K-duality for pseudomanifolds with isolated singularities

Journal of Functional Analysis 219 (2005) 109–133

K -duality for pseudomanifolds with isolated singularities

Claire Debord and Jean-Marie Lescure

Laboratoire de Mathe´matiques, Universite´ Blaise Pascal, Complexe universitaire des Ce´zeaux, 24 Av. des Landais, 63177 Aubie`re cedex, France

Received 1 March 2004; accepted 11 March 2004 Available online 17 June 2004 Communicated by A. Connes

Abstract

We associate to a pseudomanifoldXwith a conical singularity a differentiable groupoidG which plays the role of the tangent space ofX:We construct a Dirac element and a dual Dirac element which induce aK-duality between theC-algebrasCðGÞandCðXÞ:This is a first step toward an index theory for pseudomanifolds.

r2004 Elsevier Inc. All rights reserved.

Keywords:Singular manifolds; Smooth groupoids; Kasparov bivariantK-theory; Poincare´ duality

0. Introduction

A basic point in the Atiyah–Singer index theory for closed manifolds lies in the isomorphism:

KðVÞ-KðTVÞ; ð1Þ

induced by the map which assigns to the class of an elliptic pseudodifferential operator on a closed manifoldV; the class of its principal symbol[2].

To prove this isomorphism, Kasparov and Connes and Skandalis [6,15], define two elements D_VAKKðCðVÞ#C₀ðTVÞ;CÞ and l_VAKKðC;CðVÞ#C₀ðTVÞÞ

Correspondingauthor.

E-mail addresses:[email protected] (C. Debord), [email protected].

fr (J.-M. Lescure).

0022-1236/$ - see front matterr2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jfa.2004.03.017

which induce aK-duality betweenCðVÞandC0ðTVÞ;i.e.

l_V

#

CðVÞ

D_V¼1_C0ðTVÞ and l_V

#

C0ðTVÞ

D_V¼1_CðVÞ: Isomorphism (1) is then equal toðlV#CðVÞÞ:

Moreover, Connes and Skandalis recover the Atiyah–Singer index theorem using thisK-duality together with other tools coming from bivariantK-theory (wrong-way functoriality maps).

This notion of K-duality, also called Poincare´ duality in K-theory, has a quite general meaning [14,5]: two C-algebras A and B are K-dual if there exist DAKKðA#B;CÞandlAKKðC;A#BÞsuch that

l

#

A

D¼1_BAKKðB;BÞ and l

#

B

D¼1_AAKKðA;AÞ:

We usually callDa Dirac element and l a dual Dirac element. A consequence of these equalities is that for anyC-algebrasCandE;the groups homomorphisms:

l

#

A

: KKðA#C;EÞ-KKðC;B#EÞ;

l

#

B

: KKðB#C;EÞ-KKðC;A#EÞ are isomorphisms with inversesð#BDÞandð#ADÞ:

It is a natural question to look for a generalization of the K-duality between a manifold and its tangent bundle for spaces less regular than smooth manifolds.

Pseudomanifolds[9]offer a large class of interesting examples of such spaces. We have focused our attention on the model case of a pseudomanifoldXwith a conical isolated singularityc:We use bivariantK-theory, groupoids and pseudodifferential calculus on groupoids to prove a Poincare´ duality inK-theory in this context.

Let us explain our choice of the algebrasAandB:As in the smooth case we take A¼CðXÞ:ForB;we need to define an appropriate notion of tangent space forX:It should take into account the smooth structure ofX\fcg and encodes the geometry of the conical singularity. This problem finds an answer no longer in the category of vector bundles but in the larger category of groupoids. Thus, we assign to X a smooth groupoidG;the tangent space of X, and we letBbe the non-commutative C-algebraCðGÞ:

The definition of the tangent spaceG of X is actually motivated by the case of smooth manifolds. In particular, the concrete meaningof the isomorphism (1) was the initial source of inspiration.

The regular partX\fcgidentifies to an open subset ofG^ð0Þand the restriction ofG to this subset is the ordinary tangent space of the manifoldX\fcg:The tangent space

‘‘over’’ the singular point is given by a pair groupoid. Furthermore the orbits space G^ð0Þ=GofGis topologically equivalent toX;that isCðXÞCCðG^ð0Þ=GÞ:ThusCðXÞ

maps to the multiplier algebra of CðGÞ: The Dirac element is defined as the Kasparov product D¼ ½C #@ where ½C is the element of KKðCðXÞ#

CðGÞ;CðGÞÞ comingfrom the multiplication morphism C and @ is an element ofKKðCðGÞ;CÞcomingfrom a deformation groupoidGofGin a pair groupoid.

This auxiliary groupoidGis the analogue of the tangent groupoid defined by Connes for a smooth manifold[5].

The construction of the dual Dirac element l is more difficult. We let X_b be the bounded manifold with boundaryL which identifies with the closure ofX\fcg inG^ð0Þ;it satisfiesXCX_b=Land we denote byAGthe Lie algebroid of the tangent space G: We first consider a suitable K-oriented map X_b-AGX_b: This map leads to an element l⁰ of KKðC;CðAGÞ#CðXbÞÞ: The adiabatic groupoid of G (see [5,16,17]), provides an element Y of KKðCðAGÞ;CðGÞÞ and we define l⁰⁰¼l⁰#CðAGÞY: The element l⁰⁰ can be seen as a continuous family ðl⁰⁰_xÞ_xAXb where l⁰⁰_xAK0ðCðGÞÞ: An explicit description of l⁰⁰ shows that its restriction to L is the class of a constant family: that means l⁰⁰ determines an elementlAKKðC;CðGÞ#CðXÞÞ:

An alternative and naı¨ve description oflis the following. To each pointyofX is assigned an appropriate open subset Oˆy of G^ð0Þ satisfying KðCðGjOˆyÞÞCZ: We construct a continuous familyðb_yÞ_yAX;whereb_yis a generator ofKðCðGjOˆyÞÞ:This family gives rise to an element ofKðCðGXjOˆÞÞ; where Oˆ is an open subset of G^ð0ÞX:We obtainl by pushingforward this element inKðCðGXÞÞwith the help of the inclusion morphism ofCðGXjOˆÞin CðGXÞ:

The dual Dirac element has the two followingimportant properties:

(i) The set Oˆ-X₁X₁ is in the range of the exponential map. Here X₁ is the complement of a conical open neighborhood ofc:

(ii) The equality l#CðGÞ@¼1AK⁰ðXÞholds.

