The Mobius Band and the Mobius Foliation

The Mobius Band and the Mobius F oliation Ioannis P . ZOIS Sc ho ol of Natural Sciences, Departmen t of Mathematics The American College of Greece, Deree College 6 Gra vias Sstreet, GR- 153 42 Aghia P arask evi, A thens, Greece e-mail: izois@acg.gr Septem b er 4, 2021 Abstract Some years ag o we intro duced a new top ological inv a riant for fo- liated manifolds using techniques from noncommut ative geo metr y , in particular the pairing b etw een K- The o ry and cyclic c o homology . The motiv ation came from flat principa l G -bundles where the base s pace is a non simply connected manifold. The computation o f this inv ariant is quite complica ted. In this article we try to per form certa in computa- tions for the Mobius band (or Mo bius foliation) which is a n interesting nontrivial exa mple of foliations; this example has a key fea tur e: it is the simplest case of a la rge class of ex amples of foliations, that of bun- dles with discr ete structure gro ups which also includes the folia tions given by flat vector (or G -principal) bundles. W e shall see that the Mobius foliation example also helps o ne to under stand ano ther larg e class of examples of foliations coming from gr o up actions on ma nifolds which are not free. P ACS class ification: 11.1 0 .-z; 11.15.-q; 11.30.- Ly Keywords: Noncommutativ e Geo metry , F olia ted Manifolds. 1 In tro duction In this article we study the Mobious b an d in an attempt to p erform a non- trivial computation for an op erator algebraic in v ariant for foliations intro- duced in [8]. The construction of this in v ariant is quite complicat ed, it 1 in v olv es v arious stages: fi rst on e has to determine th e holonom y group oid of the foliation, then fin d its co rr esp onding C ∗ -algebra, then compute its K-Theory and its cyclic cohomolog y and construct n atural classes in b oth and fi nally apply Connes’ pairing b et we en K-Theory an cyclic cohomology . In this preliminary version of the final article, we sh all p resen t some of these steps f or the example of the Mobius band (or the Mobiu s foliation) a long with some questions. But b efore that w e s h all r ecall some basic f acts ab out foliations (throughout this w ork all manifolds are assu med to b e smo oth). Let M b e a sm ooth connected and closed manifold of dimension m . A co dim- q (and hence of dimension ( m − q )) f olia tion on M is giv en b y an in- te gr able subbund le V of the tangen t b undle T M of M where the d im en sion of the fibre of V is ( m − q ). Quite often V is called the tangen t bun d le of the foliatio n as opp osed to th e qu otient bund le T M /V whic h is called the transv erse bun dle of the foliation. The effect is that giv en a V as ab o v e, M can b e written as the disjoint union of the lea v es of the foliati on which are immersed , connected submanifolds of M , all of the same dimension equal to ( m − q ). The top ology of the lea ves m ay v ary drastically: some ma y b e compact but others not and more imp ortan tly their f undamen tal group s are different. F r om these t wo differences one can see that foliations are im- p ortan t generalisations of the total space of fibre bu ndles b ecause in a fibre bund le the total sp ace is the disjoin t union of the fi bres wh ich are essen tially the same manifold (ie homo eomorphic or diffeomorph ic) to a fixed mod el manifold called typica l fibre. If the foliation h as a trans verse bund le wh ich can b e oriente d, an equiv- alen t lo cal d efi nition of a co dim- q foliation is giv en by a n onsingular decom- p osable q -form ω s atisfying the in tegrabilit y condition ω ∧ dω = 0. The lea v es are the su bmanifolds wh ose tangent v ectors v anish on ω . By the F rob enius theorem the set of smo oth sections of V denoted C ∞ ( V ) form a Lie subalge- bra of the Lie algebra C ∞ ( T M ) of vect or fields of M (seen as sections of its tangen t bu ndle). Du ally , the annihilator ideal I ( V ) of V consisting of differ- en tial forms v anish ing on