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arXiv:1207.3514v2 [math.OA] 14 Sep 2012
A cohomological formula for the Atiyah-Patodi-Singer index
on manifolds with boundary
P. Carrillo Rouse, J.M. Lescure and B. Monthubert
September 17, 2012
Abstract
We compute a cohomological formula for the index of a fully elliptic pseudodifferential
operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an
embedding into an euclidean space to express the index as the integral of a cohomology
class depending in this case on a noncommutative symbol, the integral being over a C∞-
manifold called the singular normal bundle associated to the embedding. The formula is
based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that
is drawn from Connes’ tangent groupoid approach.
Contents
1 Introduction 1
2 Groupoids 9
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Free proper groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Crossed product groupoids by homomorphisms on RNand Connes-Thom
isomorphism.................................... 11
3 Noncommutative spaces for manifolds with boundary 13
3.1 The noncommutative tangent space of a manifold with boundary . . . . . . 13
3.2 Embeddings of manifolds with boundary and proper free groupoids . . . . . 15
3.2.1 Singular normal bundle . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 The APS classifying space . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Atiyah-Patodi-Singer theorem in K-theory 25
4.1 The Fredholm index morphism . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Atiyah-Patodi-Singer index theorem in K-theory . . . . . . . . . . . . . . . 29
5 The cohomological APS formula 30
5.1 A more explicit description of (5.3) . . . . . . . . . . . . . . . . . . . . . . . 31
A Deformation to the normal cone functor and tangent groupoids 34
1 Introduction
In the early 60’s, Atiyah and Singer gave a positive answer to a problem posed by Gelfand
about investigating the relationship between topological and analytical invariants of elliptic
2010 Mathematics Subject Classification. Primary 19K56, 58B34. Secondary 58H05, 58J28.
1
(pseudo)differential operators on closed smooth manifolds without boundary, [4, 5]. In a
series of papers, Atiyah-Singer not only gave a general cohomological formula for the
index of an elliptic (pseudo)differential operator on a closed smooth manifold, they also
gave several applications and more important they opened a entire new way of study index
problems. Since then, index theory has been at the core of interest of several domains in
mathematics and mathematical physics.
To be more descriptive, let Mbe a closed smooth manifold, let Dbe a elliptic
(pseudo)differential operator with principal symbol σD. The Atiyah-Singer index formula
states
ind D =ZT∗M
ch([σD])T(M) (1.1)
where [σD]∈K0(T∗M) is the principal symbol class in K-theory, ch([σD]) ∈Hev
dR(T∗M)
its Chern character and T(M)∈Hev
dR(M) the Todd class of (the complexified) T∗M.
A fundamental step in order to achieve such a formula was to realize that the map
D7→ ind D is completely encoded by a group morphism K0(T∗M)−→ Z,called the
analytic index of M. That is, if Ell(M) denotes the set of elliptic pseudodifferential
operators over M, then the following diagram is commutative:
Ell(M)ind //
σ
Z
K0(T∗M)
inda,M
;;
✈
✈
✈
✈
✈
✈
✈
✈
✈
✈
,
(1.2)
where Ell(M)σ
−→ K0(T∗M) is the surjective map that associates the class of the prin-
cipal symbol of the operator in K0(T∗M). The use of K-theory was a breakthrough in
the approach by Atiyah-Singer, indeed, they could use its (generalized) cohomological
properties to decompose the analytic index morphism in a composition of topologic (and
hence computable) morphisms. The idea is as follows. Consider an embedding M ֒→RN
(assume Neven for the purpose of this exposition) and the corresponding normal bundle
N(M), Atiyah-Singer showed that the analytic index decomposes as the composition of
•The Thom isomorphism
K0(T∗M)T
−→ K0(N(M))
followed by
•the canonical morphism
K0(N(M)) j!
−→ K0(RN)
induced from a identification of the normal bundle as an open subset of RN, and
followed by
•the Bott isomorphism
K0(RN)B
−→ K0({pt})≈Z.
In particular, modulo the Thom and Bott isomorphisms, the analytic index is trans-
formed in a very simple shriek map: K0(N(M)) j!
−→ K0(RN).The formula (1.1) is then
obtained as an algebraic topology exercise of comparison between K-theory and cohomol-
ogy, [5].
The composition (with obvious notations) T∗Mπ
→Mi0
→N(M) is K-oriented.
2
For the purposes of the present paper remark that the formula (1.1) can be also written
as follows:
ind D =ZN(M)
ch(T([σD])).(1.3)
Getting to the subject of the present article we need to start talking about groupoids.
In his book, [10] section II.5, Connes sketches a (conceptually) simple proof of the K-
theoretical Atiyah-Singer Index theorem for closed smooth manifolds using tangent groupoid
techniques. This proof is in fact a groupoid translation of Atiyah-Singer’s original proof.
Let Mbe a closed smooth manifold, GM=M×Mits pair groupoid. For the readers
not familiar with groupoids, one can think of the kernel algebra (convolution algebra of
the groupoid), that is, for the pair groupoid: on the algebra of smooth complex valued
functions on M×Mwith kernel convolution product.
Consider the tangent groupoid
TGM:= T M × {0}GM×M×(0,1] ⇒M×[0,1].
It is nowadays well known that the index morphism provided by this deformation groupoid
is precisely the analytic index of Atiyah-Singer, [10, 29]. In other words, there is a short
exact sequence of C∗-algebras
0//C∗(M×M×(0,1]) //C∗(TGM)e0//C0(T∗M)//0 (1.4)
and since C∗(M×M×(0,1]) is contractible, the morphism induced in K-theory by e0is
invertible. The analytic index of Mis the morphism
K0(T M )(e0)−1
∗//K0(TGM)(e1)∗//K0(M×M) := K0(K(L2(M))) ≈Z,(1.5)
where etare the obvious evaluation morphisms at t, and where we denote for a general
groupoid G:K0(G) := K0(C∗
r(G)).
Actually, all groupoids considered in this work are (at least) continuous family groupoids,
so there is a notion of reduced and envelopping C∗-algebra (using half-densities or by pick-
ing up a continuous Haar system) and amenable as well, thus the distinction between the
reduced and envelopping C∗-algebras is not even necessary.
As pointed out by Connes, if the groupoids appearing in this interpretation of the index
were equivalent to spaces then we would immediately have a geometric interpretation of the
index. Now, M×Mis equivalent to a point (hence to a space), but the other fundamental
groupoid playing a role in the previous discussion is not, that is, T M is a groupoid whose
fibers are the groups TxM, which are not equivalent (as groupoids) to spaces. The idea
of Connes is to use an appropriate action of the tangent groupoid in some RNin order
to translate the index (via a Thom isomorphism) in an index associated to a deformation
groupoid which will be equivalent to some space.
The case of manifolds with boundary
In a series of papers [1, 2, 3], Atiyah, Patodi and Singer investigated the case of non-
local elliptic boundary value problems. They showed that under some boundary conditions
(the now so-called APS boundary conditions), Dirac operators (among a larger class of
first order differential operators) become Fredholm on suitable spaces and they computed
the index. To the characteristic form from the closed smooth case they added a correction
term, called the eta invariant, which is determined by an appropriate restriction of the
operator to the boundary : this is a spectral invariant, measuring the asymmetry of the
spectrum. However, a cohomological formula expressing the APS index as an integration
of some characteristic form on the boundary is impossible. Roughly speaking, the non
locality of the chosen boundary condition prevents us to express this spectral correction
3
term with a local object, like a density on the boundary. In other words, applying directly
the methods of Atiyah-Singer can not yield the result in the APS index problem. However,
we will see that the use of some noncommutative spaces, associated with groupoids, will
give us the possibility to understand the computation of the APS index using classic
algebraic topology methods (K-theory and cohomology).
In this paper, we will follow Connes’ groupoid approach to obtain a cohomological
formula for the index of a fully elliptic (pseudo)differential operator on a closed manifold
with boundary. For the case of such a manifold, the pair groupoid does not give the same
information as the smooth case. In fact this case of manifolds with boundary becomes
more interesting since different boundary conditions can be considered and each of these
give different index problems. In this paper we will be interested in the so-called Atiyah-
Patodi-Singer boundary condition. For the moment we will not recall what this condition
is, in fact we rather describe the groupoid whose pseudodifferential calculus gives rise to
the index theory related to such a condition.
Let Xbe a manifold with boundary. We denote, as usual, ◦
Xthe interior which is a
smooth manifold and ∂X its boundary. Let
Γ(X)⇒X(1.6)
be the groupoid of the b-calculus, where
Γ(X) = ◦
X×◦
XG∂X ×∂X ×R,
with groupoid structure given as a family of pair groupoids and the (additive) group R. It
is a continuous family groupoid with the topology explicity described in [27] (see beginning
of section 3.1 below).
Consider
Γ(X)tan =A(Γ(X)) GΓ(X)×(0,1] ⇒X×[0,1]
its tangent groupoid.
Take now the open subgroupoid of Γ(X)tan obtained by restriction to XF:= X×
[0,1] \(∂X × {1})
Γ(X)F=A(Γ(X)) × {0}G◦
X×◦
X×(0,1] G∂X ×∂X ×R×(0,1) ⇒XF.
By definition ◦
X×◦
X×(0,1] is a saturated, open subgroupoid of Γ(X)F. This leads to
a complementary closed subgroupoid of Γ(X)F:
TncX⇒X∂,(1.7)
where X∂:= XF\◦
X×(0,1] = X∪
∂X ×{0}∂X ×[0,1].
The groupoid TncX, called ”The noncommutative tangent space of X”, was introduced
independently in [28] and in [12] where it was proven to be the K-Poincar´e dual of the
conic pseudomanifold associated to X.
Deformation groupoids like Γ(X)Finduce index morphisms. Indeed, its algebra comes
equipped with a restriction morphism to the algebra of TncXand an evaluation morphism
to the algebra of ◦
X×◦
X(for t= 1). Indeed, we have a short exact sequence of C∗-algebras
0//C∗(◦
X×◦
X×(0,1]) //C∗(Γ(X)F)e0//C∗(TncX)//0 (1.8)
where the algebra C∗(◦
X×◦
X×(0,1]) is contractible. Hence applying the K-theory functor
to this sequence we obtain an index morphism
indF= (e1)∗◦(e0)−1
∗:K0(TncX)−→ K0(◦
X×◦
X)≈Z.(1.9)
4
This index computes indeed the Fredholm index of those elliptic operators on X satisfying
the APS boundary condition, and hence we call it The Fredholm index morphism of X.
To be more explicit, the statement is the following:
Proposition 1.1 [19, 21, 11]. For any fully elliptic operator Don X, there is a naturally
associated ”non commutative symbol” [σD]∈K0(TncX)and
indF([σD]) = IndexAP S(D),(1.10)
where IndexAP S (D)is the Fredholm index of D. Moreover, every element in K0(Tnc X)
can be realized in this way.
A first task in order to follow the Atiyah-Singer approach would be to compute the
morphism indFby topological means. For instance, using an appropriate embedding into
a space in which the computation could follow in an easier way. This idea has been
already followed up in [14] in the framework of manifolds with conical singularities, using
aKK-equivalent version of the noncommutative tangent space TncX. There, the authors
use embeddings into euclidean spaces to extend the construction of the Atiyah-Singer
topological index map, thanks to a “Thom isomorphism“ between the noncommutative
tangent spaces of the singular manifold and of its singular normal bundle, and then get
an index theorem in the framework of K-theory. Here, we follow a different approach and
we are going to extend the Atiyah-Singer topological index map using Connes’ ideas on
tangent groupoid actions on euclidean spaces; moreover we investigate the cohomological
counterpart of the K-theoretic statement of the index theorem. Note also that the index
map considered here coincide, through K K-equivalences, with the index maps considered
in [19] and in [14].
We start in section 3.2 by considering an embedding
i:X ֒→RN−1×R+(1.11)
of Xas a neat submanifold of RN−1×R+(i.e., respecting the boundary). Following
Connes, we use it to define a continuous family groupoid morphism (see 3.4 for the explicit
definition)
h: Γ(X)→RN(1.12)
where we see RNas an additive group and we assume Neven. This morphism induces
an action of Γ(X) on X×RNand an induced deformation action of Γ(X)Fon XF×RN
(coming from an induced morphism hF). The main task in section 3.2 is to prove the
following result (proposition 3.6 below):
Proposition 1.2 The crossed product groupoid (Γ(X)F)hF:= Γ(X)F⋊hFRNis a free
proper groupoid.