These two properties ofl are crucial to obtain our main result:

Theorem. The Dirac element D and the dual-Dirac elementlinduce a Poincare´ duality between CðGÞand CðXÞ:

All our constructions are obviously equivariant under the action of a group of automorphisms of X; that is homeomorphisms of X which are smooth diffeomorphisms of X\fcg: In previous works, Julgand Kasparov and Skandalis [12,13]investigated theK-duality for simplicial complexes. Our approach is more in the spirit of [15] since it avoids the use of a simplicial decomposition of the pseudomanifold. We hope that it is better suited for applications to index theory.

Indeed, the K-duality gives an isomorphism between KKðCðXÞ;CÞ and KKðC;CðGÞÞwhich is, as in the smooth case, the map which assigns to the class of an elliptic pseudodifferential operator the class of itssymbol. This point, among connections with the analysis on manifolds with boundary or conical manifolds, will be discussed in a forthcomingpaper.

This paper is organized as follows:

Section 1 is devoted to some preliminaries aroundC-algebras of groupoids and specialKK-elements.

In Section 2, we define thetangent space Gof a conical pseudomanifoldX as well as thetangent groupoidGofX:

In Section 3, we define the Dirac element and in Section 4, we construct the dual Dirac element.

The Section 5 is devoted to the proof of the Poincare´ duality.

We want to address special thanks to Georges Skandalis for his always relevant suggestions.

1. Preliminaries

1.1. C-algebras of a groupoid

We recall in this section some useful results aboutC-algebras of groupoids[18,5].

LetG4^s

r G^ð0Þbe a smooth Hausdorff groupoid with sourcesand ranger:IfU is any subset ofG^ð0Þ;we let:

G_U :¼s¹ðUÞ; G^U :¼r¹ðUÞ and G_U^U ¼Gj_U :¼G_U-G^U:

We denote byC_c^NðGÞthe space of complex valued smooth and compactly supported functions onG:It is provided with a structure of involutive algebra as follows. Iff andgbelongtoC_c^NðGÞwe define

theinvolutionby

for gAG; fðgÞ ¼fðg¹Þ;

theconvolution product by

for gAG; f gðgÞ ¼ Z

ZAG^rðgÞ

fðZÞgðZ¹gÞ:

To give a sense to the integral above, we fix a Haar system forG;that is, a smooth familyfl^x;xAG^ð0Þgof left invariant measures onGindexed byxAG^ð0Þsuch that the support ofl^x isG^x:

Alternatively, one could replaceC_c^NðGÞ by the space C^N_c ðG;L¹²Þ of compactly supported smooth sections of the line bundle of half densities L¹² over G: If k denotes the dimension of thes(orr) fibers ofG;the fiberL

1

2govergAGis defined to be the linear space of maps:

r:L^kðTgðG^rðgÞÞÞ#L^kðTgðGsðgÞÞÞ-C;

such thatrðlvÞ ¼ jlj¹²rðvÞfor alll inRandvinL^kðTgðG^rðgÞÞÞ#L^kðTgðGsðgÞÞÞ:

Then, the convolution product makes sense as the integral of a 1-density on the manifoldG^rðgÞ:Both constructions lead to the sameC-algebra.

For eachxinG^ð0Þ;we define a-representationpxofC_c^NðGÞon the Hilbert space L²ðGxÞby

pxðfÞðxÞðgÞ ¼ Z

ZAG^rðgÞ

fðZÞxðZ¹gÞ;

wherexAL²ðGxÞ;fAC_c^NðGÞandgAG_x:

The completion ofC_c^NðGÞfor the normjjfjj_r¼sup_xAG^ð0ÞjjpxðfÞjjis aC-algebra, called the reducedC-algebra ofG and denoted byC_rðGÞ:

The maximalC-algebraCðGÞis the completion of C^N_c ðGÞfor the norm:

jjfjj ¼supfjjpðfÞjj jp Hilbert space representation of C_c^NðGÞg:

The previous constructions still hold when the groupoid G is smooth only in the orbit direction, which means that Gj_Ox is smooth for any orbit O_x¼rðs¹ðxÞÞ;

xAG^ð0Þ:In this situation one can replaceC^N_c ðGÞbyCcðGÞ:

The identity map of C_c^NðGÞ induces a surjective morphism from CðGÞ onto C_rðGÞ:The injectivity of this morphism is related to amenability of groupoids [1].

WhenG is an amenable groupoid, its reduced and maximalC-algebras are equal and, moreover, this commonC-algebra is nuclear.

1.2. Subalgebras and exact sequences of groupoid C-algebras

To an open subsetOofG^ð0Þcorresponds an inclusioni_OofC_c^NðGj_OÞintoC^N_c ðGÞ which induces an injective morphism, again denoted byiO;fromCðGj_OÞintoCðGÞ:

WhenOis saturated,CðGj_OÞis an ideal ofCðGÞ:In this case, F:¼G^ð0Þ\Ois a saturated closed subset ofG^ð0Þ and the restriction of functions induces a surjective morphism rF from CðGÞ to CðGj_FÞ: Moreover, accordingto[10], the following sequence ofC-algebras is exact:

0-CðGj_OÞ !^iO CðGÞ !^rFCðGj_FÞ-0:

1.3. C-modules arising from bundles and groupoids

Let us now consider an hermitian bundle E on G^ð0Þ: We equip the space C_c^NðG;rEÞwith theCðGÞ-valued product:

/f;gSðgÞ ¼ Z

ZAG^rðgÞ

/fðZ¹Þ;gðZ¹gÞS_sðZÞ:

This endows C_c^NðG;rEÞ with a structure of CðGÞ-pre-Hilbert module and we denote by CðG;EÞ the corresponding CðGÞ-Hilbert module. As usual, we note

LðEÞ and KðEÞ the C-algebras of (adjointable) endomorphisms and compact endomorphisms of any Hilbert moduleE:

1.4. KK-tools

This paper makes an intensive use of Kasparov’s bivariant K-theory. The unfamiliar reader may consult [3,14,19]. In this section, we recall some basic constructions and fix the notations.

