psi(D^{\varepsilon_{0}}_{G}),\mathrm{sgn}(CS_{s}(D_{G}^{\varepsilon_{0}},D_{H}\otimes
1))\right)\right]$ (7.13)

gives a homotopy, parametrized by $s$, from (7.12) to
$\varrho(D^{\varepsilon_{0}}_{M\times M})$. ∎

Observe now that the Lie groupoid $M\times_{B}M\rightrightarrows M$ is the
holonomy groupoid $\mathrm{Hol}(\pi)$ associated with the foliation $(M,\ker
d\pi)$. If the fibers are not simply connected, it differs from the monodromy
groupoid $\mathrm{Mon}(\pi)$, whose $s$-fibers are diffeomorphic to the
universal covering of the typical fiber of $\pi$. This last groupoid admits an
immersion
$\tilde{\iota}\colon\mathrm{Mon}(\pi)\looparrowright\widetilde{M}\times_{\Gamma}\widetilde{M}$
of Lie groupoids over $M$, where $\Gamma=\pi_{1}(M)$. Notice that the proof of
the previous theorem is obtained from local considerations about differential
operators which adapt almost verbatim to the higher situation through
equivariant lift. Hence, we can directly state the following more general
result.

######  Theorem 7.3.

Let $\pi\colon(M,g^{M})\to(B,g^{B})$ be a Riemannian submersion between $Spin$
manifolds. Let $g^{M/B}$ be a metric of $\ker d\pi$ with positive scalar
curvature and let $\varrho(D_{\mathrm{Mon}(\pi)})\in
KK_{n}(\mathbb{C},C^{*}(\mathrm{Mon}(\pi)_{ad}^{[0,1)}))$ be the associated
$\varrho$-class. Let $\varepsilon_{0}$ be such that $g^{M}_{\varepsilon_{0}}$
has positive scalar curvature too. Then we have that

$\varrho(D_{\mathrm{Mon}(\pi)})\otimes\tilde{\iota}_{!}^{ad}=\varrho(D^{\varepsilon_{0}}_{\widetilde{M}\times_{\Gamma}\widetilde{M}})\in
KK_{n+k}(\mathbb{C},C^{*}(\widetilde{M}\times_{\Gamma}\widetilde{M}_{ad}^{[0,1)})).$
(7.14)

###  7.2 Foliated bundles

Let $\Gamma$ be a discrete group of isometries acting on a smooth Riemannian
manifold $(\bar{M},g^{\bar{M}})$ freely, properly discontinuously and so that
$(M,g^{M}):=(\bar{M},g^{\bar{M}})/\Gamma$ is a compact smooth Riemannian
manifold. Suppose that we also have an isometric action of $\Gamma$ on a
compact Riemannian manifold $(B,g^{B})$ and a surjective Riemannian submersion
$\pi\colon\bar{M}\to B$, which is $\Gamma$-equivariant. The simple foliation
of $\bar{M}$ associated with the involutive sub-bundle $\ker d\pi\subset
T\bar{M}$ is invariant under the action of $\Gamma$ and hence it induces the
quotient foliation $\mathcal{F}(\pi,\Gamma):=\ker d\pi/\Gamma$ on $M$. Now, as
in the previous section, $g^{\bar{M}}$ is equal to
$g^{\bar{M}/B}\oplus\pi^{*}g^{B}$, where $g^{\bar{M}/B}$ is a metric on $\ker
d\pi$ which induces a metric $g^{\mathcal{F}}$ on the quotient sub-bundle
$\mathcal{F}(\pi,\Gamma)\subset TM$.

If $\Gamma_{b}\subset\Gamma$ is the isotropy group at $b\in B$ of the action
of $\Gamma$ on $B$, then the leaf of $(M,\mathcal{F}(\pi,\Gamma))$ obtained
from the leaf $\pi^{-1}(b)$ of $(\bar{M},\ker d\pi)$ is naturally
diffeomorphic to $\pi^{-1}(b)/\Gamma_{b}$.

Observe that $\Gamma$ induces a free and properly discontinuous action via
groupoid automorphisms on the Lie groupoids associated with the foliation on
$\bar{M}$ and hence we have the following identifications of Lie groupoids
over $M$

$\mathrm{Mon}(M,\mathcal{F}(\pi,\Gamma))\cong\mathrm{Mon}(\bar{M},\ker
d\pi)/\Gamma\,\,\mbox{and}\,\,\mathrm{Hol}(M,\mathcal{F}(\pi,\Gamma))\cong\mathrm{Hol}(\bar{M},\ker
d\pi)/\Gamma$

for the monodromy and the holonomy groupoid, respectively. See [20, Example
5.8], for instance. Finally, it is worthy to point out that the Lie algebroid
of both these groupoids is given by $\ker d\pi/\Gamma\to M$.