the lea ves (ie on sections of V ) is closed u nder the de Rham differen tial d , n amely since the annihilator ideal is a graded ideal w e write d ( I ∗ ( V ) ⊆ I ∗ +1 ( V ) . In the codim-1 case one can sho w that this annihilator ideal of V can b e generated by ω itself and thus the int egrabilit y relation ω ∧ dω = 0 is equ iv alen t to dω = ω ∧ θ wh ere θ is another 1-form whic h h as th e p rop ert y that it is close d when r e stricte d on every le af , thus defining a class in the fi rst cohomology group of ev ery leaf. This 1-form θ is 2 sometimes called the (partial) flat Bott connection on the tr ansv erse bu ndle of the foliatio n. Moreo v er for an y co dimension dθ ∈ I 2 ( V ) . The form ω can b e m ultiplied with a no where v anishing fu nction f without c hanging the f olia tion. The effect it w ill ha v e on θ is that w e add an exact form. Th us the cohomolog y class that θ d efines on every leaf do es not c hange. The Go dbillon-V ey class of the foliation V is the real (2 q + 1) de Rham cohomology class θ ∧ ( dθ ) q and it do es not d ep en d on the coices of ω and θ , it only d ep ends on V . I ts existence follo ws fr om Bott’s v anishing theorem: if a cod im- q subb u ndle V of T M is integ rable, then the Po ntrjagin classes of th e transv erse bund le T M /V v anish in degrees greater than t w ise the co dimension. The holonomy of the flat Bott c onne ction on every le af is the infinitesimal part of the germinal holonomy of that p articular le af ; to quan- tify th is information ab out th e in finitesimal germinal h olonom y of a leaf in the co dim-1 case one can either tak e the trace or the determinant comp osed with the logarithm; in th e later case one obtains the cohomology class that θ defines in the 1st d e R h am cohomology group of the sp ecific leaf (see [1] page 66). Moreo ve r θ also defines a 1st tangen tial cohomology class. But one has to realise that θ is closed only when restricted to a particular leaf and it carries the infinitesimal information of the holonom y of that particular leaf. The GV-class ho we ve r is a real cohomology class of M wh ic h carries information ab out the infi nitesimal h olonom y of the foliation as a whole (ie someho w the GV-class is an a ve rage o v er all lea ves of the det comp osed with log of the infinitesimal germin al holonom y of eac h one of th em and the tfcc is an av erage ov er all lea ves of the traces of the infinitesimal germinal holonom y since cyclic cohomology cont ains traces; remaining is to see what is the a v erage o ver all lea v es of the Ra y-Singer analytic torsion). Du m in y’s theorem sa ys that in the co dim-1 case only the resilient lea ve s cont ribu te to the GV-class. The simp lest example of foliated manifolds is the C artesian pro duct of t w o manifolds M × N . The second example is submersions: let P and M b e smo oth manifolds of d im en sion p an d m ≤ p resp ectiv ely and let f : P → M b e a submer- sion, namely r ank ( d f ) = m . By the implicit f unction theorem f ind uces a co dim- m foliation on P where the lea ve s are the comp onents of f − 1 ( x ) for x ∈ M . Fibr e bund les are particular examples of this s ort. The thir d nontrivial example of foliations is giv en b y bund les with dis- 3 cr ete structur e gr oup . Let π : P → M b e a differen tiable fib re bun d le with fibre F and p = dim ( P ), q = dim ( F ) and m = dim ( M ); clearly p = q + m . A bundle is defined by an open co v ering { U a } a ∈ A of M , diffeomorphisms h a : π − 1 ( U a ) → U a × F called lo cal trivialisatio ns and transition functions g ab : U a ∩ U b → Dif f ( F ) su c h that h a ◦ h − 1 b ( x, y ) = ( x, g ab ( x )( y )) and moreo v er the transition fu nctions satisfy the co cycle relation in triple in- tersections g ab ◦ g bc = g ac . If the transition functions are lo cally constan t , then the bu ndle is said to ha ve discrete structure group. Under this assump- tion the co dim- q foliations of π − 1 ( U a ) giv en b y the sub m ersions fi t nicely together to give a foliation on