When A is a C-algebra, the element 1_AAKKðA;AÞ is the class of the triple ðA;i_A;0Þ; where A is graded by A^ð1Þ¼0 and i_A:A-LðAÞ is given by iðaÞb¼ ab; a;bAA:

IfB andC are additionalC-algebras, t_C:KKðA;BÞ-KKðA#C;B#CÞis the group homomorphism defined byt_C½ðE;r;FÞ ¼ ½ðE#C;r#i_C;F#1Þ :

The heart of Kasparov theory is the existence of a product which generalizes various functorial operations in K-theory. Recall that the Kasparov product is a well defined bilinear coupling KKðA;BÞ KKðB;CÞ-KKðA;CÞ denoted ðx;yÞ/x#By which is associative, covariant in C; contravariant in A and satisfies:

* fðxÞ#Ey¼x#BfðyÞ for any -homomorphism f :B-E; xAKKðA;BÞ and yAKKðE;CÞ:

* x#B1B¼1A#Ax¼x;forxAKKðA;BÞ:

* tDðx#ByÞ ¼tDðxÞ#B#DtDðyÞ;whenxAKKðA;BÞandyAKKðB;CÞ:

In the sequel, we will denote simply x#y the product x#ByAKKðA;CÞ when xAKKðA;BÞandyAKKðB;CÞ:

The operation tC:KKðA;BÞ-KKðA#C;B#CÞ allows the construction of the general form of the Kasparov product:

KKðA1;B₁#CÞ KKðA2#C;B₂Þ-KKðA1#A₂;B₁#B₂Þ; ð2Þ

ðx;yÞ/x

#

C

y:¼t_A2ðxÞ#t_B1ðyÞ: ð3Þ ForxAKKðA;B#CÞandyAKKðB#C;EÞ; there is an ambiguity in the definition oft_BðxÞ#t_BðyÞ: it can be defined by (3) withB¼A₂¼B₁ or by t_Bðx#yÞ: These two products are different in general. Indeed, in the first case, the two copies ofB involved inxandyplay different roles, contrary to the second case. To remove this ambiguity, we adopt the following convention:

tBðx#yÞ ¼tBðxÞ#tBðyÞ

and

x

#

C

y¼t

B%ðxÞ#tBðyÞ or tBðxÞ#t

B%ðyÞ:

Moreover, letfB:B#B-B#B;a#b/b#abe theflipautomorphism and let½fB

be the correspondingelement ofKKðB#B;B#BÞ:The morphismfBexchanges the two copies ofB;so

t_BðxÞ#t_C½f_B #t_BðyÞ ¼x

#

C

y:

1.5. KK-elements associated to deformation groupoids We explain here a classical construction[5,10].

A smooth groupoidGis called adeformation groupoidif:

G¼G1 f0g,G2 0;1 4G^ð0Þ¼M ½0;1 ;

whereG1andG2are smooth groupoids with unit spaceM:That is,Gis obtained by gluingG2 0;1 4M 0;1 which is the groupoidG2overMparameterized by 0;1 with the groupoidG1 f0g4M f0g:

In this situation one can consider the saturated open subset M 0;1 of G^ð0Þ: Usingthe isomorphisms CðGj_{M 0;1} ÞCCðG2Þ#C₀ð 0;1 Þ and CðGj_Mf0gÞC CðG1Þ; we obtain the followingexact sequence ofC-algebras:

0-CðG2Þ#C₀ð 0;1 Þ^{iM 0;1} ! CðGÞ !^ev0CðG1Þ-0;

wherei_M 0;1 is the inclusion map and ev0is theevaluation mapat 0, that is ev0is the map comingfrom the restriction of functions toGj_Mf0g:

We assume now thatCðG1Þis nuclear. Since theC-algebraCðG2Þ#C0ð 0;1 Þis contractible, the longexact sequence in KK-theory shows that the group homomorphismðev0Þ¼ #½ev0 :KKðA;CðGÞÞ-KKðA;CðG1ÞÞis an isomorph- ism for eachC-algebraA:

In particular withA¼CðGÞwe get that½ev0 is invertible inKK-theory: there is an element ½ev01

in KKðCðG1Þ;CðGÞÞ such that ½ev0 #½ev01

¼1_CðGÞ and

½ev01#½ev0 ¼1_CðG1Þ:

Let ev₁:CðGÞ-CðG2Þbe the evaluation map at 1 and½ev1 the corresponding element ofKKðCðGÞ;CðG2ÞÞ:

TheKK-element associated to the deformation groupoid Gis defined by d¼ ½ev01#½ev1AKKðCðG1Þ;CðG2ÞÞ:

Example 1. (1) LetGbe a smooth groupoid and let AGbe its Lie algebroid.

The adiabatic groupoid ofG[5,16,17]:

Gad¼AG f0g,G 0;1 4G^ð0Þ ½0;1 ;

is a deformation groupoid. Here, the vector bundlep:AG-G^ð0Þis considered as a groupoid in the obvious way.

SinceC0ðAGÞ is nuclear, the previous construction applies and the associated KK-element dAKKðC0ðAGÞ;CðGÞÞgives rises to a map:

#d:K₀ðC0ðAGÞÞ-K₀ðCðGÞÞ

This map is defined in[16]as the analytic index of the groupoidG:

(2) A particular case of (1) is given by the tangent groupoid of R: TanðRÞ ¼ RR 0;1 ,TR f0g4R ½0;1 [5]. The correspondingKK-element d_B; which belongs to KKðC0ðR²Þ;KÞ is the dual Bott element. Precisely, the map ð#d_BÞ induces an isomorphism fromKðC0ðR²ÞÞintoKðKÞCZ:

1.6. Pseudodifferential calculus on groupoids

We recall here some definitions and results of[4,16,17,20](see also [6]).

LetGbe a smooth groupoid, possibly with a boundary [16].

Let Ug:C^NðGsðgÞÞ-C^NðGrðgÞÞ be the isomorphism induced by right multi- plication: U_gfðg⁰Þ ¼fðg⁰gÞ: An operator P:C_c^NðGÞ-C^NðGÞ is a G-operator if there exists a familyPx:C_c^NðGxÞ-C^NðGxÞsuch thatPðfÞðgÞ ¼PsðgÞðfj_GsðgÞÞðgÞand UgP_sðgÞ¼P_rðgÞUg:

AG-operatorPis apseudodifferential operator on G(resp. of orderm) if for any open local chart F:O-sðOÞ W of G such that s¼pr₁3F and any cut-off function wAC^N_c ðOÞ; we have ðFÞ¹ðwPwÞ_xF¼aðx;w;DwÞ where aASðsðOÞ TWÞis a classical symbol (resp. of order m).

One says thatKCGis asupport ofPif suppðPfÞCK:suppðfÞfor allfAC^N_c ðGÞ: WhenPhas a compact support we say thatPisuniformly supported.