In section 2.3 we explain how the Connes-Thom isomorphism links the K-theory of a
groupoid with the K-theory of a crossed product as above. For instance, the new groupoid
(Γ(X)F)hFdefines as well an index morphism and this one is linked with the index (1.9) by
a natural isomorphism, the so called Connes-Thom isomorphism, thus giving the following
5
commutative diagram
K0(TncX)
indF
''
C T ≈
K0(Γ(X)F)
C T ≈
e0
≈
ooe1//K0(◦
X×◦
X)≈Z
C T ≈
K0((TncX)h0)
indhF
99
K0((Γ(X)F)hF)
e0
≈
ooe1//K0(( ◦
X×◦
X)h1)
(1.13)
where h0and h1denote the respective restricted actions of TncXand ◦
X×◦
Xon RN.
Now, the proposition above tells us that the orbit space of (Γ(X)F)hFis a nice space
and moreover that this crossed product groupoid is Morita equivalent to its orbit space.
This means that the index morphism indhFcan be computed, modulo Morita equivalences,
as the deformation index of some space. More precisely, denoting by BhFthe orbit space of
(Γ(X)F)hF, by Bh0the orbit space of (TncX)h0and by Bh1the orbit space of (Γ(X)F)hF
we have an index morphism between K-theory of spaces (topological K-groups, no more
C∗-algebras if one does not like it!)
indBhF:K0(Bh0)K0(BhF)
e0
≈
ooe1//K0(Bh1) (1.14)
from which we would be able to compute the Fredholm index. This is what we achieve
next, indeed, in section 3.2.2 where we are able to explicitly identify these orbit spaces.
In order to describe them we need to introduce a new player, but let us first motivate
this by looking at the situation when ∂X =∅(following [10] II.5). In this case, the
orbit space of (Γ(X)F)hFcan be identified with the deformation to the normal cone (see
appendix A below for the C∞-structure of such deformations) of the embedding X ֒→RN,
that is a C∞-cobordism between the normal bundle to Xin RNand RNitself:
BAS := N(X)G(0,1] ×RN.(1.15)
In this picture we also see the orbit space of (T X )h0which identifies with N(X) and
the orbit space of (X×X)h1which identifies with RN.
Still, in this boundaryless case (∂X =∅), this space BAS gives in K-theory a defor-
mation index
indBAS :K0(N(X)) K0(BAS)
e0
≈
ooe1//K0(RN)
which is easily seen to be the shriek map associated to the identification of N(X) as an
open subset of RN.
In the boundary case, the normal bundle is not the right player, we know for instance
that the APS index cannot be computed by an integration over this space due to the
non-locality of the APS boundary condition. One has then to compute the orbit spaces,
in fact the orbit space of (TncX)h0identifies (lemma 3.9) with the singular normal bundle:
Nsing(X) := N(X)× {0}GRN−1×(0,1) (1.16)
is the C∞-manifold obtained by gluing N(X) and
D∂:= N(∂X)× {0}GRN−1×(0,1)
6
the deformation to the normal cone associated to the embedding ∂X ֒→RN−1, along
their common boundary (the gluing depending on a choice of a defining function of the
boundary of X). The orbit space of ( ◦
X×◦
X)h1is easily identified with RN(lemma 3.8).
Finally the orbit space of (Γ(X)F)hFis homeomorphic to a space (section 3.2.2) looking
as
BF:= Nsing(X)G(0,1] ×RN,(1.17)
where more precisely we prove the following (proposition 3.11)
Proposition 1.3 The locally compact space BFadmits an oriented C∞-manifold with
boundary structure of dimension N+ 1.
The last proposition is essential to explicitly compute the index (1.14) above once the
explicit identifications are performed, indeed BFis an oriented cobordism from Nsing(X)
to RN, we can hence apply a Stoke’s theorem argument to obtain the following (proposition
3.12):
Proposition 1.4 The index morphism of the deformation space BFcan be computed by
means of the following commutative diagram:
K0(Nsing(X))
indBF
''
RNsing(X)ch(·)
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
K0(BF)
(e0)∗
≈
oo(e1)∗//K0(RN)
RRNch(·)
xxq
q
q
q
q
q
q
q
q
q
q
q
R
Before enouncing the index theorem, we mentioned that there is a Connes-Thom iso-
morphism and a Morita equivalence
K0(TncX)C T
−→ K0((Tnc X)h0)Morita
−→ K0(Nsing (X)).
In section 2.3 we develop Connes-Thom using deformation groupoids, this allows us to
perform an explicit computation of the above morphism for ”noncommutative symbols”
[σD]∈K0(TncX).
The index theorem is the following:
Theorem 1.5 [K-theoretic APS] Let Xbe a manifold with boundary, consider an em-
bedding of Xin RNas in 1.11. The Fredholm index morphism indF:K0(TncX)→Z
decomposes as the composition of the following three morphisms
1. A Connes-Thom isomorphism C T :
K0(TncX)C T
−→ K0(Nsing (X)),
2. The index morphism of the deformation space BF:
K0(Nsing(X)) indBF//K0(RN)
3. the usual Bott periodicity isomorphism:
K0(RN)Bott
−→ Z.
7
In other terms, the following diagram is commutative
K0(TncX)
C T ≈
indF//Z
K0(Nsing(X)) indBF
//K0(RN)
Bott
≈
OO
As discussed above, the three morphisms of the last theorem are computable, and then,
exactly as in the classic Atiyah-Singer theorem the last theorem allows to conclude that,
given an embedding i:X ֒→RNas above, any fully elliptic operator Don Xwith ”non
commutative symbol” [σD]∈K0(Tnc X) gives rise to the following formula:
Cohomological formula for the APS index (corollary 5.1)
IndexAP S (D) = ZNsing(X)
Ch(C T ([σD])) (1.18)
where RNsing(X)is the integration with respect to the fundamental class of Nsing(X). In
section 5.3 we perform an explicit description for C T ([σD]) ∈K0(Nsing(X)).
The manifold Nsing (X) (see (1.16) above) already reflects an interior contribution and
a boundary contribution. In particular, picking up a differential form ωDon Nsing(X)
representing Ch(C T ([σD]), we obtain:
IndexAP S (D) = ZN(X)
ωD+ZD∂
ωD.(1.19)
The first integral above involves the restriction of ωDto N(X), which is related to the
ordinary principal symbol of D. More precisely, the principal symbol σpr(D) of Dprovides
aK-theory class of C∗(A∗(Γ(X))), that is a compactly supported K-theory class of the
dual of the Lie algebroid of Γ(X) or in other words of the b-cotangent bundle bT∗X, and
by functoriality of both the Chern character and Thom-Connes maps, we have
[(ωD)|N(X)] = Ch(C T ([σpr (D)]).
The second integral can thus be viewed as a correction term, which contains the eta invari-
ant appearing in APS formula and which also depends on the choice of the representative
ωD∈Ch(C T ([σD])).
Further developments: The same methods as above can be applied for manifolds
with corners for which we already count with the appropriate b-groupoids ([27]). The
generalization of the formula is not however immediate. Indeed, we need to explicitly
compute the orbit spaces, which will not be anymore smooth manifolds.
In another direction, as we mentioned above, there is a relation between the second
integral on (1.19) and the so called eta invariant. For deeply understanding this, we need
to explicitly describe the Chern character of the C T ([σD]∈K0(Nsing(X)), for which one
might need to use the Chern character computations done in [7] for instance. Also, the
second integral comes from the part of the b-groupoid corresponding to the boundary,
∂X ×∂X ×R×(0,1),
and this groupoid’s algebra is related with the suspended algebra of Melrose ([25]), a
relation between this integral and the Melrose and Melrose-Nistor ([23]), becomes then
very interesting to study.
8
2 Groupoids
2.1 Preliminaries
Let us recall some preliminaries on groupoids:
Definition 2.1 A groupoid consists of the following data: two sets Gand G(0), and maps
(1) s, r :G→G(0) called the source and range (or target) map respectively,
(2) m:G(2) →Gcalled the product map (where G(2) ={(γ, η)∈G×G:s(γ) = r(η)}),
such that there exist two maps, u:G(0) →G(the unit map) and i:G→G(the inverse
map), which, if we denote m(γ, η) = γ·η,u(x) = xand i(γ) = γ−1, satisfy the following
properties:
(i). r(γ·η) = r(γ)and s(γ·η) = s(η).
(ii). γ·(η·δ) = (γ·η)·δ,∀γ, η, δ ∈Gwhen this is possible.
(iii). γ·x=γand x·η=η,∀γ, η ∈Gwith s(γ) = xand r(η) = x.
(iv). γ·γ−1=u(r(γ)) and γ−1·γ=u(s(γ)),∀γ∈G.
Generally, we denote a groupoid by G⇒G(0). A morphism ffrom a groupoid H⇒H(0)
to a groupoid G⇒G(0) is given by a map ffrom Gto Hwhich preserves the groupoid
structure, i.e. fcommutes with the source, target, unit, inverse maps, and respects the
groupoid product in the sense that f(h1·h2) = f(h1)·f(h2)for any (h1, h2)∈H(2).
For A, B subsets of G(0) we use the notation GB
Afor the subset
{γ∈G:s(γ)∈A, r(γ)∈B}.
A groupoid can be endowed with a structure of topological space, or manifold, for
instance. In the case when Gand G(0) are smooth manifolds, and s, r, m, u are smooth
maps (with sand rsubimmersions), then Gis a Lie groupoid. In the case of manifolds
with boundary, or with corners, this notion can be generalized to that of continuous family
groupoids (see [32]).
One can endow a groupoid with a topological structure. A strict morphism of lo-
cally compact groupoids is a groupoid morphism which is continuous. Locally compact
groupoids form a category with strict morphisms of groupoids. It is now classical in
groupoids theory that the right category to consider is the one in which Morita equiva-
lences correspond precisely to isomorphisms. For more details about the assertions about
generalized morphisms written in this section, the reader can read [36] section 2.1, or
[17, 30, 26].
We need to introduce some basic definitions, classical when dealing with principal
bundles for groups over spaces.
We recall first the notion of groupoid action. Given a l.c. groupoid G⇒G(0), a right
G-bundle over a manifold Mis a manifold Psuch that:
•Pis endowed with maps πand qas in
P
π
q
!!
❉
❉
❉
❉
❉
❉
❉
❉G
r
s
MG(0)
9
•Pis endowed with a continuous right action µ:P×(q,r)G→P, such that if we denote
µ(p, γ) = pγ, one has π(pγ) = π(p) and p(γ1·γ2) = (pγ1)γ2for any (γ1, γ2)∈G(2).
Here P×(q,r)Gdenotes the fiber product of q:P→G(0) and r:G→G(0).
AG-bundle Pis called principal if
(i) πis a surjective submersion, and
(ii) the map P×(q,r)G→P×MP, (p, γ)7→ (p, pγ ) is a homeomorphism.
We can now define generalized morphisms between two Lie groupoids.
Definition 2.2 (Generalized morphism) Let G⇒G(0) and H⇒H(0) be two Lie
groupoids. A generalized morphism (or Hilsum-Skandalis morphism) from Gto H,f:
H//❴❴❴ G, is given by the isomorphism class of a right G−principal bundle over H,
that is, the isomorphism class of:
•A right principal G-bundle Pfover H(0) which is also a left H-bundle such that
the two actions commute, formally denoted by
H
Pf
}}
}}③
③
③
③
③
③
③
③
!!
❈
❈
❈
❈
❈
❈
❈
❈
G
H(0) G(0),
First of all, usual morphisms between groupoids are also generalized morphisms. Next,
as the word suggests it, generalized morphisms can be composed. Indeed, if Pand P′are
generalized morphisms from Hto Gand from Gto Lrespectively, then
P×GP′:= P×G(0) P′/(p, p′)∼(p·γ, γ −1·p′)
is a generalized morphism from Hto L. The composition is associative and thus we
can consider the category GrpdHS with objects l.c. groupoids and morphisms given by
generalized morphisms. There is a functor
Grpd −→ GrpdHS (2.1)
where Grpd is the category of groupoids with usual morphisms.
Definition 2.3 (Morita equivalence) Two groupoids are Morita equivalent if they are
isomorphic in GrpdHS.
2.2 Free proper groupoids
Definition 2.4 (Free proper groupoid) Let H⇒H(0) be a locally compact groupoid.
We will say that it is free and proper if it has trivial isotropy groups and it is proper.