P . Flat ve ctor or prin cipal G -bundles with G a compact and connected Lie group are of this sort, namely v ector bundles or pr incipal G -bu ndles equipp ed with a flat connection (a connection with zero curv ature). Ev ery suc h bu ndle can b e constru cted in the follo wing wa y: let φ : π 1 ( M ) → D if f ( F ) b e a group homomorph ism and w e d enote by G the image of π 1 ( M ) into D if f ( F ) via φ . Moreo v er let ˜ M denote the unive rsal co vering s pace of the base manifold M . Then π 1 ( M ) acts jointly on the pro duct sp ace ˜ M × F as follo ws: it acts via dec k transformations on ˜ M and b y φ on F . So w e can define P := ˜ M × π 1 ( M ) F . Th is action is free and prop erly d iscon tinuous, hence P is a foliated manifold of codim - q (and hence of dimension m ) where q = dim ( F ). T he lea ve s lo ok lik e many v alued cross-sections of the bund le π : P → M and in f act π restricted to an y leaf is a co vering map. T o see this, n ote that if L x is the leaf through the p oin t corresp onding to ˜ M × { x } ⊂ ˜ M × F , then L x is diffeomorphic to ˜ M /G x where G x = { g ∈ π 1 ( M ) : φ ( g )( x ) = x } d enotes th e isotrop y group at x . The simplest case of a b undle with discrete stru cture group is the Mobius b and . W e shall fo cus on this example here. W e take the base space M to b e S 1 with lo cal coord inate s and we kn o w that π 1 ( S 1 ) = Z , and the univ er- sal co v ering space of S 1 is R . Then Z acts on R via dec k transformations s 7→ s + 1 Th e fib re F will b e R with local co ordinate denoted r . Then we pic k φ ∈ D if f ( R ) to b e given b y φ ( r ) = − r for r ∈ R . Then th e p ro d uct space R × R has a Z action giv en by ( s, r ) 7→ ( s + 1 , − r ). The quotien t space by this Z action is the Mobius b and M := R × Z R . Eac h leaf L r is a circle wrapping around twic e except for the core circle (corresp onding to r = 0) whic h w raps aroun d only once. What ab out our lo cal d efinition of foliations? W ell, for th e Mobiu s band w e ha ve that its transv erse bun dle is n ot orien table so there do es n ot exist 4 a definition inv olving 1-forms. An imp ortan t remark now: the Mobius band M can b e considered in t w o wa ys: either as the total space of a vec tor b undle o ve r S 1 with fibre R . In this case if w e squeeze every fibre to a p oin t we get of course as a result S 1 as the quotien t s pace and K i ( S 1 ) = Z with i = 0 , 1. Ho w eve r if w e consider the Mobius b and as a foliated manifold as ab ov e and we squ eeze ev ery leaf to a p oin t we get R r ≥ 0 as the quotien t s pace and w e kno w that K i ( R r ≥ 0 ) = 0 for i = 0 , 1 (h ere since the space is only lo cally compact we ha v e to use K-Theory with compact supp orts). Next we wa nt to compute the holo nomy gr oup oid of the Mobius band but b efore doing that let us briefly recall the k ey notion of holonom y for foliations and ho w these data can b e organised to wh at is call ed the germinal holonom y group oid of the f olia tion. The holonom y essentia lly tell s us how lea v es assem ble together to give the foliation and it is the most imp ortan t in trinsing notion for foliations. It enco des information ab out the fu n damen tal groups of the lea ves (wh ic h as w e emphasised ab o v e th ey can v ary enormous ly from leaf to leaf in sharp con trast to the fi b res in a fi bre bund le whic h are th e same manifold) along with information ab out how the lea v es fi t nicely together in order to ha v e the manifold as their disjoint union. The k ey difference with fibr e b undles here is the spiralling phenomenon: lea v es ma y sp iral rep eatedly o v er eac h other without intersect ing. Let x b e a p oin t on a foli ated manifold sa y M and let L x denote the unique leaf trough the p oint x . Then the h olonomy group G x x o v er the p oint x is defined to b e the image of th e surj ecti ve homomorphism h x : π 1 ( L x ) → D if f ( F , x ) where F is a transversal and Dif f ( F , x ) denotes th e germs of lo cal diffeomorphisms at x wh ic h