These definitions extend immediately to the case of operators actingon sections of bundles on G^ð0Þ (pulled back to G with r), and we denote by CðG;EÞ the algebra of uniformly supported pseudodifferential operators onGactingon sections ofE:Thanks to the invariance property, each operatorPACðG;EÞhas a principal symbol sðPÞAC_c^NðSG;hompEÞ where SG is the sphere bundle associated to AG and p its natural projection onto G^ð0Þ: The followinginclusions hold:

C⁰ðG;EÞCLðCðG;EÞÞ and C¹ðG;EÞCKðCðG;EÞÞ: Moreover, the symbol map extends by continuity and gives rise to the following exact sequence of C- algebras:

0-KðCðG;EÞÞ-C0ðG;EÞ !^s C0ðSG;hompEÞ-0: ð4Þ

where C0ðG;EÞ ¼C⁰ðG;EÞ^LðC

ðG;EÞÞ

: Finally a linear section Op_G of the symbol map can be defined by the followingformula:

Op_GðaÞðuÞðgÞ ¼ Z

xAA_rðgÞG;

g⁰AGsðgÞ

e^{i/E1Gðg0g1Þ;xS}aðrðgÞ;xÞfðg⁰g¹Þprðg⁰Þ;rðgÞuðg⁰Þdg⁰dx:

HerefAC^NðGÞis supported in the range of an exponential mapEG:VðAGÞ-G whereVðAGÞdenotes a small neighborhood of the zero section inAG;moreoverf is assumed to be equal to one on a neighborhood of V in G: We have used a parallel transportp to get local trivializations of the bundleE:It is implicit in the formula that the symbolaAC_c^NðSG;hompEÞhas been extended in the usual way toAG:

2. The geometry

Let X₁ be an m-dimensional compact manifold with boundary L: We attach to each connected componentL_i of the boundary the cone cL_i¼L_i ½0;1 =Li f0g;

usingthe obvious map Li f1g-LiC@X1: The new space X¼0^pi¼1cLi,X1 is a compact pseudomanifold with isolated singularities [9]. In general, there is no manifold structure around the vertices of the cones. From now on, we assume thatL is connected, i.e.X has only one singularity denoted byc:The general case follows by exactly the same methods.

For anyeA 0;1 ;we will refer toc_eL¼L ½0;e =L f0gas a compact cone overL;

toce

o L¼L ½0;e½=L f0gas an open cone overLand we letXe¼L ½e;1 ,X1: We define the manifoldMby attachingtoX1a cylinderL 1;1 :We fix onMa Riemannian metric which is of product type onL 1;1 and we assume that its injectivity radius is bigger than 1.

We will use the followingnotations:M^þdenotesL 0;1 ,X₁;M_þ its closure in M andM¼L 1;0½:Ifyis a point of the cylindrical part ofM orX\fcg;we will write y¼ ðyL;k_yÞ where y_LAL and k_yA 1;1 are the tangential and radial coordinates. We extend the map k on X1 to a smooth definingfunction for its boundary; in particular,k¹ð1Þ ¼@X₁ andkðX1ÞC½1;þN½:

2.1. The tangent space of the conical pseudomanifold X

Let us considerTM^þ;the restriction toM^þof the tangent bundle ofM:As aC^N vector bundle, it is a smooth groupoid with unit spaceM^þ:We define the groupoid Gas the disjoint union:

G¼MM,T M^þ4^s

r M;

whereMM4M is the pair groupoid.

In order to endowGwith a smooth structure, compatible with the usual smooth structure onMM and onTM^þ;we have to take care of what happens around points ofTM^þj_@Mþ:

Lettbe a smooth positive function on 1;þN½such thatt¹ðf0gÞ ¼R^þ:We let

*

tbe the smooth map fromM toR^þ given bytðyÞ ¼* tðkyÞ:

Let ðU;fÞ be a local chart for M around zA@M^þ: Setting U¼U-M and U^þ¼U-M^þ;we define a local chart ofGby

f* :UU,T U^þ-R^mR^m;

fðx;* yÞ ¼ fðxÞ;fðyÞ fðxÞ

* tðxÞ

if ðx;yÞAUU ð5Þ and

fðx;* VÞ ¼ ðfðxÞ;ðfÞðx;VÞÞ elsewhere:

Let us explain why the range of ffe is open. We can assume that fðUÞ ¼R^m and fðUÞ ¼R^m¼R^m1 N;0½:LetBbe a open ball inR^m:Sinceettvanishes onU^þ there exists an open neighborhood W of @U^þ such that fettðxÞpþfðxÞ jxA W-U; pABgCR^m:ThenffðT@Ue ^þÞ-R^mBCfðWÞ BCImeff:

We define in this way a structure of smooth groupoid onG:

Remark 2. (1) IftisC^l then the atlas defined above providesGwith a structure of C^l1 groupoid (it is easy to see that the source, target and inversion maps have the same regularity as the atlas).

(2) At the topological level, the space of orbitsM=GofGis equivalent toX: there is a canonical isomorphism between the algebrasCðXÞandCðM=GÞ:

Definition 3. The smooth groupoidG4M is called a tangent space ofX:

It is important to remark that the Lie algebroid ofG4Mis the bundleAG¼TM over M with anchor p_G:AG¼TM-TM;ðx;VÞ/ðx;tðxÞV* Þ; in particular pG is the zero map in restriction to TM^þ: The exponential map exp of the Riemannian manifold M provides an exponential map EG for the groupoid G ( for a description of exponential maps for groupoids, see e.g. [17,7]). More precisely:

E_G:VðTMÞ-G;

EGðy;VÞ ¼ ðy;VÞ when yAM^þ

and

EGðy;VÞ ¼ ðy;expyðtðyÞV* ÞÞwhen yAM;

whereVðTMÞ ¼ fðy;VÞATMj jj*tðyÞVjjo1 and exp_yðtðyÞV* ÞÞAM if yAMg:

The map EG is a diffeomorphism onto a neighborhood of the unit space M in G: In fact, we could have defined the smooth structure of G usingthe mapEG:

Remark 4. (1) There exists a slightly different groupoid which could naturally play the role of the tangent space ofX: We will call it the tangent space with tail. It is defined by

G_q¼LLTð 1;0½Þ,T M^þ4M:

As a groupoid, G_q is the union of two groupoids: the bundleTM^þ4M^þ and the groupoid LLTð 1;0½Þ4L 1;0½¼M which is the product of the pair groupoid over L with the vector bundle Tð 1;0½Þ4 1;0½: One can equip Gq

with a smooth structure similarly as we did forG:We will see that theC-algebras of GandG_q areKK-equivalent.