Given a groupoid H⇒H(0), its orbit space is O(H) := H(0)/∼, where x∼yiff
there is γ∈Hsuch that s(γ) = xand r(γ) = y.
The following fact is well known. In particular in can be deduced from propositions
2.11, 2.12 and 2.29 in [35].
Proposition 2.5 If H⇒H(0) is a free proper (Lie) groupoid, then His Morita equiv-
alent to the locally compact space (manifold) O(H).
10
In fact the Morita bibundle which gives the Morita equivalence is the unit space H(0):
H
H(0)
Id
{{
{{①
①
①
①
①
①
①
①
①π
$$
❍
❍
❍
❍
❍
❍
❍
❍
❍
❍O(H)
H(0) O(H),
(2.2)
It is obvious that Hacts on its units freely and properly if His free and proper, and
for the same reason O(H) is a nice locally compact space (even a manifold if the groupoid
is Lie).
In particular there is an invertible Hilsum-Skandalis generalized morphism
O(H)− −− >H,(2.3)
that can also be given as a 1-cocycle from O(H) with values in H. This point of view
will be very useful for us in the sequel.
2.3 Crossed product groupoids by homomorphisms on RNand Connes-
Thom isomorphism
Let G⇒Mbe a locally compact groupoid.
We consider RNas an additive group, hence a one unit groupoid. Suppose we have an
homomorphism of groupoids
Gh
−→ RN.(2.4)
This gives rise to an action (say, a right one) of Gon the space M×RNand thus to
a new groupoid:
Gh:= (M×RN)⋊G:G×RN⇒M×RN(2.5)
which has the following structural maps:
•The source and target maps are given by
s(γ, X ) = (s(γ), X +h(γ)) and r(γ, X ) = (r(γ), X)
•The multiplication is defined on composable arrows by the formula
(γ, X )·(η, X +h(γ)) := (γ·η, X ).
Then it is obviously a groupoid with unit map u(m, X) = (m, X) (h(m) = 0 since his an
homomorphism), and inverse given by (γ, X )−1= (γ−1, X +h(γ)) (again since we have a
homomorphism, h(γ) + h(γ−1) = 0).
Remark 2.6 For the trivial homomorphism h0= 0, the associated groupoid is just the
product groupoid
G×RN⇒M×RN.
If Gis provided with a Haar system, then Ghinherits a natural Haar system such
that the C∗-algebra C∗(Gh) is isomorphic to the crossed product algebra C∗(G)⋊hRN
where RNacts on C∗(G) by automorphisms by the formula: αX(f)(γ) = ei·(X·h(γ)) f(γ),
∀f∈Cc(G), (see [10], propostion II.5.7 for details). In particular, in the case Nis even,
we have a Connes-Thom isomorphism in K-theory ([10], II.C)
K0(G)C T
≈//K0(Gh) (2.6)
11
which generalizes the classical Thom isomorphism, and which is natural with respect to
morphisms of algebras.
Since we will need to compute explicitly the morphism induced by the homomorphism
we propose an alternative construction of Connes-Thom which can be computed in our
context. More precisely we want to work directly with the groupoid algebras C∗(Gh)
without passing through the isomorphism with C∗(G)⋊hRN.
Given the morphism Gh
−→ RNwe consider the product groupoid G×[0,1] ⇒M×[0,1]
of the groupoid Gwith the space [0,1] and we define
H:G×[0,1] −→ RN,
the homomorphism given by
H(γ, t) := t·h(γ).
This homomorphism gives a deformation between the trivial homomorphism and h.
Consider the semi-direct groupoid associated to the homomorphism H. Since the
action is trivial at 0, it can be identified as
GH=G×RN× {0}GGh×(0,1].(2.7)
This decomposition in an open, saturated subgroupoid and a closed one gives rise to
a short exact sequence of C∗-algebras ([17, 34]):
0→C∗(Gh×(0,1]) →C∗(GH)e0
→C∗(G×RN)→0,(2.8)
where e0is induced by the evaluation at zero. This defines a deformation index
Dh:K∗(G×RN)→K∗(Gh).(2.9)
The natural map GH→[0,1] gives to GHthe structure of a continuous field of groupoids
over [0,1] and if Gis assumed to be amenable, we get by [18] that C∗(GH) is the space
of continuous sections of a continuous field of C∗-algebras. Then, the deformation index
above coincides with the morphism of theorem 3.1 in [15].
Definition 2.7 Let (G, h)be a groupoid together with a homomorphism on RN(with N
even). Consider the morphism in K-theory
K∗(G)C T h
−→ K∗(Gh),(2.10)
given by the composition of the Bott morphism
K∗(G)B
−→ K∗(G×RN),
and the deformation index
K∗(G×RN)Dh
−→ K∗(Gh).
We will refer to this morphism as the Connes-Thom map associated to h.
In fact, Elliot, Natsume and Nest proved that this morphism coincides with the usual
Connes-Thom isomorphism, theorem 5.1 in [15]. We can restate their result in our frame-
work as follows:
Proposition 2.8 (Elliot-Natsume-Nest) Let (G, h)be an amenable continuous family
groupoid (or amenable locally compact groupoid with a continuous Haar system) together
with a homomorphism on RN(Neven). Then the morphism C T h:K∗(G)→K∗(Gh)
coincides with the Connes-Thom isomorphism. In particular, it satisfies the following
properties:
1. Naturality.
2. If Gis a space (the groupoid equals its set of units), then C T his the Bott morphism.
12
3 Noncommutative spaces for manifolds with boundary
3.1 The noncommutative tangent space of a manifold with boundary
Let Xbe a manifold with boundary. We denote, as usual, ◦
Xthe interior which is a smooth
manifold and ∂X its boundary. Let
Γ(X)⇒X(3.1)
be the groupoid of the b-calculus ([29, 19, 27]). This groupoid has a pseudodifferential
calculus which coincides with Melrose’s b-calculus. There is a canonical definition, but
in our case we need to choose a defining function of the boundary, making the definition
simpler. A defining function of the boundary is a smooth function ρ:X→R+which is
zero on the boundary and only there, and whose differential is non zero on the boundary.
Definition 3.1 The b-groupoid of Xis
Γ(X) = {(x, y, α)∈X×X×R, ρ(x) = eαρ(y)}.
this implies that
Γ(X) = ◦
X×◦
XG∂X ×∂X ×R⇒X,
with groupoid structure compatible with those of ◦
X×◦
Xand ∂X ×∂X ×R(Ras an
additive group). It is a continuous family groupoid, see [27, 19] for details. For instance,
(xn, yn)→(x, y, α)
if and only if
xn→x, yn→yand log(ρ(xn)
ρ(yn))→α.
For such a groupoid it is possible to construct an algebra of pseudodifferential op-
erators. Although we do not need it in this article, we recall this background to help
the reader relate our work to usual index theory. See [29, 31, 27, 19, 37] for a detailed
presentation of pseudodifferential calculus on groupoids.
A pseudodifferential operator on a Lie groupoid (or more generally a continuous family
groupoid) Gis a family of peudodifferential operators on the fibers of G(which are smooth
manifolds without boundary), the family being equivariant under the natural action of G.
Compactly supported pseudodifferential operators form an algebra, denoted by Ψ∞(G).
The algebra of order 0 pseudodifferential operators can be completed into a C∗-algebra,
Ψ0(G). There exists a symbol map, σ, whose kernel is C∗(G). This gives rise to the
following exact sequence:
0→C∗(G)→Ψ0(G)→C0(S∗(G))
where S∗(G) is the cosphere bundle of the Lie algebroid of G.
In the general context of index theory on groupoids, there is an analytic index which
can be defined in two ways. The first way, classical, is to consider the boundary map of
the 6-terms exact sequence in K-theory induced by the short exact sequence above:
inda:K1(C0(S∗(G))) →K0(C∗(G)).
Actually, an alternative is to define it through the tangent groupoid of Connes, which
was originally defined for the groupoid of a smooth manifold and later extended to the
case of continuous family groupoids ([29, 19]). In general,
Gtan =A(G)GG×(0,1] ⇒G(0) ×[0,1]
13
Using the evaluation maps, one has two K-theory maps, e0:K∗(Gtan)→K∗(A(G))
which is an isomorphism (since K∗(G×(0,1]) = 0), and e1:K∗(Gtan)→K∗(G). The
analytic index can be defined as
inda=e1◦e−1
0:K∗(A(G)) →K∗(G).
It is thus possible to work on index problems without using the algebra of pseudodifferential
operators. In the rest of this article, we will only use deformation groupoids like the tangent
groupoid, and not pseudodifferential algebras.
But in general, this analytic index is not the Fredholm index. In certain cases, it is
possible to define the latter using a refinement of the tangent groupoid.
In order to use explicitly the tangent groupoid in our case, a little discussion on the
algebroid is needed. For this, remark that we can define the vector bundle T X over X
as the restriction of the tangent space of a smooth manifold ˜
X, for example we will use
the double of X(gluing along the boundary with the defining function ρ), on which Xis
included. So, T X := TX˜
Xand we can use the defining function to trivialize the normal
bundle of ∂X in ˜
Xto identify TX˜
Xwith the bundle with fibers
TX˜
Xx=Tx
◦
Xif x∈◦
Xand TX˜
Xx=Tx∂X ×Rif x∈∂X.
Now, Γ(X) is a continuous family groupoid and as such one can define T(Γ(X)), see
[29, 19, 33] for more details. What it is important for us is its restriction to X, where
X ֒→Γ(X) as the groupoid’s units. This vector bundle TX(Γ(X)) over Xhas fibers
TX(Γ(X))x=T(x,x)(◦
X×◦
X) if x∈◦
Xand
TXΓ(X)x=T(x,x)(∂X ×∂X)×R×Rif x∈∂X.
The algebroid A(Γ(X)) →Xis defined as the restriction to the unit space Xof the vector
bundle TΓ(X) = ∪x∈XTΓ(X)xover the groupoid Γ(X), in particular:
A(Γ(X))x≃Tx
◦
Xif x∈◦
Xand A(Γ(X))x≃Tx∂X ×Rif x∈∂X.
Take the tangent groupoid associated to Γ(X):
Γ(X)tan =A(Γ(X)) GΓ(X)×(0,1] ⇒X×[0,1]
Let us now consider the open subgroupoid of Γ(X)tan given by the restriction to
XF:= X×[0,1] \(∂X × {1}) :
Γ(X)F:= A(Γ(X)) G∂X ×∂X ×R×(0,1) G◦
X×◦
X×(0,1] ⇒XF.
As we will discuss below, the deformation index associated to this groupoid gives pre-
cisely the Fredholm index. But let us continue with our construction of noncommutative
spaces.
By definition ◦
X×◦
X×(0,1] ⊂Γ(X)F
as a saturated open subgroupoid. We can then obtain a complementary closed subgroupoid
of Γ(X)F:
TncX⇒X∂,(3.2)
where X∂:= XF\◦
X×(0,1].
To be more descriptive, the groupoid looks like
TncX=A(Γ(X)) G∂X ×∂X ×R×(0,1).
Definition 3.2 The groupoid TncXwill be called ”The noncommutative tangent space of
X”. We will also refer to Γ(X)Fas ”The Fredholm tangent groupoid of X”.
14
3.2 Embeddings of manifolds with boundary and proper free groupoids
To define a homomorphism Γ(X)Fh
−→ RNwe will need as in the nonboundary case an
appropiate embedding. We recall the construction outlined in [9]. Consider an embedding
i∂:X ֒→RN−1,
in which Nis an even integer. The embedding we are going to use is
i:X ֒→RN−1×R+(3.3)
where
i(x) = (i∂(x), ρ(x)).
We can then use it to define a homomorphism h: Γ(X)→RNas follows.
h:
h(x, y) = (i∂(x)−i∂(y), log(ρ(x)
ρ(y))) on ◦
X×◦
X
h(x, y, α) = (i∂(x)−i∂(y), α) on ∂X ×∂X ×R
(3.4)
By the definition of the topology on Γ(X), it is clearly a continuous family groupoid
morphism.
The interest of defining a good morphism his that the induced groupoid will be free
and proper, like in the case of a smooth manifold, as it was shown by Connes. The freeness
follows from a general principle: if we have a homomorphism Gh
−→ RNthen the isotropy
groups of the induced groupoid Ghare
(Gh)(x,X)
(x,X)={(γ , X)∈G×RN/γ ∈Gx
x, h(γ) = 0}.