fi x x . Th is map h is precisely the holonom y of th e f olia tion which giv es diffeomorphism s b et w een transversals follo wing the plaque to plaque p ro cess using regular foliated atlases (“sliding transver- sals along lea v es”). Note the similarities w ith the holonom y of a connection on a v ector bun dle: the fibr es are the tr an s v ers als and the plaque to plaque pro cess corresp onds to parallel transp ort of ve ctors using the connection (for more details see for example [1]). No w one can simply consider the disj oint union of all the holonom y groups G x x for all p oin ts x of the foliated manif old sa y M and this forms a group oid called the germinal holonomy gr oup oid of the foliation. A group oid can b e d efined as a small category with in v erses and the space of ob jects is 5 the foliated manifold M itself (for more details see [2] or [4]). P erhaps a more concise notation is the follo w ing: the holonom y group oid of a foliation V on a manifold M as a set consists of trip les G := { ( x, h, y ) : x ∈ M , y ∈ L x , h the holonom y class of a path (u s ually a lo op) fr om x to y } . W e no w turn to the case of a b u ndle with discrete structure group P , with fibr e F and base smanifold M with discrete structure group giv en by the group homomorphism φ : π 1 ( M ) → D if f ( F ) as describ ed ab o v e; th e total space is defin ed via P := ˜ M × π 1 ( M ) F where π 1 ( M ) acts via de ck tr ans- formations on ˜ M and by φ on F . Let G denote the image of π 1 ( M ) into D if f ( F ) und er φ and for eac h x ∈ F let G x := { g ∈ G : g x = x } d enote the isotrop y group at x a nd let G x := { g ∈ G : g y = y for all y in some neigh b orho o d of x in F } denote the stable isotrop y group at x . Th e leaf L x through x (whic h is the image of ˜ M × { x } in P ) can b e expressed as L x = ˜ M /G x where G x acts on ˜ M by dec k transformation. Then G x is a normal subgroup of G x and the holonom y group G x x of the foliation o ver x is simply G x x = G x /G x . Let now m ∈ M b e a basep oin t and ˜ m ∈ ˜ M b e a preimage of m and let N b e the image of ˜ m × F in P . Th e map ˜ m × F → N is a diffeomorph ism since π 1 ( M ) acts freely on ˜ M , so N is a cop y of the fib re F sitting as a complete transv ers al in the foliated manif old P . Thus w e see th at all bundles with discrete structure group admit a complete transversal . This is imp ortan t b ecause if a f oliation admits a co mplete transv ers al sa y N , b oth its h olonom y group oid and its C ∗ -algebra c ompletion simplify drastically by th e Hilsum-Sk andalis th eorem: namely the holonom y group oid reduces to G N N whic h is simply the restriction of the full h olonom y group oid o v er the co mplete tr an s v ersal N (if w e see the h olonom y group oid of the foliation as a small category with in v erses with ob jects P , then G N N is a full su b categ ory with ob j ects N ) and th e C ∗ -completion of the holonom y group oid is Morita Equiv alen t to the C ∗ - completion of just G N N . Hence all w e h a ve to unders tand is G N N . In fact G N N is completely determined by the action of G on F (remem b er that G is the image of π 1 ( M ) in to D if f ( F ) under φ ). More pr ecisely one has the follo w in g homo eomorphism of top ological group oids: G N N ∼ = ( F × G ) / ∼ where the equiv alence relation is give n by ( x, γ ) ∼ ( y , δ ) if and only if x = y and δ − 1 γ lies in the stable isotrop y group G x . Perhaps a b etter w a y to 6 rewrite the ab o ve w ould b e that G N N = F ⋊ φ G and th us it is clear that the corresp ondin g C ∗ -algebra to this foliation will b e Morita equiv alent to C 0 ( F ) ⋊ φ G . If w e hav e the particular case of a flat principal H -bu ndle o v er M where H is the stru cture Lie group, th en the holono my of th e flat connection d efi nes a map h : π 1 ( M ) → H and clearly in our discussion ab o v e G w ill b e the homomorphic image of the fund amen tal group on to H via h , n amely G = h ( π 1 ( M )) ⊂ H and hence th e corresp onding C ∗ -algebra