Observe now that there is a commutative diagram of Lie groupoid morphisms

$\textstyle{\mathrm{Mon}(\bar{M},\ker
d\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\iota}}$$\textstyle{\mathrm{Mon}(\bar{M},T\bar{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mon}(M,\mathcal{F}(\pi,\Gamma))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\mathrm{Mon}(M,TM)}$
(7.15)

where the horizontal arrows are immersions and the vertical ones are quotient
maps.

######  Theorem 7.4.

In the previous geometric situation, let the metric $g^{\mathcal{F}}$ on $\ker
d\pi/\Gamma$ be a metric with positive scalar curvature. Let us assume that
both $M$ and $B$ are endowed with a $\Gamma$-invariant $Spin$ structure. Then
there exists an $\varepsilon>0$ such that

$\varrho(D_{g^{\mathcal{F}}})\otimes(\iota_{ad}^{[0,1)})_{!}=\varrho(D_{g_{\varepsilon}^{M}})\in
KK(C,C^{*}(\mathrm{Mon}(M,TM)_{ad}^{[0,1)}))$ (7.16)

where $g_{\varepsilon}^{M}$ is induced by
$g_{\varepsilon}^{\bar{M}}=g^{\bar{M}/B}\oplus\varepsilon^{-1}g^{B}$.

###### Proof.

First, let us fix the following notation $G:=\mathrm{Mon}(\bar{M},\ker d\pi)$,
$G/\Gamma:=\mathrm{Mon}(M,\mathcal{F}(\pi,\Gamma))$,
$H:=\mathrm{Mon}(\bar{M},T\bar{M})$ and $H/\Gamma:=\mathrm{Mon}(M,TM)$.
Observe that, thanks to Proposition 6.9 and Proposition 6.10 we have that

$\begin{split}&\varrho(D_{G/\Gamma})\otimes(\iota_{ad}^{[0,1)})_{!}=\\\
=&\varrho(D_{G/\Gamma})\otimes(\mathcal{M}_{G}^{\Gamma})^{-1}\otimes\mathcal{M}_{G}^{\Gamma}\otimes(\iota_{ad}^{[0,1)})_{!}\otimes(\mathcal{M}_{H}^{\Gamma})^{-1}\otimes\mathcal{M}_{H}^{\Gamma}=\\\
=&\varrho(D_{G}^{\Gamma})\otimes
j^{\Gamma}((\bar{\iota}_{ad}^{[0,1)})_{!})\otimes\mathcal{M}_{H}^{\Gamma}=\\\
=&\varrho(D_{H}^{\Gamma})\otimes\mathcal{M}_{H}^{\Gamma}=\varrho(D_{H}),\end{split}$
(7.17)

where the only thing to prove is the equality

$\varrho(D_{G}^{\Gamma})\otimes
j^{\Gamma}((\bar{\iota}_{ad}^{[0,1)})_{!})=\varrho(D_{H}^{\Gamma})\in
KK(\mathbb{C},C^{*}(H_{ad}^{[0,1})\rtimes\Gamma).$ (7.18)

Thanks to (6.8) and (6.9), by using the functoriality of the descent map
$j^{\Gamma}$, it is easy to see that $\varrho(D_{G}^{\Gamma})\otimes
j^{\Gamma}((\bar{\iota}_{ad}^{[0,1)})_{!})$ is represented, as in (3.2), by
the pair of bimodules

$\left[\left(pt^{*}\mathcal{E}(H)_{ad}^{\rtimes},\psi(D^{\Gamma}_{H_{ad}})\right),\left(pt^{*}\mathcal{E}^{\rtimes}(H)\otimes
C[0,1],CS_{t}(\psi(D^{\Gamma}_{H}),\mathrm{sgn}(D^{\Gamma}_{G})\otimes
1\right)\right]$ (7.19)

From now the proof follows as from (7.11) on and we get the desired result. ∎

######  Remark 7.5.

Let us consider the more general situation where we have a sequence of
$\Gamma$-equivariant surjective Riemannian submersions

$\textstyle{\bar{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{B^{\prime}}$
(7.20)

and let denote by $\pi^{\prime}$ the composition $q\circ\pi$. Then we have an
inclusion of foliations $\ker d\pi/\Gamma\hookrightarrow\ker
d\pi^{\prime}/\Gamma$ over $M$ and then an immersion of Lie groupoids

$\iota\colon\mathrm{Mon}(M,\mathcal{F}(\pi,\Gamma))\looparrowright\mathrm{Mon}(M,\mathcal{F}(\pi^{\prime},\Gamma)).$

Observe that if $B^{\prime}=pt$ we recover the situation of Theorem 7.4. Let
$g^{\pi}$ a longitudinal metric of positive scalar curvature on $\ker
d\pi/\Gamma$ and let $g^{\pi^{\prime}}:=g^{\pi}\oplus g^{B^{\prime}}$ be a
metric on $\ker d\pi^{\prime}/\Gamma$ with positive scalar curvature too, then
the proof of the following product formula

$\varrho(D_{g^{\pi^{\prime}}})=\varrho(D_{g^{\pi}})\otimes(\iota_{ad}^{[0,1)})_{!}\in
KK_{*}\left(\mathbb{C},C^{*}(\mathrm{Mon}(M,\mathcal{F}(\pi^{\prime},\Gamma))_{ad}^{[0,1)})\right)$
(7.21)

follows without substantial changes as the proof of Theorem 7.4. The only
thing to observe is that the estimation of the commutator between the operator
of the smaller foliation and the bigger one works exactly as in [5, Equation
(4.46)].