In particular, if his a monomorphism (h(γ) = 0 iff γis a unit), it implies that the isotropy
groups described above are trivial, (Gh)(x,X )
(x,X)={(x, X )}.
We will now check the properness of the groupoids Ghwe will be using. For that we
will use the general properness condition (ii) of proposition 2.14 in [35] which tell us that
we need to check two things:
(A) The map
Gh
(t,s)
−→ (G(0) ×RN)×(G(0) ×RN)
is closed, and
(B) For every (a, X)∈G(0) ×RNthe stabilizers (Gh)(a,X):= {γ∈G:t(γ) = a=
s(γ)and X =X+h(γ)}are quasi-compact.
In our case property (B) is immediately verified, indeed, as we mentioned above, h
is a monomorphism so the stabilizers are trivial, (Γ(X)F)(a,X )={(a, X)}, hence quasi-
compact.
Lemma 3.3 The induced crossed product groupoid Γ(X)his a free proper groupoid.
Proof : As mentioned above we have to verify only property (A), that is, we have to
check that the map
Γ(X)×RN(t,s)
−→ (X×RN)×(X×RN)
given by
(t, s) :
(x, y, X )7→ ((x, X),(y, X + (i∂(x)−i∂(y), log(ρ(x)
ρ(y)))) on ◦
X×◦
X×RN
(x, y, α, X )7→ ((x, X),(y, X + (i∂(x)−i∂(y), α)) on ∂X ×∂X ×R×RN
(3.5)
15
is closed.
Let (An)n:= (γn, Xn)na sequence in Γ(X)×RNsuch that
limn→∞(t, s)(γn, Xn) = P(3.6)
with Pa point in (X×RN)×(X×RN). It is enough to justify that there is a subsequence
of (An)nconverging to an antecedent of P: We will separate the analysis in four cases
(a) Suppose P= ((x, X),(y, Y )) with x∈◦
Xand y∈∂X . Since ◦
Xis open in Xwe
have that γn∈◦
X×◦
Xfrom a certain large enough n, that is, we might suppose that
for each n,γn= (xn, yn) for some xn, yn∈◦
Xand in particular
(t, s)(γn, Xn) = ((xn, Xn),(yn, Xn+ (i∂(xn)−i∂(yn), log(ρ(xn)
ρ(yn))))).
Now, the limit (3.6) above implies the following convergences: xn→x,yn→y,
Xn→Xand Xn+ (i∂(xn)−i∂(yn), log(ρ(xn)
ρ(yn))) →Y. Hence we also have that
log(ρ(xn)
ρ(yn)) converges, but this is impossible since ρ(xn)→ρ(x)>0 and ρ(yn)→
ρ(y) = 0. This case is thus not possible.
(b) Suppose P= ((x, X),(y, Y )) with x∈∂X and y∈◦
X. This case is symmetric to
the precedent one, the same analysis shows is empty.
(c) Suppose P= ((x, X),(y, Y )) with x∈◦
Xand y∈◦
X. We might suppose again that
for each n,γn= (xn, yn) for some xn, yn∈◦
Xand in particular
(t, s)(γn, Xn) = ((xn, Xn),(yn, Xn+ (i∂(xn)−i∂(yn), log(ρ(xn)
ρ(yn))))).
Let A= (x, y, X ), A∈Γ(X)×RNand (t, s)(A) = P. The limit (3.6) implies that
we have the convergence An→A.
(d) Finally, suppose P= ((x, X),(y, Y )) with x∈∂X and y∈∂X . In this case we have
two possibilities, (d1) : either Anhas a subsequence completely contained in ◦
X×
◦
X×RN, or (d2) : Anhas a subsequence completely contained in ∂X ×∂X ×R×RN.
In the case (d1) we might suppose again that for each n,γn= (xn, yn) for some
xn, yn∈◦
X. The limit (3.6) above implies the following convergences: xn→x,
yn→y,Xn→Xand Xn+ (i∂(xn)−i∂(yn), log(ρ(xn)
ρ(yn))) →Y. In particular we also
have log(ρ(xn)
ρ(yn)) converges to a certain α∈R. Hence, letting A= (x, y, α, X) we have
that (t, s)(A) = Pand Anconverges to A.
In the case (d2), we might suppose that for each n,γn= (xn, yn, αn) with xn, yn∈∂X
and αn∈R. The limit (3.6) above implies the following convergences: xn→x,
yn→y,Xn→Xand Xn+ (i∂(xn)−i∂(yn), αn)→Y. Thus αnconverges too to a
α∈R. Hence, letting A= (x, y, α, X ) we have that (t, s)(A) = Pand Anconverges
to A.
✷
the two options may coexist.
16
Now, the morphism hinduces a morphism between the algebroids
A(h) : A(Γ(X)) →A(RN) = RN.
With the identification that we have for the algebroid, we explicitly have
A(h)(x, V ) = dxi+(V) (3.7)
if x∈◦
Xand V∈Tx
◦
X, where i+:◦
X→RNis defined as i+(x) := (i∂(x), log(ρ(x))); and
A(h)((x, ξ), α) = (dxi∂(ξ), α) (3.8)
if x∈∂X,ξ∈Tx∂X and α∈R, where i∂is the restriction of i∂to ∂X. We also have the
properness of the respective action.
Lemma 3.4 The induced crossed product groupoid A(Γ(X))A(h)is a free proper groupoid.
Proof : Again, we have to verify only property (A), that is, we have to check that the
map
A(Γ(X)) ×RN(t,s)
−→ (X×RN)×(X×RN)
given by
(t, s) : (((x, V ), X )7→ ((x, X),(x, X +dxi+(V)) on T◦
X×RN
((x, ξ), X )7→ ((x, X),(x, X + (dxi∂(ξ), α)) on T ∂ X ×R×RN(3.9)
is closed.
Let (An)n:= (γn, Xn)na sequence in A(Γ(X)) ×RNsuch that
limn→∞(t, s)(γn, Xn) = P(3.10)
with Pa point in (X×RN)×(X×RN). It is enough to justify that there is a subsequence
of (An)nconverging to an antecedent of P: We will separate the analysis in four cases
(a) Suppose P= ((x, X),(y, Y )) with x∈◦
Xand y∈∂X . Since ◦
Xis open in Xwe
have that γn∈T◦
Xfrom a certain large enough n, that is, we might suppose that
for each n,γn= (xn, Vn) for some Vn∈Txn
◦
Xand in particular
(t, s)(γn, Xn) = ((xn, Xn),(xn, Xn+dxni+(Vn))).
Now, the limit (3.10) above implies in particular the following convergences: xn→x
and xn→y. This case is thus not possible.
(b) The case P= ((x, X),(y, Y )) with x∈∂X and y∈◦
Xis empty, the argument of (a)
above applies as well.
(c) Suppose P= ((x, X),(y, Y )) with x∈◦
Xand y∈◦
X. We might suppose again that
for each n,γn= (xn, Vn) for some ξn∈Txn
◦
Xand in particular
(t, s)(γn, Xn) = ((xn, Xn),(xn, Xn+dxni+(Vn))).
The limit (3.10) implies that xn→x,Xn→Xand Xn+dxni+(Vn)→Y, in
particular (dxni+(Vn))nconverges in RNtoo. Now, since i+is an embedding we
have that di+is a closed embedding, in other words there is a V∈Tx
◦
Xsuch that
dxi+(V) is the limit of (dxni+(Vn))n. Hence, letting A= ((x, V ), X )∈T◦
X×RNwe
have that (t, s)(A) = Pand Anconverges to A.
17
(d) Finally, suppose P= ((x, X ),(y, Y )) with x∈∂X and y∈∂X. In this case we
have two possibilities, (d1) : either Anhas a subsequence completely contained in
T◦
X×RN, or (d2) : Anhas a subsequence completely contained in T ∂X ×R×RN.
In the case (d1) we might suppose again that for each n,γn= (xn, Vn) for some
Vn∈Txn
◦
X. The limit (3.10) above implies the following convergences: xn→x,
Xn→Xand Xn+dxni+(Vn)→Y. In particular we also have dxni+(Vn) converges in
RN. Again, there is then a V∈Tx
◦
Xsuch that dxi+(V) is the limit of (dxni+(Vn))n.
Hence, letting A= ((x, V ), X ) we have that (t, s)(A) = Pand Anconverges to A.
In the case (d2), we might suppose that for each n,γn= ((xn, ξn), αn) with ξn∈
Txn∂X and αn∈R. The limit (3.10) above implies the following convergences:
xn→x,Xn→Xand Xn+ (dxni∂(ξn), αn)→Y. Thus dxni∂(ξn) and αnconverge
too in RN−1and in Rrespectively. Again, because di∂is a closed embedding,
there is a ξ∈Tx∂X such that dxi∂(ξ) is the limit of (dxni∂(ξn))n. Letting A=
((x, ξ), α, X )∈T∂X×R×RNwith αthe limit of (αn)n, we have that (t, s)(A) = P
and Anconverges to A.
✷
Let us now apply to hthe tangent groupoid functor to obtain a continuous family
groupoid morphism
htan : Γ(X)tan →(RN)tan,
explicitly given by
htan :(htan(γ, ε) = (h(γ), ε) on Γ(X)tan ×(0,1]
htan(x, ξ ) = A(h)(x, ξ) on A(Γ(X)) (3.11)
Remember now that the tangent groupoid of RN(as an additive group) is diffeomorphic
to RN×[0,1] by the diffeormorphism (RN)tan →RN×[0,1] given by
((X, 0) 7→ (X, 0) on RN× {0}
(X, ε)7→ (X
ε, ε) on RN×(0,1] (3.12)
As a corollary of proposition A.5 and the two lemmas above we have
Corollary 3.5 Consider the continuous family groupoids morphism Γ(X)tan hT
→RNgiven
as the composition of htan composed with the diffeomorphism (RN)tan ≈RN×[0,1] and
finally with the projection on RN.
Then the crossed product groupoid (Γ(X)tan)hTis a free proper groupoid.
We consider finally the morphism on the Fredholm groupoid
hF: Γ(X)F→RN(3.13)
given by the restriction of hTto Γ(X)F.
We have obtained in particular the following result:
Proposition 3.6 hF: Γ(X)F→RNdefines a homomorphism of continuous family
groupoids and the groupoid (Γ(X)F)hFis a free proper groupoid.
Remark 3.7 As an immediate consequence of the proposition above, the groupoid (Γ(X)F)h
is Morita equivalent to its space of orbits, (see proposition 2.5).
18
The index morphism of the orbit space of (Γ(X)F)h
Let us denote by Bh=XF×RN/∼hthe space of orbits of (Γ(X)F)h. Remember
that to define the deformation index associated to the Fredholm groupoid we considered
the saturated open subgroupoid ◦
X×◦
X×(0,1] ⇒
◦
X×(0,1] and its closed complement
Tnc(X)⇒X∂. The restrictions of hto these two subgroupoids have the same properties as
h, the induced actions are free and proper. Moreover, since we are dealing with saturated
subgroupoids, we have a good behaviour at the level of orbit spaces. That is, denoting
by Bh0=X∂×RN/∼h0and Bh1=◦
X×RN/∼h1the orbit spaces of (Tnc(X))h0and
(◦
X×◦
X)h1respectively, we have an index morphism
K0(Bh0)K0(Bh)
(e0)∗
≈
oo(e1)∗//K0(Bh1) (3.14)
by considering the open saturated subset Bh(0,1] =◦
X×(0,1] ×RN/∼of Bh.
We want next to fully understand this index morphism. Since we are now dealing with
spaces this should be possible.
Lemma 3.8 We have an homeomorphism between the open dense subset Bh(0,1] =◦
X×
(0,1] ×RN/∼of Bhand (0,1] ×RN. More explicitly, the map
(x, ε, X)7→ (ε, ε ·X+i(x))
passes to the quotient into an homeomorphism
e◦
q:◦
X×(0,1] ×RN/∼−→ (0,1] ×RN
Proof : The map ◦
q: (x, ε, X)7→ (ε, ε ·X+i(x))
is obviously a continuous open surjection from ◦
X×(0,1] ×RNto (0,1] ×RN. Moreover,
by definition ◦
q(x, ε, X) = ◦
q(y, ε′, Y )
if and only if
ε=ε′and Y =X+x−y
ε
that is, if and only if
(x, ε, X)∼h(y , ε′, Y ).
The conclusion follows now immediately.
✷
3.2.1 Singular normal bundle
We will need to describe as well the closed complement of the open subset considered
above.