to the foliation will b e Morita equ iv alen t to C 0 ( H ) ⋊ h G . W ould lik e to see what groups can app ear as holonom y group s of flat connections and if the action of th e holonom y h they has fi xed p oints (well, it can hav e as we s ee from teh Mobius foliatio n b elo w), s ince b oth these issues are imp ortan t in order to compute the K 0 group of the corresp onding crossed pro duct al gebra. No w for the Mobius band M := R × Z R foliated by circles, these circles corresp ond to the images of R × { r } f or v arious v alues of r : if r 6 = 0 then π 1 ( L r ) = π 1 ( S 1 ) = Z acts trivially on D if f ( R , r ) and hence G r r = 0 (note these circles wr ap arroun d twice b efore they close). Ho w ev er the holon- om y group G 0 0 of th e m iddle circle which wraps around only once is th e group Z 2 since the diffeomorphism φ ( r ) = − r wh ich creates M do es lie in D if f ( R , 0) and φ 2 = 1. Thus th e group G = Z 2 acts non-fr e e ly since 0 is a fixed p oin t. Ho w ev er this fixed p oin t is isolate d with no interior and th us we h a ve that G N N = R × Z 2 and this is the holonom y group oid of the Mobius foliation as a top ological s p ace. No w Z 2 = {± 1 } b ut w e d enote these t wo elemen ts as e = 1 for the identit y elemen t and ǫ = − 1 the other one. If w e w ant to tak e th e m ultiplication into acc ount as w ell, this will b e Γ = R ⋊ φ Z 2 where the action φ of Z 2 on to R is giv en by “flipping the sign”. Its C ∗ -algebra completion is then A := C 0 ( R ) ⋊ φ Z 2 where C 0 ( R ) denotes con tinuous complex v alued functions defin ed on R whic h v anish at 0 and infin ity . Let us recall that as a linear sp ace C 0 ( R ) ⋊ φ Z 2 consists of con tin uous maps F : Z 2 → C 0 ( R ). T he p rod uct in C 0 ( R ) ⋊ φ Z 2 is giv en by ( F 0 ∗ F 1 )( ξ ) P n ∈ Z / 2 F 0 ( n ) φ ( n ) F 1 ( n − 1 ξ ) where n, ξ ∈ Z 2 , and φ ( e ) = e f or the identit y elemen t wh ereas φ ( ǫ )( F )( x ) = F ( − x ). Let us b e a little more exp licit : since M lo call y looks lik e S 1 × R , w e c ho ose local coord inates ( s, r ) as ab o ve. Th en if π : M → S 1 is th e bun dle pro jection, we pic k as a complete tr ansv ersal N the s p ace π − 1 (0) wh ich is just a cop y of R . Th en the holonomy group oid G of the Mobius foliation according to what we mentio ned ab o v e for arbitrary b u ndles with discrete structure group is homo eomorphic to simply G N N where N is a complete 7 transv ersal. The next order of bu s siness is to compu te the K-Th eory of the group oid C ∗ -algebra completio n. Th is is complicated b ecause t his algebra is n on unital and h ence we ha v e to attac h a unit and then thro w it aw a y . W e shall use the fact that in general an y short exact sequence of algebras 0 → J → E → B := E /J → 0 giv es r ise to a 6-term long exact sequence in K-Theory: K 0 ( J ) − − − − → K 0 ( E ) − − − − → K 0 ( E /J )   y   y   y exp K 1 ( E /J ) ← − − − − K 1 ( E ) ← − − − − K 1 ( J ) (1) W e shall apply this in oredr to compute the 0th K-group of the algebra C 0 ( R ) ⋊ φ Z 2 corresp onding to the Mobius foliation. Remem b er that the group Z 2 action on R has 0 as a fixed p oint; consider the map ev 0 : C 0 ( R ) → C whic h is giv en b y e v aluating functions at the (fixed p oin t) zero. Then one h as th e follo wing short exact sequ ence (1): 0 → C 0 ( R − ) ⊕ C 0 ( R + ) ֒ → C 0 ( R ) → C → 0 where R − = ( −∞ , 0), R + = (0 , ∞ ), C 0 ( R + ) denotes contin u ou s fun ctions v anishing b oth at 0 and + ∞ (and similarly for th e − sign). Clearly sin ce R − ∼ = R + ∼ = R are homo eomorphic, then C 0 ( R − ) ⊕ C 0 ( R + ) ∼ = C 0 ( R ) ⊕ C 0 ( R ). Using the follo wing we ll-kno wn results that K 0 ( C ) = Z , K 1 ( C ) = 0, K 0 ( C 0 ( R )) = 0 and K 1 ( C 0 ( R )) = Z along with the additivit y prop ert y of the K-functor w e get the follo wing corresp onding K-Theory 6-term l.e.s.: 0 − − − − → 0 − − − − → Z   y   y   y exp 0 ← − − − − Z ← − − − − Z 2 (2) Since 0 is a fixed p oin t we can readily inco rp orate the Z 2 action on to the first s.e.s. and w