(2) The groupoidGis obtained by gluing along their common boundaryTLR the groupoids T M^þ and a groupoid isomorphic to TanðLÞsR_þ obtained by the action of R_þ (by multiplication on the real parameter) on the tangent groupoid TanðLÞ ¼LLR_þ,TL f0gofL:The groupoidG_qis defined in the same way except that we consider the trivial action ofR_þ:

2.2. The tangent groupoid of the pseudomanifold X

The following construction is a natural generalization of the tangent groupoid of a manifold defined by Connes [5]. We define the tangent groupoid G of the pseudomanifold X as a deformation of the pair groupoid over M into the groupoid G: This deformation process has a nice description at the level of Lie algebroids. Indeed, the Lie algebroid of G should be the (unique) Lie algebroid given by the fiber bundle AG¼ ½0;1 AG¼ ½0;1 TM over ½0;1 M; with anchor map

p_G:AG¼ ½0;1 TM - Tð½0;1 MÞ ¼T½0;1 TM

ðl;x;VÞ / ðl;0;x;p_Gðx;VÞ þlVÞ ¼ ðl;0;x;ð*tðxÞ þlÞVÞ:

Such a Lie algebroid is almost injective, thus it is integrable[7,8].

We now define thetangent groupoid:

G¼MM 0;1 ,G f0g4M ½0;1 ; whose smooth structure is described hereafter.

Since G¼ ðMM ½0;1 \ðM^þM,MM^þÞ f0gÞ,T M^þ f0g; we keep the smooth structure on MM ½0;1 \ðM^þM,MM^þÞ f0g as an open subset in the manifold with boundaryMM ½0;1 :We consider the followingmap:

r:VðTM ½0;1 Þ-G¼MM 0;1 ,G f0g;

rðz;V;lÞ ¼ ðz;V;0Þ if zAM^þ and l¼0;

ðz;expzðð*tðzÞ þlÞ:VÞ;lÞ elsewhere;

(

where VðTM ½0;1 Þ is an open subset in TM ½0;1 such that VðTM

½0;1 Þ-TM f0g ¼VðTMÞ; and which is small enough so the exponential in the definition ofris well defined. ThenTM^þ f0gis in the image ofrand we equipG aroundTM^þ f0gwith the smooth structure for whichris a diffeomorphism onto its image. One can easily check that it is compatible with the smooth structure of MM ½0;1 \ðM^þM,MM^þÞ f0g:

The Lie algebroid ofGisAGandris an exponential map forG:

2.3. The C-algebras

Letmbe the Riemannian measure onM and letnbe the correspondingLebesgue measure on the fibers ofTM:The familyfl^x;xAMg;wheredl^xðyÞ ¼_*tðyÞ^1mdmðyÞifx belongs toM;anddl^xðVÞ ¼dnðVÞifxis inM^þ;is a Haar system forG:We use this Haar system to define the convolution algebra ofG:

Remark 5. (1) Gis a continuous field of amenable groupoids parameterized by X: More precisely,G¼0xAXp¹ðxÞwherep:G-X is the obvious projection map. If xac; p¹ðxÞ ¼TxM is amenable. If x¼c; p¹ðcÞ ¼MM,TM^þj_@Mþ is isomorphic to the groupoid H ¼TanðLÞsR of an action of R on the tangent groupoid TanðLÞ ¼LL 0;1½,TL f0g ofL: The groupoidH is an extension of the groupRby TanðLÞ;both of them beingamenable, accordingto[1, Theorem 5.3.14],H is amenable. Finally accordingto[1, Proposition 5.3.4],G is amenable.

In the same way, G and G_q are amenable. Hence their reduced and maximal C-algebras are equal and they are nuclear.

(2) UsingtheKK-equivalence betweenKandC0ðR²Þ(cf. Example 1(2)) one can establish aKK-equivalence betweenCðGÞandCðGqÞ:

3. The Dirac element

The tangent groupoidG4M ½0;1 is a deformation groupoid and itsC-algebra is nuclear, thus it defines aKK-element. We let@@ebe theKK-element associated toG:

More precisely:

e

@

@¼ ½e01#½e1AKKðCðGÞ;KÞ;

where e₀:CðGÞ-CðGj_Mf0gÞ ¼CðGÞ; the evaluation map at 0 is K-invertible, ande₁:CðGÞ-CðGj_Mf1gÞ ¼KðL²ðMÞÞis the evaluation map at 1. Letbbe the (positive) generator ofKKðK;CÞCZ: We set@¼@@e#b:

The algebra CðXÞ is isomorphic to the algebra of continuous functions on the orbits spaceM=GofG:ThusCðXÞmaps to the multiplier algebra ofCðGÞand we letCbe the morphismC:CðGÞ#CðXÞ-CðGÞinduced by the multiplication. In other words, ifaAC_c^NðGÞandfACðXÞ;Cða;fÞACcðGÞis defined by

Cða;fÞðgÞ ¼ aðgÞfðrðgÞÞ ¼aðgÞfðsðgÞÞ if gAT M^þ; aðgÞfðcÞ if gAMM: (

We denote by½C the correspondingelement inKKðCðGÞ#CðXÞ;CðGÞÞ:

Definition 6. The Dirac element is

D¼ ½C #@AKKðCðGÞ#CðXÞ;CÞ:

4. The dual Dirac element

We first recall the construction of the dual Dirac element for a compact manifold V [6,15].

Let V be a smooth compact n dimensional Riemannian manifold, whose injectivity radius is at least 1. We denote by L the bundle of complex valued differential forms on V; and we keep this notation for its pull-back to TV and its restrictions to various subsets of V and TV: For xAV; we denote by Ox the geodesic ball with radius¹₄;Hxthe Hilbert space L²ðOx;LÞand we writeH for the continuous field of Hilbert spacesS

xAVH_x:

With a model operator onRⁿ;for instance those given in Theorem (19.2.12) of [11], we define a continuous family P¼ ðPxÞ_xAV of pseudodifferential operators P_xAC⁰ðOx;LÞof order 0 satisfyingthe followingconditions:

(1) P_x istrivial at infinity of O_x; which means that P_x is the sum of a compactly supported pseudodifferential operator and a smooth bounded section of the bundle EndL-O_x;(in particular, this ensures boundedness onH_x).