Let us consider
Nsing(X) := N(X)× {0}GRN−1×(0,1) (3.15)
with the topology such that N(X) is a closed subset and RN−1×(0,1) is an open subset
whose closure looks like
D∂:= N(∂X)× {0}GRN−1×(0,1)
19
the deformation to the normal cone associated to the embedding ∂X ֒→RN−1. In fact
we will be more precise in this statement, we will describe Nsing (X) as a C∞-manifold
of dimension N. The structure is such that N(X)|◦
Xand RN−1×(0,1) are two open
submanifolds. We have then to describe the structure around N(X)|∂X :
Let V⊂Rdim∂X be an open subset, consider
W−
V:= V×RN−1−dim∂X ×(−1,0].
Take now, U:= V×(−1,1)N−1−dim∂X ⊂RN−1and consider
W+
V:= {(a, Y, ε)∈Rdim∂X ×RN−1−dim∂X ×[0,1) : a+εY ∈U}.
In particular remark that W+
VTRdim∂X ×RN−1−dim∂X × {0}=V× {0} × {0}
We can consider the open subset of RN:
WV:= W−
V[W+
V.
We define charts
WVΨ
−→ WV⊂Nsing (X) (3.16)
around N(X)|∂X by taking charts Vφ
≈ V covering ∂X and such that we have trivializations
V×RN−1−dim∂X ×(−1,0] ≈ V × RN−1−dim∂X ×(−1,0] ≈N(X)|V×(−1,0]
Thus obtaining in this way
W−
V
Ψ−
−→ W−
V:= N(X)|V×(−1,0] .(3.17)
For W+
V, we might suppose that φgives a slice chart of ∂X in RN−1diffeomorphic to
U:= V×(−1,1)N−1−dim∂X .
Then the W+
Vare precisely the open subsets ΩU
Vconsidered in [8] section 3. We consider
the deformation to the normal cone charts explicitly described in [8] section 3, proposition
3.1 (see also [10] section II.5).
W+
V
Ψ+
−→ W+
V⊂D∂.(3.18)
Locally they look like:
ΩU
V→DU
V≈DU
V⊂D∂
(a, Y, 0) 7→ (a, Y , 0), and
(a, Y, ε)7→ (a+εY , ε)for ε 6= 0.
The fact that {(WV,Ψ)}are compatible with N(X)|◦
Xis immediate and the fact that
they are compatible with RN−1×(0,1) follows from proposition 3.1 in [8].
Lemma 3.9 We have an homeomorphism between the closed subset Bh0=X∂×RN/∼
of Bhand Nsing(X). More explicitly, the map
((x, ε, X)7→ (ε, ε ·XN−1+i∂(x))
(x, 0, X)7→ [X]x∈Nx(X)(3.19)
passes to the quotient into an homeomorphism
eq∂:X∂×RN/∼−→ Nsing (X)
20
Proof : Let us denote by q∂:X∂×RN→Nsing (X) the map (3.19) above. We will
show that it is an open continuous surjection. The fact that is surjective is immediate.
For proving that it is open and continuous, it is enough to check that for every point p∈
X∂×RNthere is an open neighborhood Vpsuch that q∂(Vp) is open and q∂:Vp→q∂(Vp)
is continuous.
For points in ∂X ×(0,1) ×RN, the restriction
∂X ×(0,1) ×RN→q∂(∂X ×(0,1) ×RN) = (0,1) ×RN−1
is given by (x, ε, X)7→ (ε, ε ·XN−1+i∂(x)) which is open and continuous (the proof is
completely analog to lemma 3.8).
For points in ◦
X× {0} × RN, the restriction
◦
X× {0} × RN→q∂(◦
X× {0} × RN) = N(X)|◦
X
is given by (x, 0, X)7→ [X]x∈Nx(X) which is clearly open and continuous.
Finally, for points in ∂X ×{0}×RNwe need to have more careful. Let x0∈∂X, we take
a chart Vφ
≈ V as above, together with the correspondant slice chart U∈RN−1. Around
a point (x, 0, X0)∈XF×RNwe can consider a chart (with boundary) diffeomorphic to
U:= {(x, t, ε, X)∈V×(−1,0] ×[0,1) ×RN:x+εXN−1−dim∂X ∈U}.
Take U∂the corresponding intersection with X∂×RN. In these local coordinates, the map
q∂looks like:
U → WV
given by
(x, 0, ε, X)7→ (x, XN−1−dim∂ X , ε)∈W+
V
and
(x, t, 0, X)7→ (x, XN−1−dim∂X , t)∈W−
V.
It is evidently an open continuous map. Hence the map q∂is an open continuous surjection
for which q∂(a) = q∂(b) if and only if a∼hbin X∂×RN. We conclude then that the
induced map is an homeomorphism.
✷
3.2.2 The APS classifying space
Consider
BF:= Nsing(X)G(0,1] ×RN.(3.20)
By the two precedent lemmas we can conclude that there is an unique locally compact
topology on BFsuch that the bijection
fqF:= eq∂Ge◦
q:Bh−→ BF
induced by eq∂and e◦
qis an homeomorphism.
With the new identifications, the index morphism (3.14) takes the following form:
K0(Nsing(X)) K0(BF)
(e0)∗
≈
oo(e1)∗//K0(RN).(3.21)
The APS index theorem proposed below, will be useful only if we can compute this index
morphism. In the case ∂X =∅the topology of BFgives immediately that the above index
21
is just the shriek map induced by an open inclusion. In the case with boundary, Nsing(X)
is not quite an open subset of RN, but still we can prove the following by analyzing the
topology of BF:
To describe the topology of this space we will describe three big open subsets that cover
BFand that generate the entire topology (see proposition below). Let us first define the
subjacent subsets U0
F,U1
Fand U2
F:
U0
F=RN×(0,1],
U1
F= (RN−1×(0,1)) G(RN−1×(0,1) ×(0,1)),
where the first component of the disjoint union belongs to Nsing(X) and the second one
is the open subset (RN−1×(0,1)) ×(0,1) of RN×(0,1). For introduce U2
F, we will need
to chose a (good) tubular neighborhood of Xin RN, that is a diffeomorphism
N(X)f
≈//W⊂RN−1×R−
X
OO
id //X
OO(3.22)
from N(X) to Wan open neighborhood of X(or i∂(X) to be formal) in RNsuch that f
is the identity in X(identifying Xwith the zero section), and the restriction to ∂X gives
a tubular neighborhood of ∂X in RN−1:
N(X)|∂X
f∂
≈//W∂⊂RN−1× {0}
∂X
OO
id //∂X
OO(3.23)
where W∂=WTRN−1× {0}. Consider the open subset of RNconsisting of putting a
collar to Win the positive direction:
Wsing := W[W∂×[0,1)
We will introduce the subset U2
Fby its intersections with Nsing(X) and with RN×(0,1]:
U2
F\Nsing(X) = N(X)× {0}GW∂×(0,1),
and
U2
F\RN×(0,1] = Wsing ×(0,1).
Lemma 3.10 The subsets U0
F,U1
Fand U2
Fof BFare open.
Proof : By definition BFinduced from the quotient topology on Bh. Hence if we
consider the map
qF:XF×RN→BF
given by qF:= q∂F◦
q(with the notations of lemmas 3.8 and 3.9), it is enough to observe
that q−1
F(V) is an open subset of XF×RNto conclude that Vis open in BF. In the case
we are dealing with, we have:
•q−1
F(U0
F) = ◦
X×(0,1] ×RN,which is clearly open in XF×RN.
•q−1
F(U1
F) = X×(0,1) ×RN,which is clearly open in XF×RN.
22
•For the last one, an explicit description of the inverse image allows to verify it is open
by a direct analysis at each point, we leave this as direct excercice computation,
q−1
F(U2
F) = X× {0} × RN
G{(x, ε, X)∈∂X ×(0,1) ×RN:i∂(x) + ε·XN−1∈W∂}
G{(x, ε, X)∈X×(0,1) ×RN:i(x) + ε·X∈Wsing}
✷
Proposition 3.11 The locally compact space BFadmits an oriented C∞-manifold with
boundary structure of dimension N+ 1.
Proof : We will cover BFwith three explicit charts by using the three open subsets of
BF,U0
F,U1
Fand U2
Fof the lemma above.
Chart (U0
F,Ψ0
F):We let W0
F=RN×(0,1] and Ψ0
Fto be the identity:
W0
F
Ψ0
F=id
−→ U0
F.
Chart (U1
F,Ψ1
F):We let W1
F=RN−1×[0,1)t×(0,1)sand Ψ1
F,
W1
F
Ψ1
F
−→ U1
F
defined by
((XN−1,0, s)7→ (XN−1, s)∈RN−1×(0,1) ⊂Nsing
(XN−1, t, s)7→ (XN−1, s, t)∈(RN−1×(0,1)) ×(0,1) ⊂RN×(0,1].(3.24)
Chart (U2
F,Ψ2
F):We let W2
F=Wsing ×[0,1)tand Ψ2
F,
W2
F
Ψ2
F
−→ U2
F
defined by ((w, 0) ∈W7→ (f−1(w),0) ∈N(X)× {0} ⊂ Nsing
(w∂, s, 0) 7→ (f∂(s·f−1
∂(w∂)), s)∈W∂×(0,1) ⊂Nsing.(3.25)
if t= 0, and by
((w, t)7→ (f(t·f−1(w)), t)∈W×(0,1) ⊂RN×(0,1]
(w∂, s, t)7→ (f∂((s+t)·f−1
∂(w∂)), s, t)∈(W∂×[0,1)) ×(0,1) ⊂RN×(0,1].(3.26)
for t6= 0.
We will check now the compatibility of the charts, together with the fact that the
changes of coordinates have positive sign:
•((Ψ0
F)−1◦Ψ1
F): This is the easiest case, indeed we have that
(Ψ1
F)−1(U0
F\U1
F) = (RN−1×(0,1)) ×(0,1),
(Ψ0
F)−1(U0
F\U1
F) = (RN−1×(0,1)) ×(0,1)
and
(Ψ1
F)−1(U0
FTU1
F)(Ψ0
F)−1◦Ψ1
F//(Ψ0
F)−1(U0
FTU1
F)
is the identity.
23
•((Ψ0
F)−1◦Ψ2
F):
(Ψ2
F)−1(U0
F\U2
F) = Wsing ×(0,1),
(Ψ0
F)−1(U0
F\U2
F) = Wsing ×(0,1)
and
(Ψ2
F)−1(U0
FTU2
F)(Ψ0
F)−1◦Ψ2
F//(Ψ0
F)−1(U0
FTU2
F)
is given by ((w, t)7→ (f(t·f−1(w)), t)
(w∂, s, t)7→ (f∂((s+t)·f−1
∂(w∂)), s, t)
which is evidently a diffeomorphism with positive determinant.
•((Ψ1
F)−1◦Ψ2
F):
(Ψ2
F)−1(U1
F\U2
F) = W∂×(0,1) ×[0,1),
(Ψ1
F)−1(U1
F\U2
F) = W∂×[0,1) ×(0,1)
and
(Ψ2
F)−1(U1
FTU2
F)(Ψ1
F)−1◦Ψ2
F//(Ψ1
F)−1(U1
FTU2
F)
is given by
(w∂, s, t)7→ (f∂((s+t)·f−1
∂(w∂)), t, s)
which is evidently a diffeomorphism with positive determinant.
✷
In particular, we have obtained an oriented cobordism BFfrom Nsing(X) to RN.
From now on, we orient BFsuch that the induced orientation on the boundary is
∂BF=−Nsing(X)[RN.
We can hence apply a Stoke’s theorem argument to obtain the following proposition:
Proposition 3.12 The following diagram is commutative
K0(Nsing(X))
indBF
''
RNsing(X)ch(·)
''
P
P
P
P
P
P
P
P
P
P
P
P
P
P
K0(BF)
(e0)∗
≈
oo(e1)∗//K0(RN)
RRNch(·)
xxq
q
q
q
q
q
q
q
q
q
q
q
R
Proof : By definition, the algebra morphisms e0:C0(BF)→C0(Nsing(X)) and
e1:C0(BF)→C0(RN) are induced by the closed embeddings i0:Nsing(X)֒→BFand
It is a diffeomorphism whose linear representation is of the form A·λ·A−1with positive λ.
same as previous footnote.