e directly ge t the seco nd s.e.s (namely that the map ev aluation at p oin t 0 is compatible with the Z 2 -action): 0 → ( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 ֒ → C 0 ( R ) ⋊ φ Z 2 → C ⋊ φ Z 2 → 0 8 W e k n o w that C ⋊ φ Z 2 is isomorphic to C ⊕ C and hen ce their K-groups are equal. F r om what w e men tioned ab o v e ab out the K-groups of C and the additivit y of the K-fu nctor we can hence deduce that K 0 ( C ⋊ φ Z 2 ) = Z 2 = Z ⊕ Z and that K 1 ( C ⋊ φ Z 2 ) = 0. Moreo ve r ( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 is isomorph ic to M 2 ( C 0 ( R )), the isomorphism b eing denoted Ψ wh ic h tak es an F ∈ C 0 ( R − ) ⊕ C 0 ( R + ) ⋊ φ Z 2 and it is mapp ed to Ψ( F ) =  ψ − ( F 1 ( e )) ψ − ( F 1 ( ǫ )) ψ + ( F 2 ( ǫ )) ψ + ( F 2 ( e ))  where ψ ± : C 0 ( R ± ) ∼ = C 0 ( R ) with ψ + ( f ) = f ◦ exp and ψ − ( f ) = f ◦ ( − exp ). Y et M 2 ( C 0 ( R )) is Morita E q u iv alen t to C 0 ( R ) and th us they ha v e the same K-groups, so we get that K 0 (( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 ) = 0 and that K 1 (( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 ) = Z . Then w e apply the corresp ond ing 6- term K-Theory l.e.s. to the second s.e.s. w h ic h incorp orates the Z 2 -action and we get: 0 − − − − → K 0 ( C 0 ( R ) ⋊ φ Z 2 ) − − − − → Z ⊕ Z   y   y   y exp 0 ← − − − − K 1 ( C 0 ( R ) ⋊ φ Z 2 ) ← − − − − Z (3) whic h giv es the result that K 0 ( C 0 ( R ) ⋊ φ Z 2 ) = K er ( exp ) and K 1 ( C 0 ( R ) ⋊ φ Z 2 ) = I m ( exp ) In order to try to compute the groups explicitly we need more work and b y p erformin g the computation we shall also determine the generators of the groups as well. Let u s start b y recalling s ome known facts: If A is an asso ciati ve algebra, p ∈ A is called a pro jection if p 2 = p w ith 0 b eing the trivial pro j ecti on; for t wo pro jections p , q w e write p < q if pq = p . A pro jection is called minimal if w e cannot find a smaller one. If A is u n ital 9 w e can easily constr u ct pr o jections in M n ( A ) (the algebra of n × n matrices with entries from A ) just by considering the d iagonal matices with th e unit in the diagonal and eac h one will correspp ond to th e f ree mo dule o ver A of rank n . Let p + = 1 2  1 1 1 1  and p − = 1 2  1 − 1 − 1 1  denote minimal pro jections in C ⊕ C (these can b e used also as g enerators of the algebra C ⊕ C ) which un der the isomorph ism corresp ond to m inimal pro jections 1 2 (1 e + 1 ǫ ) an d 1 2 (1 e − 1 ǫ ) in C ⋊ φ Z 2 where evidently 1 e : Z 2 → C denotes the fu nction giving 1 on the iden tit y elemen t i.e. 1 e ( e ) = 1 and 1 e ( ǫ ) = 0 and similarly for 1 ǫ . Their corresp ondin g K -classes wil b e den oted [ p + ] and [ p − ] and these are the t wo generators of K 0 ( C ⋊ φ Z 2 ) = Z 2 , hence an y elemen t in the 0th K-group can b e written as a finite integ er linear com- bination of these tw o elemen ts. W e w ould lik e to see u nder th e exp onential map exp : K 0 ( C ⋊ φ Z 2 )(= Z 2 ) → K 1 (( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 )(= Z )what happ ens to the generators. So we w an t to calculate exp ([ p ± ]) = [ e 2 iπ χ 1 2 (1 e ± 1 ǫ ) ] = [ u ± ], where χ : R → C and χ 1 e : Z 2 → C 0 ( R ) suc h that χ 1 e ( e ) = χ and χ 1 e ( ǫ ) = 0 and moreo v er ev 0 χ 1 e = χ (0)1 e = 1 e , χ ( −∞ ) = χ ( ∞ ) = 0 and χ (0) = 1 and simi- larly for χ 1 ǫ . Not e here that χ 1 2 (1 e ± 1 ǫ ) denotes the lifting of the pr o jections p ± = 1 2 (1 e ± 1 ǫ ) originally in C ⋊ φ Z 2 to self-adjoint elemen ts in C 0 ( R ) ⋊ φ Z 2 . More concretely one can understand u : Z 2 × R → C thinking it a s a function from Z 2 → C 0 ( R ) with u ± ( −∞ ) = u ± ( ∞ ) = 1 = u ± (0). Hence we h a ve to calculate exp ([1 e ]) = [ e 2 iπ χ 1 e ]. No w we said that for an y asso ciativ e algebra A , this is Morita equ iv alen t to M 2 ( A ) hence K 1 ( M 2 ( A )) ∼ = K 1 ( A ) an d if [ u ] is in K 1 ( A ) th e corresp ond ing elemen t in K 1 ( M 2 ( A )) is: 10 [  u 0 0 1  ] . W e men tion the well -known fact that the f ollo wing elemen ts are