(2) Pxis selfadjoint onHx;and has degree one (i.e.Px¼ _P^0þ

x

P_x 0

Þwith respect to the grading induced byL¼L^ev"L^odd:

(3) P²_xId is a compactly supported pseudodifferential operator of order 1; in particular it is compact on Hx:

(4) the familyP¼ ðPxÞ_xAV has a trivial index bundle of rank one. In fact, for allx;

P^þ_x is onto, it has a one-dimensional kernel and there exists a continuous section V{x/exAkerP^þ_xCL²ðM;LÞ:

Here the continuity of the family means thatP is an endomorphism of theCðVÞ- Hilbert moduleH:

We let ax be the principal symbol of Px: Under the assumptions above, the Kasparov module

lx¼ ½ðC0ðTOx;LÞ;1;axÞ is a generator ofK₀ðC0ðTO_xÞÞCZ:The followingelement:

lV ¼ ½ðC0ðTOx;LÞ;1;axÞ_xAVAK0ðC0ðTVVÞÞ

is the dual Dirac element used in the proof the Poincare´ duality betweenCðVÞand C₀ðTVÞ[15].

There is an alternative elegant description of l_V [6]. Let us consider the map f:V-TVV;x/ððx;0Þ;xÞ:This map isK-oriented so it gives rise to an element f!AKKðCðVÞ;C₀ðTVVÞÞ:Ifpdenotes the obvious mapC-CðVÞ;then:

lV ¼ ½p #f!:

With the previous example in mind and keepingthe same notations, we shall define an elementDAK₀ðCðGXÞÞwhose evaluationl¼ ðe0ÞðDÞAK₀ðCðGXÞÞwill be the appropriate dual Dirac element of the pseudomanifoldX:

We define a maph:X\fcgCM^þ-Mwhich pushes points inM:More precisely, hðyÞ ¼y whenkyX1;andhðyÞ ¼ ðyL;lðkyÞÞotherwise, where:

lðkÞ ¼ 3k2 if 1=2pkp1;

1=2 if 0okp1=2:

From now we fixeA 0;1=2½:Recall thatXe¼ fxAXjkxXeg:We set:

d¼ ðlhðxÞÞtA½0;1 ;xAXeAK0ðC0ðAðGÞ XeÞÞ:

Here AðGÞCTM ½0;1 and d corresponds to the K-oriented map: ½0;1 Xe-TM ½0;1 Xe;ðt;xÞ/ððhðxÞ;0Þ;t;xÞ:

Next, let us consider the adiabatic groupoid ofG[5,16,17] (see Example 1(1)):

H¼ f0g AðGÞ, 0;1 G4½0;1 G^ð0Þ:

We letYAKKðC0ðAðGÞÞ;CðGÞÞbe theKK-element associated toH; i.e.

Y¼ ½ev01#½ev1 ;

where ev0:CðHÞ-C0ðTM ½0;1 Þ is the evaluation map at 0 composed with the Fourier transform CðAðGÞÞ !^CC0ðAðGÞÞ; and ev1:CðHÞ-CðGÞ is the

evaluation at 1. We defineDeAK0ðCðGXeÞÞby D_e¼d

#

C0ðAðGÞÞ

Y:

Proposition 7. (1)The elementDe satisfies

ðe1ÞðDeÞ ¼1XeAK⁰ðXeÞCK₀ðK#CðXeÞÞ; where e1:CðGÞ-CðGj_Mf1gÞCKis the evaluation map at 1.

(2)There existsD0AK0ðCðGXj_O½0;1 ÞÞextending De;that is, r3ðiO½0;1 ÞðD0Þ ¼D_e;

whereOis the open subsetS

xAM^þOhðxÞ fxg,Mc^oLofMX;iO½0;1 :CðG Xj_O½0;1 Þ-CðGXÞ is the inclusion morphism and r:CðGXÞ-CðGXeÞ is the restriction morphism.

Proof. (1) Let us noteOe¼S

xAXe O_hðxÞ fxg:This is an open subset ofMX_ethe unit space of the groupoidGXe:LetOOeee be its lift toM ½0;1 ²Xewhich is the unit space of the groupoidHXe:We letObe the groupoid:

O¼HX_ejOOeee

:

In fact,Oidentifies with the adiabatic groupoid of O1¼GXej_Oe½0;1 :

We shall use the pseudodifferential calculus onOto get an explicit representant of D_e: The family ðahðxÞÞ_xAXe depends smoothly on x and defines a symbol aAS⁰ðAðOÞ;EndLÞ: Note that this symbol is independent of the two real parameters comingfrom the lift ofOe to OOeee:Let Op_O be a quantification map for O: Thanks to the properties of this calculus and the fact that each ahðxÞðy;xÞis of order 0, trivial at infinity (that is independent ofx near the infinity of O_hðxÞ) and a²_hðxÞðy;xÞ 1 is of order1 and vanishes near the infinity ofO_hðxÞ;we deduce from the exact sequence (4):

Op_OðaÞALðCðO;LÞÞ and Op²_OðaÞ IdAKðCðO;LÞÞ:

Hence, we get an element ½ðCðO;LÞ;Op_OðaÞÞ AK₀ðCðOÞÞ which gives using the inclusionOCHX_e an elementDDeAK₀ðCðHX_eÞÞsatisfying

ðev0ÞðeDDÞ ¼dAK0ðC0ðTM ½0;1 XeÞÞ:

Hence,

De¼ ðev1ÞðeDDÞ ¼ ½ðCðO1;LÞ;Op_O1ðaÞÞ ;

whereO1¼GXej_Oe½0;1 :Now consider the evaluation mape1:CðGÞ-K:We get ðe1ÞðDeÞ ¼ ½ðCðO1;1;LÞ;Op_O1;1ðaÞÞ AK0ðK#CðXeÞÞ;

where we have setO_1;1¼GX_ej_Oef1g:Note that O_1;1¼ [

xAXe

O_hðxÞO_hðxÞ fxgCðMMÞ X_e

and Op_O1;1 is an ordinary quantification map which assigns to a symbol living on TO_hðxÞCAðOhðxÞO_hðxÞÞ a pseudodifferential operator on O_hðxÞ: Since Pj_Xe¼ ðPhðxÞÞ_xAXehas symbol equal toðahðxÞÞ_xAXeand has a trivial index bundle of rank one, the followingholds:

ðe1ÞðDeÞ ¼ ½ðCðO1;1;LÞ;Op_O1;1ðaÞÞ ¼ ½ðHj_Xe;Pj_XeÞ ¼1XeAK⁰ðXeÞ:

(2) The existence ofD₀ follows immediately from:

Lemma 8. If r_L:CðXeÞ-CðLÞ ðL¼@X_eÞdenotes the restriction homomorphism,and i_M½0;1 :K#Cð½0;1 ÞCCðGj_M½0;1 Þ-CðGÞthe inclusion morphism,then