24
i1:RN֒→BFrespectively. The Chern character being natural we have that the following
diagram is commutative
K0(Nsing(X))
ch
K0(BF)
ch
(e0)∗
≈
oo(e1)∗//K0(RN)
ch
Hev
dR(Nsing (X)) Hev
dR(BF)
(i0)∗
oo(i1)∗
//Hev
dR(RN)
(3.27)
The result now follows from Stoke’s theorem, indeed, for any ω N -closed differential form
on BFwith compact support, we have by Stoke’s that
Z∂BF
ω=O
and hence RNsing(X)ω=RRNω.
✷
4 Atiyah-Patodi-Singer theorem in K-theory
4.1 The Fredholm index morphism
Deformation groupoids induce index morphisms. The groupoid Γ(X)Fis parametrized by
the closed interval [0,1]. Its algebra comes equipped with evaluations to the algebra of
TncX(at t=0) and to the algebra of ◦
X×◦
X(for t6= 0). We have a short exact sequence
of C∗-algebras
0→C∗(◦
X×◦
X×(0,1]) →C∗(Γ(X)F)→C∗(TncX)→0 (4.1)
where the algebra C∗(◦
X×◦
X×(0,1]) is contractible. Hence applying the K-theory functor
to this sequence we obtain an index morphism
indF= (e1)∗◦(e0)−1
∗:K0(TncX)−→ K0(◦
X×◦
X)≈Z.
Proposition 4.1 [19, 21, 11]. For any fully elliptic operator Don X, there is a ”non-
commutative symbol” [σD]∈K0(TncX)and
indF(σD) = IndexAP S (D) (4.2)
Proof : For the sake of completeness, we briefly explain the proof. Let D∈Ψ0(Γ(X); E, F )
be a zero-order fully elliptic b-operator. Here E , F are hermitian bundles on X, pulled-
back to Γ(X) with the target map. Let Q∈Ψ0(Γ(X); F, E) be a full parametrix of P.
This means:
P Q −1∈Ψ−1(Γ(X); F, F ), QP −1∈Ψ−1(Γ(X); E, E ) (4.3)
and that, moreover,
P∂Q∂= 1, Q∂P∂= 1 (4.4)
where we have denoted by A∂the ∂X ×∂X ×R-pseudodifferential operator obtained by
restriction of any Γ(X)-operator A. The equations (4.3) reflect the interior ellipticity while
(4.4) reflect the boundary ellipticity. It is well known that P:L2
b(X, E)−→ L2
b(X, F ) is
bounded and Fredholm [24, 27] where the hermitian structure of E, F as bundles over X
is used together with the measure on Xassociated with a b-metric, and
IndexAP S (P) = IndexFred(P) = dim ker P−dimcokerP.
25
This can also be recovered as the analytical index of a K-homology class of a compact
space. Let Xc=X/∂X be the conical space associated with Xand we note πthe canonical
projection map. We represent C(Xc) into L2
b(X, G), where Gis any hermitian bundle over
X, as follows:
f∈C(Xc), ξ ∈L2
b(X, G); ∀y∈X, m(f)(ξ)(y) = f(π(y))ξ(y).
It is immediate to check that
(P) = L2
b(X, E ⊕F); m;0Q
P0 (4.5)
is a Kasparov (C(Xc),C)-module and that the resulting K-homology class [P]∈K0(Xc)
does not depend on the choices of the parametrix Qnor the particular b-metric. Then
IndexAPS(P) = Indexana(P) = p∗([P]) where p:Xc→ {point}.
We define σP∈K0(Tnc X) to be the Poincar´e dual class of P. Let us describe this
element more explicitly.
Let e
P,e
Qbe any elliptic operators on Γ(X)tan such that e
P|t=1 =P,e
Q|t=1 =Qand:
e
Pe
Q−1,e
Qe
P−1∈Ψ−1(Γ(X)tan).
We have by construction:
(e
Pt=1)∂(e
Qt=1)∂−1 = 0 = ( e
Qt=1)∂(e
Pt=1)∂−1,
Hence:
(e
P) = C∗(Γ(X)F, E ⊕F),1, 0e
Q
e
P0!! (4.6)
is a Kasparov (C, C∗(Γ(X)F))-module, and the restriction:
σnc(e
P) := e
P|TncX(4.7)
provides a Kasparov (C, C∗(Tnc X))-module:
(σnc(e
P)) := C∗(TncX, E ⊕F),1, 0σnc(e
Q)
σnc(e
P) 0 !!.(4.8)
Denoting by e0the ∗-homomorphism C∗(Γ(X)F)→C∗(TncX), the previous class satisfies:
(e0)∗[e
P] = [σnc(e
P)] ∈K0(TncX) (4.9)
By construction the Poincar´e duality isomorphism [13, 21] sends [P] to [σnc (e
P)], and we
thus set:
σP= [σnc(e
P)]
Now, µ0denoting the Morita equivalence ◦
X×◦
X∼point, we compute:
indF(σP) = µ0◦(e1)∗◦(e0)−1
∗([σnc(e
P)] (4.10)
=µ0◦(e1)∗(e
P) (4.11)
=µ0C∗(◦
X×◦
X, E ⊕F),1,0Q
P0 (4.12)
=L2
b(X, E ⊕F),1,0Q
P0 (4.13)
= Indexana(P) = p∗([P]) = IndexAPS(P) (4.14)
26
✷
It is also interesting to manage geometric operators (for instance, Dirac type operators
on Xequipped with an exact b-metric gb) instead of abstract 0-order pseudodifferential
operators. Under appropriate assumptions, they also give rise to K-homology classes of
Xcand thus one may look for a geometric representative of their Poincar´e dual class in
K0(TncX).
Rather than dealing exaclty with Dirac operators on (X, gb), we shall consider the
following class of differential operators on Xcontaining them. Let Ebe a smooth hermitian
vector bundle over Xendowed with an orthogonal decomposition E=E0⊕E1and
an isomorphism U:E|∂X ×(0,ε)→(p1)∗E∂⊕(p1)∗E∂where p1is the first projection
subordonated to a collar identification near the boundary and E∂=E0|∂X . Then we
consider first order elliptic differential operators D, which have, after conjugation by U,
the following expression near the boundary:
UDU −1=0−x∂
∂x +S
x∂
∂x +S0=0D−
D+0(4.15)
where S∈Diff1(∂X, E∂). We require Sto be elliptic, symmetric, and to simplify in-
dependant of x < ε. It follows from these assumptions that D2+ 1 is invertible, as
a linear map, on C∞(X, E) [24] and since Ψ(Γ(X)) is spectrally invariant (it is un-
derstood that the appropriate Schwartz algebra is added in the calculus [20]), we have
(D2+ 1)−1/2∈Ψ−1(Γ(X)). Now, we moreover require the invertibility of S, which is
equivalent here to the full ellipticity of D. Observe that this assumption is not sufficient
to let Dinto an unbounded (C(Xc),C)-Kasparov module in the sense of [6], but it nev-
ertheless implies that W(D) := D(D2+ 1)−1/2∈Ψ0(Γ(X), E) is fully elliptic since the
indicial family map I: Ψ(Γ(X)) →Ψ(∂X, R) is a homomorphism of algebras:
I(W(D), τ ) = 0S−iτ
(S2+τ2+1)1/2
S+iτ
(S2+τ2+1)1/20!=0W(D)−
W(D)+0.
Thus one can already associate to Dthe (bounded) K-homology class [W(D)+], with
Poincar´e dual given by the (bounded) K-theory class σW(D)+as above. Alternatively, we
may look for a more geometric representative of these classes.
For that purpose, we define a lift e
D∈Ψ(Γ(X)tan) of Das follows. Let sbe the
complete symbol of Swith respect to the exponential map of the metric on the boundary
(see [16]), and dbe the complete symbol of Dwith respect to the exponential map of the
metric on X. We rescale sand das follows:
∀0< ε ≤1,∀(x, ξ)∈T∗∂X, sad(y , ξ) = s(y, εξ) and ∀(z, ζ )∈T∗X, dad(z, ζ ) = d(z, εζ )
Setting Sad =sad(y, Dy), Dad =dad(z, Dz) and using positive functions ϕ, ψ ∈C∞(X)
such that ϕ+ψ= 1 and ϕ= 1 if x < ε,ϕ= 0 if x≥2ε, we let
e
D|xε>0=ϕU−10−εx ∂
∂x +Sad
εx ∂
∂x +Sad 0U+ψDad,
e
D|x=0,ε>0=U−10−∂
∂λ +Sad
∂
∂λ +Sad 0U,
and
e
D|ε=0 =ϕU−10−∂
∂λ +s(y, DY)
∂
∂λ +s(y, DY) 0 U+ψd(z, DZ),
where DY, DZstand for the differentiation in the fibers coordinates of T ∂ X and T X
respectively.
27
On the other hand, we can also rescale e
D|x=0,ε>0into:
σ∂−ubd(D) = U−10−∂
∂λ +1
1−εSad
∂
∂λ +1
1−εSad 0U
and define σubd(D)∈Diff1(TncX) by
σubd(D) = σ∂−ubd(D) if x= 0 and σubd (D) = e
D|ε=0 if x > 0
We also note e
Ethe pull back of Efor the map X×[0,1] p1
−→ Xand then Ead its re-
striction to {xε = 0}. By construction, we get an elliptic symmetric element σubd(D)∈
Diff1(TncX, r∗Ead), whose closure σubd(D) as an unbounded operator on the Hilbert mod-
ule E:= C∗(TncX, r∗Ead) with domain C∞,0(Tnc X, r∗Ead) is a regular selfadjoint operator
[6]. To prove that assertion, we can not directly apply proposition 3.6.2 and lemma 3.6.3
in [37] since the unit space of TncXis not compact. Nevertheless, we can pick up a suitable
parametrix q∈Ψ−1(TncX , r∗Ead) of σubd (D), where Tnc X:= Γ(X)tan|xε=0, in such a way
that the proofs given in [37] apply verbatim : we define qby combining, using cut-off
functions, a parametrix given by the inversion of the principal symbol of σubd(D) with the
true inverse of σ∂−ubd(D)|εwhen 1 −α < ε < 1 and extended by 0 at ε= 1. Then, we
have by construction:
σubd(D)q= 1+k1, qσubd(D) = 1+k2with q, ki∈Ψ−1(TncX , r∗Ead), q|ε=1 = 0, ki|ε=1 = 0.
The first of the two conditions on the operators q, k1, k2implies that they extend into
compact morphisms on C∗(TncX , r∗Ead) and the second that they actually are compact
on C∗(TncX, r ∗Ead).
It then follows that (σubd(D)2+ 1)−1is a compact morphism on C∗(TncX, r∗Ead ).
Thus, we get an unbounded (C, C ∗(TncX))-Kasparov class in the sense of [6]:
σubd,D := (C∗(TncX, r∗Ead), σubd(D)) ∈ Eubd (C, C∗(TncX)).
In the equality above, Eubd(A, B) denotes the family of unbounded Kasparov A-B-bimodules
as defined in [6].
To check that the latter is an unbounded representative of the Poincar´e dual of
[W(D)+] (and thus can be used for the computation of the index of D+), we have to
prove the equality :
[σW(D+)] = [W(σubd(D))] ∈KK(C, C∗(TncX)),
which can be achieved by comparing the operator part of these KK-classes. When ε, they
coincide. When ε > 0, the operator part in [σW(D+)] can be represented by:
0Sad−i∂λ
(S2
ad+∂2
λ+1)1/2
Sad+i∂λ
(S2
ad+∂2
λ+1)1/20!,
and for [W(σubd(D))] we have:
0Sad−i(1−ε)∂λ
(S2
ad+(1−ε)2(∂2
λ+1))1/2
Sad+i(1−ε)∂λ
(S2
ad+(1−ε)2(∂2
λ+1))1/20
.
Homotoping the numerical factor (1 −ε) with 1 provides an operator homotopy between
both, and this proves the assertion.
Observe also that:
W(σubd(D))|ε=1 = 0S
|S|
S
|S|0!,
and thus that, playing again with homotopies, this value at ε= 1 can be conserved for
ε∈]α, 1] for arbitrary α > 0.
28
4.2 Atiyah-Patodi-Singer index theorem in K-theory
Definition 4.2 [Atiyah-Patodi-Singer topological index morphism for a manifold with
boundary] Let Xbe a manifold with boundary consider an embedding of Xin RNas
in section 3.2. The topological index morphism of Xis the morphism
indX
t:K0(TncX)−→ Z
defined as the composition of the following three morphisms
1. The Connes-Thom isomorphism C T 0followed by the Morita equivalence M0:
K0(TncX)C T 0
−→ K0((Tnc X)h0)M0
−→ K0(Nsing (X))
2. The index morphism of the deformation space BF(proposition 3.12):
K0(Nsing(X))
indBF
''
K0(BF)
(e0)∗
≈
oo(e1)∗//K0(RN)
and
3. the usual Bott periodicity morphism: K0(RN)Bott
−→ Z.