equal (as K-classes, h ence homotopic as pro jections): [  u 0 0 1  ] = [  1 0 0 u  ] , whic h w e shall use later on. W e pic k a fun ctio n θ : R → C whic h is 0 at −∞ and 1 at ∞ s o e 2 iπ θ ∈ C 0 ( R ). Thus using the explicit isomorphism Ψ b et w een C 0 ( R − ) ⊕ C 0 ( R + ) ⋊ φ Z 2 and M 2 ( C 0 ( R )) plu s the relation b et w een the generators of the K 1 ’s of the Morita equiv alen t algebras A and M 2 ( A ) w e ded uce that the generator of K 1 (( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 ) is: [  ( ψ − ) − 1 ( e 2 iπ θ ) 0 0 1  ] = [  1 0 0 ( ψ + ) − 1 ( e 2 iπ θ )  ] . Hence we conclude that (using also the relation b etw een θ and χ ) exp ([1 e ]) = [  ( ψ − ) − 1 ( e 2 iπ θ ) 0 0 1  ] + [  1 0 0 ( ψ + ) − 1 ( e 2 iπ θ )  ] = 2 , namely exp ([ p + ] + [ p − ]) = 2[ e 2 iπ θ ], so the sum of the generetors of K 0 ( C ⋊ φ Z 2 ) equals 2 and hence [ p + ] + [ p − ] is NOT in K er ( exp ). So we still wan t to find the k ernel of the exp onen tial m ap. W e know that ev 0 χp ± = p ± then exp [ p ± ] = [ exp (2 iπ 1 2 (1 e ± 1 ǫ ) χ )] bu t exp (2 iπ 1 2 (1 e ± 1 ǫ ) χ ) = [ e 2 iπ χ 1 2 (1 e ± 1 ǫ )] + 1 2 (1 e ± 1 ǫ ) ⊥ (whic h follo ws from the general identit y for a pro jection p commuting with f th at e 2 ipf π = e 2 if π p + (1 − p )). Hence ( e 2 iπ χ − 1) ∈ C 0 ( R − ) ⊕ C 0 ( R + ) b ecause it v anishes at 0 and at ±∞ since χ (0) = 1 and χ ( ±∞ ) = 0. This can b e b etter written as a ro w v ector with comp onen ts (( e 2 iπ χ | R − − 1) , ( e 2 iπ χ | R + − 1)) and the first component u nder the isomorphism ψ − corresp onds to ( e − 2 iπ θ − 1) ∈ C 0 ( R ) and the second comp onen t u nder the isomorphism ψ + corresp onds to ( e − 2 iπ θ − 1) ∈ C 0 ( R ). W e can see th e ab ov e element exp [ p ± ] as an elemen t in ( C 0 ( R − ) ⊕ C 0 ( R + )) ⋊ φ Z 2 ∼ = M 2 ( C 0 ( R )) as the follo wing 2 × 2 matrix 1 2  ( e − 2 iπ θ + 1) ± e − 2 iπ θ ∓ ± e − 2 iπ θ ∓ 1 e − 2 iπ θ + 1  11 Let us call this matrix W and for simplicit y denote it as W :=  a b c d  Then the + corresp ond to th e pro j ecti on p + under the isomorphism and similarly for the − s ign. No w w e incorp orate the homotopies in order to see to w h at exp licit elemen ts the pr o jections p ± corresp ond to. W e kn o w that there exists a homotop y b et ween 1 2  1 1 1 1  and  1 0 0 0  and b et we en 1 2  1 − 1 − 1 1  and  0 0 0 1  giv en b y u 0 =  1 0 0 1  and u 1 = 1 √ 2  1 − 1 1 1  with u t 1 2  1 1 1 0  u ∗ t whic h is equal to  1 0 0 0  for t = 1 and to u 0 for t = 0. This can b e also wr itten as  cos ( t π 4 ) − s in ( t π 4 ) sin ( t π 4 ) cos ( t π 4 )  . Hence u 1 W u ∗ 1 = 1 2 1 2  1 1 − 1 1   a b b a   1 − 1 1 1   1 1 − 1 1   a − b b + a b − a a + b  = =  0 2( a + b ) 2( b − 1) 0  T o the last matrix we substitute the v alues of a , b from our earlier com- putations of the m atrix W and get th at  0 2( a + b ) 2( b − 1) 0  = 1 2  0 e − 2 iπ θ (1 ± 1) + (1 ∓ 1) ( ± − 1) e − 2 iπ θ + ( ∓ 1 − 1) 0  So we can see that [ p + ] 7→ [  0 e − 2 iπ θ − 1 0  ] 12 and [ p − ] 7→ [  0 − 1 − e − 2 iπ θ 0  ] and we see that their images ar e homotopic , n amely exp ([ p + ]) is homo- topic to exp ([ p − ]) th us by this homotop y w e lose one generator and so the generator of K 0 ( C 0 ( R ) ⋊ φ Z 2 ) = Z is [  0 e − 2 iπ θ − 1 0  ] W e now turn our atten tion to the cyclic cohomology of the algebra A := C 0 ( R ) ⋊ φ Z 2 whic h is the corresp onding C ∗ -algebra to the Mobius foliation after having pr o ved ab o ve that K 0 ( A ) = Z =. The cyclic co cycles are traces of A . One usually lo oks at traces which are inv ariant und er the holonom y action. In our case we ta ke a holo nomy inv arian t transverse mea sur e, namely a measur e µ on R wh ic h is the tr ansv ersal su c h that µ ◦ φ = µ , n amely it is inv ariant u n der the action φ of the holonom y group Z 2 on R . In general an inv arian t measure has the prop ert y that giv en an y map f : R → C one has R dµ ( x ) f ( x ) = R dµ ( x ) f ( − x ). Suc h measures are very common indeed (e.g. the Leb esgue measure has this pr op ert