ðrLÞðDeÞ ¼ ði_M½0;1 Þð1LÞ;

where1Lis the unit of the ring K0ðK#Cð½0;1 LÞÞCK⁰ðLÞ:

Proof. The elementðrLÞðDeÞis represented by

ðCð@O1;LÞ; @PÞAEðC;CðGLÞÞ where @O₁¼S

ðt;xÞA½0;1 LO_hðxÞO_hðxÞ fðt;xÞgCðMMÞ ½0;1 L and

@P¼ ðPhðxÞÞ_xAL: SinceCð@O1;LÞ is also a K#Cð½0;1 LÞ-Hilbert module, we observe that

x¼ ½ðCð@O1;LÞ; @PÞ AK0ðK#Cð½0;1 LÞÞ

is such that ði_M½0;1 ÞðxÞ ¼ ðrLÞðDeÞ: Moreover, under the isomorphism K⁰ðLÞCK0ðK#Cð½0;1 LÞÞ; the element x is represented by: ðHhðxÞ;PhðxÞÞ_xAL; which also represents the unit element 1LAK⁰ðLÞthanks to the triviality of the index bundle of the familyðPhðxÞÞ_xAL: &

By a slight abuse of notation, GX_ej_O½0;1 ; Gc_eLj_O½0;1 and GLj_O½0;1 will denote, respectively, the restrictions of GX to O ½0;1 -MX_e ½0;1 ; O

½0;1 -Mc_eL ½0;1 ¼Mc_eL ½0;1 and O ½0;1 -M@X_e ½0;1 ¼ ML ½0;1 :

It is obvious from the concrete description ofDe that it comes from an element De;OAK0ðCðGXej_O½0;1 ÞÞ via the inclusion morphism. Now let x0AK0ðCðG

ceLj_O½0;1 ÞÞbe the pushforward of 1AK0ðCÞvia the obvious homomorphism:

KCCðMMÞ-CðMMÞ#Cð½0;1 c_eLÞCCðGc_eLj_O½0;1 ÞÞ:

The precedinglemma and the Mayer-Vietoris exact sequence inK-theory associated to the followingcommutative diagram:

show that there exists D₀AK₀ðCðGXj_O½0;1 ÞÞ satisfying rðD0Þ ¼D_e;O and ðrcLÞðD0Þ ¼x₀: &

Remark 9. In fact, ðrLÞðDe;OÞ may be represented by the trivial vector bundle kerPj_L¼S

xALkerP_hðxÞ-L while x0 may be represented by the product vector bundle cLC-cL: The element D₀ is obtained by gluing these bundles along L¼L f1gCcL:This involves a bundle isomorphism kerPj_LCLC;which yields a continuous mapc:L-GL₁ðCÞand a class½c AK¹ðLÞ:One could be more careful with the construction of the family ðPxÞ to make sure that ½c ¼0; otherwise one may perturb D₀ by elements comingfrom K¹ðLÞ: That will be done in the next proposition.

Proposition 10. There existsD_OAK₀ðCðGXj_O½0;1 ÞÞsuch that:

(1) r3ðiO½0;1 ÞðD_OÞ ¼De;

(2) ðe1Þ3ðiO½0;1 ÞðD_OÞ ¼1XAK⁰ðXÞCK0ðCðXÞ#KÞ:

Proof. Firstly, we note that

r3ðe1Þ3ðiO½0;1 ÞðD0Þ ¼ ðe1Þ3r3ðiO½0;1 ÞðD0Þ ¼ ðe1ÞðDeÞ ¼1_Xe ¼rð1XÞ:

From the exact sequence:

0-C₀ðce

o LÞ !^j CðXÞ !^r CðXeÞ-0 and the previous computation, we deduce

ðe1ÞðD0Þ 1XAImðjÞ:

We choosey₀AK¹ðLÞCK₀ðCðce

o LÞ#KÞsuch that jðy0Þ ¼ ðe1ÞðD0Þ 1X:

Moreover one deduces from the fact that the inclusion KðL²ðMÞÞCKðL²ðMÞÞ induces an isomorphism inK-theory, that

ðe1Þ3i:K0ðCðGce

o Lj_O½0;1 ÞÞ-K0ðK#C0ðce o LÞÞ

is an isomorphism. Here Gce

o Lj_O½0;1 is the restriction of GX to O ½0;1 - M ½0;1 c^o_e Landi:CðGc^o_e Lj_O½0;1 Þ-CðGXÞis the inclusion morphism.

Now let yye00AK0ðCðGce

o Lj_O½0;1 ÞÞ be the unique element such that ðe1Þ3iðyye00Þ ¼ y0:We set:

D_O¼D₀þjðyye₀₀Þ;

where we have still denoted byjthe inclusion morphism fromCðGce

o Lj_O½0;1 Þto CðGXj_O½0;1 Þ:ThenD_OAK0ðCðGXj_O½0;1 ÞÞsatisfies (1) and (2). &

We define

D¼ ðiO½0;1 ÞðDOÞAK0ðCðGXÞÞ:

Definition 11. The dual Dirac element lAK0ðCðGÞ#CðXÞÞ of the singular manifoldX is defined by

l¼ ðe0ÞðDÞ;

wheree0:CðGÞ-CðGÞis the evaluation homomorphism at 0.

We have proved the following:

Proposition 12. (1) The following equality holds:

l

#

CðGÞ

@¼1XAK0ðCðXÞÞCK⁰ðXÞ;

where1X is the unit of the ring K0ðCðXÞÞ:

(2)For each open subsetOa;0oao1;of MX defined by

Oa¼ fðx;yÞAMXjdðx;yÞo1;ky40g,fðx;yÞAMXjkxoa;kyoag;

the dual Dirac elementl belongs to the range of:

ðiOaÞ:K0ðCðGXj_OaÞÞ-K0ðCðGXÞÞ:

Roughly speaking, everyOa contains the ‘‘support’’ ofl:From now on, we choose O¼O1=2 andella preimage of lforðiOÞ:

5. The Poincare´ duality

This section is devoted to the proof of our main result:

Theorem 13. The Dirac element D and the dual Dirac element l induce a Poincare´

duality between CðGÞand CðXÞ;that is (1) l

#

CðGÞ

D¼1CðXÞAKKðCðXÞ;CðXÞÞ;

(2) l

#

CðXÞ

D¼1CðGÞAKKðCðGÞ;CðGÞÞ:

5.1. Computation ofl#CðGÞD

Let m:CðXÞ#CðXÞ-CðXÞ be the morphism of C-algebras induced by multiplication of functions.