In other terms, the topological index fits by definition in the following commutative
diagram
K0(TncX)
C T ≈
indX
t//Z
K0(Nsing(X)) indBF
//K0(RN)
Bott
≈
OO
Remark 4.3 The topological index defined above is a natural generalisation of the topo-
logical index theorem defined by Atiyah-Singer. Indeed, in the smooth case, they coincide.
We now prove, as it was outlined in [9], the index theorem.
Theorem 4.4 [K-theoretic APS] Let Xbe a manifold with boundary, consider an embed-
ding of Xin RNas in section 3.2. The Fredholm index equals the topological index.
Proof : The morphism h: Γ(X)F→RNis by definition also parametrized by [0,1],
i.e., we have morphisms h0:TncX→RNand h1:◦
X×◦
X→RN, for t= 1. We can
consider the associated groupoids, which are free and proper.
The following diagram, in which the morphisms C T and Mare the Connes-Thom and
Morita isomorphisms respectively, is trivially commutative by naturality of the Connes-
Thom isomorphism:
K0(TncX)
≈
C T
K0(Γ(X)F)
≈
C T
e0
≈
ooe1//K0(
◦
X×
◦
X)
≈
C T
K0((TncX)h0)
≈
M
K0((Γ(X)F)h)
≈
M
e0
≈
ooe1//K0((
◦
X×
◦
X))h1)
≈
M
K0(Nsing(X)) K0(BF)
e0
≈
ooe1//K0(RN).
(4.16)
29
The left vertical line gives the first part of the topological index map. The bottom
line is the morphism induced by the deformation space B. And the right vertical line is
precisely the inverse of the Bott isomorphism Z=K0({pt})≈K0(◦
X×◦
X)→K0(RN).
Since the top line gives indF, we obtain the result.
✷
Corollary 4.5 The topological index does not depend on the choice of the embedding.
5 The cohomological APS formula
The theorem 4.4 (see also diagram (4.16)) tells us that the computation of the index can
be performed (modulo Connes-Thom and Morita) as the computation of the index of a
deformation space :
K0(Nsing(X)) K0(BF)
(e0)∗
≈
oo(e1)∗//K0(RN)
Now, consider the following diagram
K0(RN)Ch //
Bott
H∗
dR(RN)
RRN·
Z//C,
(5.1)
where RRNis the integration with respect to the fundamental class of RN. It is well known
that this diagram is commutative.
We can summarize the previous statements in the following result, which is then an
immediate consequence of theorem 4.4 (see again diagram (4.16)):
Corollary 5.1 Let (X, ∂X )be a manifold with boundary, and let i:X ֒→RNbe an
embedding as in section 3.2; we use the notations of last sections.
The index morphism indFfits in the following commutative diagram
K0(Tnc(X))
indF
((
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
❘
C T h0//K0((Tnc(X))h0)M orita
//K0(Nsing(X))
Ch
H∗(Nsing(X))
RNsing(X)
C
(5.2)
For keeping short notations we will denote by C T the composition of C T hwith the Morita
equivalence induced isomorphism M.
In particular, for any fully elliptic operator Don Xwith ”non commutative symbol”
[σD]∈K0(TncX)we have the following cohomological formula for the APS index:
IndexAP S (D) = ZNsing(X)
Ch((C T ([σD]))) (5.3)
Remember that the space Nsing (X) already splits in two, exhibiting in this way the
contributions from the interior and from the boundary. The interior contribution looks
classic but an explicit comparison between the Thom isomorphism and the Connes-Thom
isomorphism is needed. This will be detailed elsewhere.
30
In particular, picking up a differential form ωDon Nsing (X) representing Ch(C T ([σD]),
we obtain:
IndexAP S (D) = ZN(X)
ωD+ZD∂
ωD.(5.4)
The first integral above involves the restriction of ωDto N(X), which is related to the
ordinary principal symbol of D. More precisely, the principal symbol σpr(D) of Dprovides
aK-theory class of C∗(A∗(Γ(X))), that is a compactly supported K-theory class of the
dual of the Lie algebroid of Γ(X) or in other words of the b-cotangent bundle bT∗X, and
by functoriality of both the Chern character and Thom-Connes maps, we have
[(ωD)|N(X)] = Ch(C T ([σpr (D)]).
The second integral can thus be viewed as a correction term, which contains the eta invari-
ant appearing in APS formula and which also depends on the choice of the representative
ωD∈Ch(C T ([σD])).
5.1 A more explicit description of (5.3)
Let Dbe a fully elliptic b-operator. We wish to give an explicit representative of the
K-theory class Ch((C T ([σD]))) ∈K0(Nsing (X)). We may equally work in the bounded
setting, that is, Da 0-order pseudodifferential operator and [σD] or in the unbounded
setting, with Da first order differential operator and its non commutative symbol rep-
resented by unbounded KK-element σubd,D as defined in paragraph 4.1. We choose the
second option (the computation is similar for the first one) and we are going to give an
explicit representative of (see section 2.3):
C T ([σubd,D]) := M◦(e1,t)∗◦(e0,t)−1
∗◦B(σubd,D)∈K0(Nsing(X)).
We assume that N= 2Mis even, we identify RNwith CM, we denote by Λ∗(CM)
the exterior algebra of CM, by c(v) = v∧ · − vx·the Clifford multiplication by v, and
we consider the following unbounded representative of the Mth power of Bott class β∈
KK(C, C0(R2)):
βM=C0(RN,Λ∗(CM)),1, c∈ Eubd(C, C0(RN)) (5.5)
with the grading given by even/odd forms.
Then B(σubd,D) is represented by
Σubd,D := C∗(TncX×RN, r∗Ead ⊗Λ∗(CM)),1,Σubd(D)∈ Eubd(C, C ∗(TncX×RN))
(5.6)
where we have set Σubd(D) := σubd(D)ˆ
⊗1 + 1 ˆ
⊗c. Next, we look for a representative of
(e1,t)∗◦(e0,t )−1
∗◦Σubd,D . For that purpose, we introduce the following C∞,0-diffeomorphism
between TncX×RNand TncX⋉ RN:
φ:TncX×RN−→ TncX⋉ RN(5.7)
(γ, X )7−→(γ, X −h(γ))
which commutes with source maps of both groupoids, that is : s=s◦φ. Using φ, we
see that any given right Haar system dλ on TncX, we get a right Haar system dλhon
TncX⋉ RNby the formula:
Z(TncX⋉RN)(x,X )
fdλh
(x,X)=Z(Tnc X)x
f◦φdλx(5.8)
This leads to the following formulas for the reduced C∗-norms:
kfkC∗(TncX⋉RN)=kf◦φkC∗(Tnc X×RN)= sup
X∈RN
kf◦φ(·, X)kC∗(Tnc X)(5.9)
31
We also observe that a straight computation proves that if ais any TncX×RN-operator,
then ah:= (φ−1)∗◦a◦(φ)∗is a TncX⋉ RN-operator. Thus, we let:
Σubd −h(D) := (Σubd(D))h(5.10)
that is, for all f∈C∞,0
c(TncX⋉ RN, r∗Ead ×Λ∗(CM)) and ∀(γ, X )∈ TncX⋉ RN:
Σubd −h(D)(f)(γ, X ) := (σubd(D)ˆ
⊗1)(f◦φ)(γ, X +h(γ)) + (1 ˆ
⊗c(X+h(γ)))(f)(γ, X ).
(5.11)
We prove that:
Proposition 5.2 The operator Σubd −h(D)belongs to Diff1(Tnc X⋉RN, r∗Ead ×Λ∗(CM)),
it is symmetric, elliptic, and the following assertion holds:
Σh:= C∗(TncX⋉ RN, r∗Ead ⊗Λ∗(CM)),1,Σubd −h(D)∈ Eubd(C, C ∗(TncX⋉ RN)).
(5.12)
and we have the equality:
[Σh] = (e1,t)∗◦(e0,t )−1
∗◦B(σubd,D)∈KK(C, C ∗(TncX⋉ RN)).
Proof : Let us consider p:= Σubd (D)2+ 1 = σubd (D)2+|X|2+ 1 ∈Diff 2(TncX×
RN, r∗Ead ×Λ∗(CM)), where we have taken into account the identity c2(f)(γ, X ) =
|X|2.f(γ, X). It is self-adjoint and for any (x, X) belonging to the unit space (TncX×
RN)(0) =X∂×RN, the operator Σubd (D)2
(x,X)+ 1 is invertible by classical arguments.
Thus, pitself is invertible, which means that p−1exists and belongs to Ψ−2(TncX×
RN, r∗Ead ×Λ∗(CM)). As a direct consequence, Σubd(D) is regular. Moreover, let us con-
sider the groupoid Tnc X×BN⇒X∂×BNwhere BNis the compactification of RNby a
sphere at infinity and we recall that TncX=Tnc X∪∂X ×∂X ×R× {ε= 1}⇒X×[0,1]ε
. The groupoid structure of TncX×BNis the obvious one and it inherits a natural
C∞,0-structure in such a way that it contains TncX×RNas an open saturated C∞,0-
subgroupoid. We also extend the vector bundles r∗Ead and Λ∗(CM) onto X∂and BN
without changing the notations. Thanks to the decay of p−1when ε→1 or |X| → ∞, we
see that the extension of p−1by 0 gives an element of Ψ−2(TncX×BN, r∗Ead ×Λ∗(CM)).
Since TncX×BNhas a compact unit space, we know by [29, 37] that p−1is a compact
morphism of the C∗(TncX×BN)-Hilbert module E:= C∗(TncX×BN, r∗Ead ×Λ∗(CM)).
Considering the closed saturated C∞,0-subgroupoid ∂(TncX×BN) given by the equation
z:= (1 −ε)(1/(|X|+ 1)) = 0 and the corresponding Hilbert submodule ∂E, we get an
exact sequence of C∗-algebras
0−→ K(E)−→ K(E)rest|z=0
−→ K(∂E)−→ 0.(5.13)
Since rest|z=0(p−1) = 0, we conclude that p−1belongs to K(E).
Replacing hby H= (th)t∈[0,1] and extending the previous construction to the groupoid
TncX⋊HRN⇒X∂×RN×[0,1], we get and unbounded Kasparov class ΣH:= (C∗(TncX⋊H
RN, r∗Ead ⊗Λ∗(CM)),1,Σubd −H(D)) ∈Ψ1(C, C ∗(TncX⋊HRN)) such that (e0,t)∗([ΣH]) =
B(σubd,D) and (e1,t)∗([ΣH]) = [Σh].
✷
It thus remains to apply the Morita equivalence Mto [Σh], since in our previous nota-
tions : C T ([σubd,D]) = M(Σh). Following [22, 35], we implement the Morita isomorphism
M:K0(C∗(TncX⋉ RN)) →K0(C0(Nsing (X))) by the Kasparov product · ⊗ Mwith:
M= (EM, π, 0) ∈KK(C∗(TncX⋉ RN), C0(Nsing (X))).(5.14)
32
Let us recall its content. We note Ginstead of TncX⋉ RN,Oinstead of G(0)/G and λ
the Haar system previously introduced to lighten the reading. The space C0(G(0)) inherits
the following C0(O)-pre hilbertian module structure:
< ξ , η > (o) =
∀z s.t [z]=oZGz
ξ(r(γ))η(r(γ))dλz(γ),(5.15)
and the right C0(O)-module structure of C0(G(0)) is given by
ξ.a(z) = ξ(z)a([z]).(5.16)
Then we take EMas the Hilbert module completion of this pre-hilbertian module. Finally,
the following formula:
π(b).ξ(z) = ZGz
b(γ−1)ξ(r(γ))dλz(γ).(5.17)
defines the appropriate ∗-homomorphism π:C∗(G)→ K(EM).