y) but there are also man y measures on R whic h do not ha v e this p rop ert y . Ha ving pic k ed such a holonom y in v ariant measure on th e tr an s v ers al R of the Mobius f olia tion w e can d efine a trace τ µ on A (namely a closed cyclic 0-cocycle of A d enoted τ µ ∈ H C 1 ( A )) as follo ws: τ µ ( F ) = Z F ( e ) dµ where F : Z 2 → C 0 ( R ). (Aside: But w e ha v e to see wh at is the trans- v erse fu ndamen tal cyclic co cycle of the foliation used in [8] to defin e a nu- merical in v ariant for foliations; this transv erse fundamenta l cyclic cocycle has dim en sion equal to the co dimension of the foliation, clearly the cod im of the Mobius f oliaiton is 1. But we ha v e to under s tand the cyclic cohomol- ogy of the crossed pr od uct algebra A firs t. Here we discuss cyclic 0-cocycles b ecause they app ear in the gap lab elling problem, the gap lab ells come as pairings b et wee n K-classes and cyclic 0-cocycles). Ho w ev er this is not a con v enien t c h oice since the pairing of τ µ with the generator [ p + ] (and hence an y elemen t since an y elemen t can b e written as a multiple of the generator) of K 0 ( A ) v anishes. W e note here that Z (whic h is equal to K 0 ( A )) has only 2 generators, 13 the one we h a ve pic k ed and minus that, but they b oth v anish when p aired with τ µ . W e can do something elese though : sin ce we ha v e the map ev aluation at the fixed p oin t 0 ev 0 : A → C ⋊ φ Z 2 ∼ = C 0 Z 2 where C 0 Z 2 denotes the group algebra of Z 2 , instead of pic king a holonom y inv arian t measure, w e can pic k a representa tion ( ρ, V ) of the group algebra C 0 Z 2 on to a vec tor space V with ρ : C 0 Z 2 → E nd ( V ) and this in turn has a w ell defined matrix-li ke trace T r : E nd ( V ) → C . So the comp osition T r ◦ ρ ◦ ev 0 : A → C is also a trace. W e p ic k for example the follo wing rep resen tation ρ of Z 2 on to C : e 7→ 1 and ǫ 7→ − 1. Using this we can easily see that the pairing b et ween K-classes and the cyclic 0-cocycle τ ρ giv es < ([ p + ]) , ( τ ρ ) > = − 1 . No w in [8] we u sed th e transverse f undamen tal cyclic co cycle of the foli- ation in order to take pairings with K-classes. T he transve rse fund amen tal cyclic co cycle is a cyclic q -co cycle where q is the co dimension of the foliation and one needs th e transverse bund le of the foliatio n to b e orientable in order to b e able to define it. F or the Mo biu s foliati on the co dimension is 1 b ut the tr ansverse b und le is not orientable b ecause of the map φ wh ic h flips the sign and thus the transv erse fundamen tal cyclic co cycle d oes not exist for the Mobius foliatio n. W e need to find another example, either the folaition of the annulus or th e Reeb foliation of the 2-torus. Understand restrictions th at transv erse bundle must b e orient able and existence of a holonom y inv ariant transverse measure. T he (partial) Bott connection θ is flat when restricted to th e lea ves bu t it still ma y h a ve holon- om y; what is the relation b et we en the holonom y of it an d the holonomy G x x ? Are they the s ame? It seems to b e so... Ac kno wledgemen t: W e w ould lik e to th ank Professor Johannes Kellen- donk f or his v aluable help and guidan ce in p erformin g this K-Theoretic computation. 14 References [1] A. Candel, L. Conlon: “F oliatio ns I, I I”, Graduate Studies in Mathe- matics, t w o v olumes, V ol 23, AMS, Oxford Universit y Press (2000) and (2002 ). [2] A. C onnes: “Noncomm utativ e Geometry”, Academic Press, (1994 ). [3] R. Bott: “Lectures on Characteristic Classes and F oliatio ns”, LNM 279, Springer, (1972). [4] C.C. Mo ore, C. S chochet: “Glo bal Analysis on F oliated Spaces”, MS RI v ol 9, Springer, (1988 ). [5] H.B. La w son: “F oliatio ns”, Bull. AMS V ol 80 No 3, (1974). [6] B. Blac k adar: “K-Theory for Op erator Algebras”, Cam br id ge Univer- sit y pr ess (1998). [7] W egge Olsen: ”K-Theory for C ∗ -algebras”, Oxford Univ ersit y Press (1992 ). [8] I.P . Z ois: “A new in v ariant for σ mo dels”, C omm u n. Math. Phys. 209.3 (2000 ). 15