Lemma 14. The following equality holds:

l

#

CðGÞ

½C ¼l

#

CðXÞ

½m :

Proof. Accordingto Proposition 12 we have thatl#CðGÞ½C ¼tCðXÞðlÞ#½H* 0 and l#CðXÞ½m ¼t_CðXÞðlÞ#½* H˜₁; where H₀ and H˜₁ are the morphisms from CðG Xj_OÞ#CðXÞtoCðGXÞdefined by

H0ðB#fÞðg;yÞ ¼fðrðgÞÞBðg;yÞ and H˜1ðB#fÞðg;yÞ ¼fðyÞBðg;yÞ;

when fACðXÞ; BAC_cðGXj_OÞ; ðg;yÞAGX; and CðXÞ is identified with the algebra of continuous functions onM which are constant onM:

There is an obvious homotopy betweenH˜1 andH1 defined by H1ðB#fÞðg;yÞ ¼fðhðyÞÞBðg;yÞ;

wherehis the map constructed in Section 4 to push points inM:

LetW¼ fðx;yÞAMX;dðx;yÞo1 and kyXegviewed as a subset ofM^þM^þ: Let c be the continuous function W ½0;1 -M^þCX such that cðx;y;Þ is the geodesic path going fromxtoy whenðx;yÞAW:

We obtain an homotopy betweenH0 andH1 by settingfor eachtA½0;1 : H_tðB#fÞðg;yÞ ¼ fðcðrðgÞ;hðxÞ;tÞÞBðg;xÞ if kxX1=2;

fðcÞBðg;xÞ elsewhere:

&

Now we are able to compute the productl#CðGÞD:

l

#

CðGÞ

D¼ l

#

CðGÞ

½C

!

#

CðGÞ

@¼ l

#

CðXÞ

½m

!

#

CðGÞ

@

¼t_CðXÞðlÞ# ½m

#

C

@

¼t_CðXÞðlÞ# @

#

C

½m

¼tCðXÞ l

#

CðGÞ

@

!

#½m ¼1CðXÞ:

The equality of the first line results from the previous lemma, the equality of the second line comes from the commutativity of the Kasparov product overCand the last equality follows from Proposition 12. This finishes the proof of the first part of Theorem 13.

Let us notice the followingconsequence: for everyC-algebrasAandB;we have:

#

CðGÞ

D

!

3 l

#

CðXÞ

!

¼Id_{KKðCðXÞ#A;BÞ} and l

#

CðGÞ

!

3

#

CðXÞ

D

!

¼Id_{KKðA;B#CðXÞÞ}: In particular, this implies the followinguseful remark:

Remark 15. The equality ð#CðGÞDÞðl#CðXÞDÞ ¼ ð#CðGÞDÞ3ðl#CðXÞÞðDÞ ¼ D ensures that l#CðXÞD1_CðGÞ belongs to the kernel of the map ð#CðGÞDÞ:KKðCðGÞ;CðGÞÞ-KKðCðGÞ#CðXÞ;CÞ:

5.2. Computation ofl#CðXÞD

The purpose here is to prove that l

#

CðXÞ

D¼t_CðGÞðlÞ#t_CðGÞð½C #@Þ ¼1_CðGÞ:

This problem leads us to study the invariance of l#CðXÞ½C under the flip automorphismf˜ofCðGGÞCCðGÞ#CðGÞ:

Indeed, we have

t_CðGÞð½C Þ#½f˜ #tCðGÞð@Þ ¼ ½C

#

C

@¼@

#

C

½C ¼t_CðGÞ#CðXÞð@Þ#½C ;

which implies (cf. Proposition 12):

l

#

CðXÞ

½C

!

#½f˜

!

#tCðGÞð@Þ ¼ l

#

CðGÞ

@

!

#

CðXÞ

½C ¼1_CðGÞ:

Hence,

l

#

CðXÞ

D1_CðGÞ¼ l

#

CðXÞ

½C

!

#ð½id ½f˜ Þ

!

#tCðGÞð@Þ ð6Þ

and the invariance ofl#CðXÞ½C under½f˜ would enable us to conclude the proof.

Such an invariance would be analogous to Lemma 4.6 of[15]. Unfortunately, we are not able to prove thatl#CðXÞ½C is invariant under the flip.

PutC¼L 1;1½andF¼MM\CC:

We letO* be the inverse image ofOby the canonical projection ofMM-M X:We denoteði_O*Þ:KKðCðGÞ;CðGGj_O*ÞÞ-KKðCðGÞ;CðGGÞÞthe morph- ism correspondingto the inclusioni_O*:A simple computation shows that:

Lemma 16. The elementl#CðXÞ½C belongs to the image of ði_O*Þ:

The setF-O* is an open and symmetric subset ofMM; hence the flip makes sense onCðGGj_F-O*Þ:

Lemma 17. The flip automorphism of CðGGj_F-O*Þis homotopic to identity.

Proof. The setF-O* ¼ fðx;yÞAMMjdðx;yÞo1; k_xX1 or k_yX1gis a subset of M^þM^þ;so the algebraCðGGj_F-O*Þis isomorphic toC0ðTðF-OÞÞ* :To prove the lemma, it is sufficient to find a proper homotopy between the flip f_F-O* of TðF-OÞ* and id_TðF-OÞ* :

The exponential map ofM provides an isomorphismf between TðF-OÞ* and

ðTMlÞ^"3;whereMl¼ fxAMjkxXlgfor some 0olo1:Via this isomorphism, the

flip becomes the automorphism of C₀ððTM_lÞ^"3Þ induced by the map g: ðx;X;Y;ZÞAðT_xM_lÞ³/ðx;X;Z;YÞ:One can take for example:f:ðx;y;X;YÞ/

ðmðx;yÞ;exp¹_mðx;yÞðxÞ exp¹_mðx;yÞðzÞ;Tðx;y;XÞ;Tðy;x;YÞÞ;wheremðx;yÞ ¼exp_xð^exp₂^{1x y}Þ is the middle point of the geodesic joiningxto yandTðx;y;Þ: TxM-Tmðx;yÞM is the parallel transport alongthe geodesic joiningxtomðx;yÞ:

LetA:½0;1 -SO₃ðRÞbe a continuous path from

1 0 0

0 0 1

0 1 0

0

@

1 Ato

1 0 0 0 1 0 0 0 1 0

@

1 A:

The map½0;1 ðTMlÞ^"3-ðTMlÞ^"3;ðt;x;VÞ/ðx;At:VÞis a proper homotopy between identity andg: &