The Kasparov product Σh⊗
C∗(TncX⋉RN)
Mis given by:
C∗(TncX⋉ RN, r∗Ead ⊗Λ∗(CM)) ⊗
πEM,1,Σubd −h(D)b
⊗1∈KK(C, C0(Nsing (X)))
(5.18)
We use again the shorcut notations G,Ofor TncX⋉ RN,Nsing (X) and we moreover set
E=Ead ⊗Λ∗(CM). We extend to Cc(G(0), E) the previous C0(O)-prehilbertian module
structure:
< ξ , η > ([z]) = ZGz
< ξ(r(γ)), η(r(γ)) >Er(γ)dλz(γ).(5.19)
and we denote by EEits Hilbert module completion. We also observe that (5.17) makes
sense for b∈Cc(G, r∗E) since in the integral b(γ−1)∈Er(γ−1)=Ez, yielding a map again
denoted by π:
π:Cc(G, r∗E)−→ L(Cc(G(0)), Cc(G(0), E ))
We then introduce:
U:Cc(G, r∗E)⊗Cc(G(0))−→ Cc(G(0) , E) (5.20)
a⊗f−→ π(a)f(5.21)
Proposition 5.3 The map Udefined in (5.20) extends to a unitary of C0(O)-Hilbert
module:
U:C∗(G, r∗E)⊗
πEM−→ EE,(5.22)
providing a unitary equivalence between C T ([σubd,D ]) = Σh⊗
C∗(TncX⋉RN)
Mand:
(EE,1, r∗(Σubd −h(D))) (5.23)
where rdenotes the target map of the groupoid TncX⋉ RNand r∗is the map defined by
r∗(P)(f)(z) = P(f◦r)(γ)for any γsuch that r(γ) = zand any TncX⋉ RN-operator P.
Proof : If E=X×Cis the product rank one bundle, then the fact that Uis isometric
directly follows from the fact that π:C∗(G)→ L(EM) is a ∗-homomorphism:
< U(a⊗f)|U(b⊗g)>EM=< π(a)f|π(b)g >EM=< f |π(a)∗π(b)g >EM
=< f|π(a∗b)g >EM
=< f|π(< a|b >C∗(G))g >EM=:< a ⊗f|b⊗g >C∗(G)⊗
πEM
33
Since for any b∈Cc(G), ξ∈Cc(G(0)), we have π(b)(ξ) = (b.r∗(ξ))|G(0) where .denotes
the convolution product of functions on G(well defined thanks to the compactness of the
support of b), we get, taking an approximate unit for C∗(G), the density of the image of
Uand thus its surjectivity.
In the general case, we imbed E ֒→X×Ckinto a trivial bundle in such a way that
we get an imbedding C∗(G, r∗E)֒→C∗(G)kof C∗(G)-Hilbert module. Then we use the
previous arguments coordinatewise.
Now, a straight computation shows that
U◦(Σubd −h(D)b
⊗1) = r∗(Σubd −h(D)) ◦U.
✷
What we have obtained above is a concrete representative of the K-theory class
M(CTh([σD])) whose Chern character is used to compute the index of Din formula (5.3).
This representative is a family of elliptic operators parametrized by the singular normal
bundle Nsing (X). A final step should be to compute the Chern character of this K-theory
class, using for instance the method of [7]. This will be achieved in a forthcoming paper.
A Deformation to the normal cone functor and tangent
groupoids
The tangent groupoid is a particular case of a geometric construction that we describe
here.
Let Mbe a C∞manifold and X⊂Mbe a C∞submanifold. We denote by NM
Xthe
normal bundle to Xin M. We define the following set
DM
X:= NM
X×0GM×R∗(A.1)
The purpose of this section is to recall how to define a C∞-structure in DM
X. This is more
or less classical, for example it was extensively used in [17].
Let us first consider the case where M=Rp×Rqand X=Rp× {0}( here we identify
Xcanonically with Rp). We denote by q=n−pand by Dn
pfor DRn
Rpas above. In this case
we have that Dn
p=Rp×Rq×R(as a set). Consider the bijection ψ:Rp×Rq×R→Dn
p
given by
ψ(x, ξ, t) = (x, ξ , 0) if t= 0
(x, tξ, t) if t6= 0 (A.2)
whose inverse is given explicitly by
ψ−1(x, ξ, t) = (x, ξ , 0) if t= 0
(x, 1
tξ, t) if t6= 0
We can consider the C∞-structure on Dn
pinduced by this bijection.
We now switch to the general case. A local chart (U, φ) in Mis said to be a X-slice if
1) φ:U → U⊂Rp×Rqis a diffeomorphsim.
2) If V=U∩(Rp× {0}), then φ−1(V) = U ∩ X, denoted by V.
With this notation, DU
V⊂Dn
pas an open subset. We may define a function
˜
φ:DU
V→DU
V(A.3)
in the following way: For x∈ V we have φ(x)∈Rp× {0}. If we write φ(x) = (φ1(x),0),
then
φ1:V → V⊂Rp
34
is a diffeomorphism. We set ˜
φ(v, ξ, 0) = (φ1(v), dNφv(ξ),0) and ˜
φ(u, t) = (φ(u), t) for
t6= 0. Here dNφv:Nv→Rqis the normal component of the derivative dφvfor v∈ V. It
is clear that ˜
φis also a bijection (in particular it induces a C∞structure on DU
V).
Let us define, with the same notations as above, the following set
ΩU
V={(x, ξ, t)∈Rp×Rq×R: (x, t ·ξ)∈U}.(A.4)
which is an open subset of Rp×Rq×[0,1] and thus a C∞
cmanifold (with border). It is
immediate that DU
Vis diffeomorphic to ΩU
Vthrough the restriction of Ψ, used in (A.2).
Now we consider an atlas {(Uα, φα)}α∈∆of Mconsisting of X−slices. It is clear that
DM
X=∪α∈∆DUα
Vα(A.5)
and if we take DUα
Vα
ϕα
→ΩUα
Vαdefined as the composition
DUα
Vα
˜
φα
→DUα
Vα
Ψ−1
α
→ΩUα
Vα
then we have (proposition 3.1 in [8]).
Proposition A.1 {(DUα
Vα, ϕα)}α∈∆is a C∞atlas over DM
X.
Definition A.2 (Deformation to the normal cone) Let X⊂Mbe as above. The
set DM
Xequipped with the C∞structure induced by the atlas of X-slices is called the
deformation to the normal cone associated to the embedding X⊂M.
One important feature about the deformation to the normal cone is the functoriality.
More explicitly, let f: (M, X )→(M′, X′) be a C∞map f:M→M′with f(X)⊂X′.
Define D(f) : DM
X→DM′
X′by the following formulas:
1) D(f)(m, t) = (f(m), t) for t6= 0,
2) D(f)(x, ξ, 0) = (f(x), dNfx(ξ),0), where dNfxis by definition the map
(NM
X)x
dNfx
−→ (NM′
X′)f(x)
induced by TxMdfx
−→ Tf(x)M′.
Then we have, (proposition 3.4 in [8]),
Proposition A.3 D(f) : DM
X→DM′
X′is a C∞-map. In the language of categories, the
deformation to the normal cone construction defines a functor
D:C∞
2−→ C∞,(A.6)
where C∞is the category of C∞-manifolds and C∞
2is the category of pairs of C∞-
manifolds.
In [33], Paterson properly defined the notion of continuous family groupoids, for which
all above considerations and concepts apply immediately. For more details on this the
reader might consult Paterson original paper or [19] where the authors also developed the
appropriate pseudodifferential calculus. In particular, we can define the Connes tangent
groupoid in this context:
Definition A.4 (Tangent groupoid of a continuous family groupoid) Let G⇒G(0)
be a continuous family groupoid. The tangent groupoid associated to Gis the groupoid that
has
DG
G(0) =NG
G(0) × {0}GG×R∗
as the set of arrows and G(0) ×Ras the units, with:
35
1. sT(x, η, 0) = (x, 0) and rT(x, η, 0) = (x, 0) at t= 0.
2. sT(γ, t) = (s(γ), t)and rT(γ, t) = (r(γ), t)at t6= 0.
3. The product is given by mT((x, η, 0),(x, ξ , 0)) = (x, η +ξ, 0) and mT((γ, t),(β, t)) =
(m(γ, β ), t)if t6= 0 and if r(β) = s(γ).
4. The unit map uT:G(0) →GTis given by uT(x, 0) = (x, 0) and uT(x, t) = (u(x), t)
for t6= 0.
We denote GT=DG
G(0) and AG=NG
G(0) as a vector bundle over G(0). Then we have
a family of continuous family groupoids parametrized by R, which itself is a continuous
family groupoid
GT=AG× {0}GG×R∗⇒G(0) ×R.
As a consequence of the functoriality of the deformation to the normal cone, one can show
that the tangent groupoid is in fact a continuous family groupoid compatible with the
continuous family groupoid structures of Gand AG(considered as a continuous family
groupoid with its vector bundle structure).
Proposition A.5 Let Gbe a continuous family groupoid together with an injective con-
tinuous family groupoid morphism Gh
−→ RN. Consider also the induced infinitesimal
continuous family groupoid morphism A(G)A(h)
−→ RN. Assume that A(h)is also injective
and that both crossed product groupoids, Ghand A(G)A(h)are free and proper. Then the
induced morphism Gtan hT
−→ RNgives a free proper crossed groupoid as well.
Proof : We will use again properness caracterization (ii) of proposition 2.14 in [35].
In particular we have to verify only property (A) (of section 3.2 above), that is, in our
case we have to check that the map
Gtan ×RN(t,s)//(G(0) ×[0,1] ×RN)×(G(0) ×[0,1] ×RN) (A.7)
given by (((γ, ǫ), X )7→ ((t(γ), ǫ, X),(s(γ), ǫ, X +h(γ)
ǫ))
((x, ξ), X )7→ ((x, 0, X),(x, 0, X +A(h)(x, ξ))) (A.8)
is closed.
Let (An)n:= ( ˜γn, Xn)na sequence in Gtan ×RNsuch that
limn→∞(t, s)( ˜γn, Xn) = P(A.9)
with Pa point in (G(0) ×[0,1] ×RN)×(G(0) ×[0,1] ×RN). It is enough to justify
that there is a subsequence of (An)nconverging to an antecedent of P: The point Pis
of the form ((x, ǫ1, X),(y, ǫ2, Y )). The first consequence of (A.9) is that ǫ1=ǫ2, hence
P= ((x, ǫ, X),(y, ǫ, Y ))
We will separate the analysis in two cases:
(a) The case ǫ6= 0: By the explicit form of (A.8), we can assume (or there is a subse-
quence) that (An)n⊂G×(0,1] ×RN, i.e., that the elements os the sequence are of
the form An= (γn, ǫn, Xn) with ǫ6= 0. But then we have the following convergences:
t(γn)→x,ǫn→ǫ,Xn→X,s(γn)→yand Xn+h(γn)
ǫn.
In particular we obtain that h(γn)→ǫ·(Y−X), and since Ghis proper we have
that there is a subsequence of (γnk)kof (γn)nand a γ∈Gsuch that γnk→γwith
t(γ) = x,s(γ) = yand h(γ) = ǫ·(Y−X). In particular, letting A= (γ , ǫ, X) we
have that Ank→Aand (t, s)(A) = P.
36
(b) The case ǫ= 0: In this case we have two subcases:
(b1) There is a subsequence of (An)nentirely contained in A(G)×RN. In this
case we might assume that An= ((xn, ξn), Xn)∈Axn(G)×RN. Then, (A.9)
implies that xn→x=y,Xn→Xand Xn+A(h)(xn, ξn)→Y. In particular,
A(h)(xn, ξn)→Y−Xand since A(G)A(h)is proper we have that there is a
subsequence (xnk, ξnk) converging in A(G) to an element (x, ξ)∈Ax(G). Then
letting A= ((x, ξ), X ) we have that Ank→Aand (t, s)(A) = P.
(b2) There is a subsequence of (An)nentirely contained in G×(0,1] ×RN. In this
case we might assume that An= (γn, ǫ, Xn)∈G×(0,1] ×RN. Then, (A.9)
implies that t(γn)→x,ǫn→0, Xn→X,s(γn)→yand Xn+h(γn)
ǫn→Y.
This implies h(γn)
ǫn→Y−Xin RNor in other words (h(γn), ǫn)→(Y−X, 0)
in (RN)tan. In particular we have also that h(γn)→0 in RNand since Ghis
proper and his injective, we deduce that x=yand that there is a subsequence
(γnk)kof (γn)nsuch that γnk→x. Now, from the injectivity of A(h) and the
fact that h(γn)
ǫn→Y−Xwe deduce that there is an unique ξ∈Ax(G) such that
A(h)(x, ξ) = Y−X. Finally, letting A= ((x, ξ), X ) we have that Ank→A
and (t, s)(A) = P.
✷
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