###  7.3 Functoriality of $\varrho$-classes through étale surjections

In this section we give a very conceptual proof of a generalization to the
setting of foliations of [10, Theorem 1.1]. Let $M$ be a smooth compact
manifold and let $\mathcal{F}$ be an involutive sub-bundle of $TM$, namely the
tangent bundle of a regular foliation on $M$. Consider two $s$-connected Lie
groupoids $G$ and $H$ over $M$ integrating the Lie algebroid $\mathcal{F}$ and
assume that there exists a Lie groupoid surjective homomorphism $\varphi\colon
G\to H$ which integrates the identity map on $\mathcal{F}$. Notice that
$\varphi$ is an étale immersion of Lie groupoids, then the normal bundle
groupoid of $\varphi$ is $G$ it-self. So we have the following equality of
asymptotic morphism classes

$(\varphi_{ad}^{[0,1)})_{!}=[\mathrm{ev}_{0}]^{-1}\otimes[\mathrm{ev}_{1}]\in
E\left(C^{*}(G_{ad}^{[0,1)}),C^{*}(H_{ad}^{[0,1)})\right),$ (7.22)

where $\mathrm{ev}_{0}\colon DNC^{[0,1)}(G_{ad}^{[0,1)},H_{ad}^{[0,1)})\to
G_{ad}^{[0,1)}$ and $\mathrm{ev}_{1}\colon
DNC^{[0,1)}(G_{ad}^{[0,1)},H_{ad}^{[0,1)})\to H_{ad}^{[0,1)}$. Assume that
$\mathcal{F}$ is $Spin$ and that it is endowed with a metric $g^{\mathcal{F}}$
with positive scalar curvature. Then we have the following functoriality
result.

######  Theorem 7.6.

Let $D_{G}$ and $D_{H}$ the $Spin$ Dirac operators associated with the metric
$g^{\mathcal{F}}$ on $G$ and $H$ respectively, then the following equality
holds

$\varrho(D_{G})\otimes(\varphi_{ad}^{[0,1)})_{!}=\varrho(D_{H}).$ (7.23)

###### Proof.

The result follows immediately by noticing that the invertible Dirac operator
on the deformation groupoid $DNC(G_{ad}^{[0,1)},H_{ad}^{[0,1)})$, associated
with the metric given by the pull-back of $g^{\mathcal{F}}$, restricts to
$D_{G}$ at $0$ and to $D_{H}$ at $1$. ∎

######  Remark 7.7.

When $\mathcal{F}=TM$ the Lie groupoids $G$ and $H$ are of the form
$\bar{M}\times_{\Lambda}\bar{M}\rightrightarrows M$, where $\bar{M}$ is a
Galois $\Lambda$-covering of $M$ and we recover [10, Theorem 1.1] when $M$ is
compact.

##  8 Open problems

In this section we are going to list a series of open questions which arise
from this paper.

  1. 1.

The first open question comes from the fact that the main result of this
paper, namely Theorem 7.4, is proved for Riemannian foliated bundles. So it is
natural to ask for a proof in the context of general foliations which non
necessarily admit an isometric normal structure. Notice that in Section 5.2 we
just give the construction of the adiabatic transverse class for foliations
endowed with an almost isometric structure, but the general case could be
treated as usual by passing to the Connes’ fibration, as it is done in [13].

  2. 2.

Observe now that Definition 4.6 involves a perturbation which did not appear
in the applications of the present work. Indeed, an open problem which will be
treated in a future work is to provide a proof of the product formula (7.16)
for $\varrho$-classes associated with perturbed generalized Dirac operators,
such as those given by the signature operator on homotopy equivalent
foliations, see [34, Section 3.4].

  3. 3.

The natural application of these product formulas are given by stability
results as in [34, Section 3.6.1]: namely asking if two longitudinal metric
with positive scalar curvature which are not longitudinally concordant stay so
if completed to metric with positive scalar curvature on the whole manifold.
Similar questions could be asked about foliated homotopy equivalences. In
order to prove this kind of results, it is necessary to prove the injectivity
of the Kasparov product in (7.16), namely finding suitable hypotheses for the
existence of a right inverse (in the notation of Kasparov product) for the
lower shriek class between adiabatic deformation groupoids.

  4. 4.

Finally, it would be extremely interesting to construct concrete geometrical
examples corresponding to non trivial factorizations of $\varrho$-classes as
in (7.16).

Acknowledgments I would like to thank Paolo Piazza and Georges Skandalis for
interesting discussions about this subject.

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