Net bundles over posets and K-theory

Net bundles o v er p osets and K-theory John E. Rob erts (1) , Giusepp e Ruzzi (1) , Ezio V asselli (2) (1) Dipartimento di Matematica, Universit` a di Roma “T or V ergata” Via della Ricerca Scient ifica I-0013 3, Roma, Italy (2) Dipartimento di Matematica, Universit` a di Roma “La Sapienza ” Piazzale Aldo Moro 5, I-0 0 185 Roma, Italy robert s@mat.u niroma2.it , ruzzi@ mat.uni roma2.it , ezio.vas selli@g mail.com Abstract W e contin ue studying net bundles ov er partially order ed sets (posets), defined as the analo gues of o rdinary fibre bundles. T o this end, we analyze the co nnection betw een homoto py , net homology and net co homology of a p oset, giving versions of classical Hurewicz theo rems. F o cusing our attent ion on Hilb ert net bundles, we define the K -theory of a p oset and introduce functions ov er the homotopy group oid satisfying the same formal prop erties as Chern classes. As when the given po set is a base for the top ology of a space, o ur results apply to the c ategory of lo cally constant bundles. MSC-class : 1 3D15; 05E 25; 06A11 . Keyw ords : po set; K-theor y; homoto py; cohomolo gy . Con ten ts 1 In tro duction 2 2 Bac kground and Notations. 3 3 Ab elian (co)homology . 4 3.1 Net cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Net homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Homotop y and net cohomology of net bundles. 9 4.1 Net stru ctures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 P oset n et bun dles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 K-theory of a p oset. 15 5.1 Net bu ndles of Banac h spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Basic Prop erties o f K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1 5.3 Chern classes for Hilb ert net bundles. . . . . . . . . . . . . . . . . . . . . 21 5.3.1 The fi rst Chern c lass. . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3.2 Chern K-classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3.3 Chern fu nctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.4 A ske tc h of equiv arian t K-theory . . . . . . . . . . . . . . . . . . . . . . . . 26 6 Lo cally constan t bundles a nd comparison with ordinary K -theory . 27 A Simplicial sets. 30 A.1 Homotop y of pro ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.2 Simplicial s ets of a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 In tro duction This paper co ntin ues the d iscussion of inv arian ts of a partially ordered set b egun in [12, 13], comparin g them with the analogous top ological inv ariants when the partially ordered set is a suitable base for the top ology of a top ologica l space ordered un der i n- clusion. Whereas previously , t he emphasis has b een on th e fun damen tal group oid, co n- nections, cu r v ature, s implicial cohomology and ˇ Cec h cohomology , it is now on homology , lo cally constan t cohomology and K-theory with the fund amen tal group oid cont inuing its ubiquitous role . Th us w e p ro v e an analogue of the Hurewicz isomorph ism sh o wing that th e first homology group of a poset with v alues in Z is isomorphic to its Abelianized h omotop y group. W e further consider an arcwise lo cally con tractible sp ace and a p oset made u p of a base of con tractible op en sets ord ered under in clusion and sho w th at the fi rst homology groups with v alues in Z of the top olog ical space coincides with that of the p oset as d o the fi rst cohomology g roups with v alues in an Ab elian group. The piv otal notions are those o f of net bundle and quasinet bundle. These are bundles o v er a p oset where the fibres are ob jects in a c ategory to gether with a fun ctor from the p oset m apping an elemen t of the p oset to the corresp onding fibr e. In a quasinet b undle, the f unctor tak es v alues in the monomorphisms, in a net bun dle in the isomorph isms. W e sho w that a p oset n et bu ndle with p ath wise connected fibr es giv es r ise to a sh ort exact sequence of homotop y groups as do es a net bund le of top ological sp aces with connected and locally con tractible fibres. W e define a fun ctor from the category of n et bu ndles of top ologica l spaces to the catego ry of fibr e bun dles ov er the p oset with the Alexandroff top ology . Th e category of 1–cocycles with v alues in a group of the global space of the original n et bu ndle regarded as a net bundle o ve r the p oset with the opp osite ordering is e quiv alen t to the cat egory of homomorph isms of th e h omotop y group of the fi bre bund le w ith v alues in the g roup . The fib re bundle associated with a principal G –net bun dle has lo cally constan t transition functions. The remainder of the pap er is largely devote d to stud ying Hilb ert net bundles and their K-theory . Here there is a fun ctor from Hilb er t net bund les to ve ctor bun dles 2 o v er the p oset with the Alexandroff top olog y . The categ ory of Hilb ert net b undles is equiv alent to th e category of unitary fi nite-dimensional representat ions of the homotop y group of the p oset. Consequen tly , its K-r ing is isomorphic to the r epresen tation ring of its homotop y group. But it is also equiv alen t to the category of locally constan t v ector bund les ov er the p oset with the A lexandroff top olog y . This le ads to an isomorphism of the corr esp onding K-rings. W e prov e an analogue of the Thom isomorp hism asserting that the firs t cohomology of the p oset with v alues in a group is isomorp hic to the fi rst cohomology of a p ro jectiv e net b undle with v alues in the s ame group. The first Chern class of a Hilb ert net bundle is defined as an element of the net cohomology g roup H 1 ( K, T ). After this, the Chern K-c lasses are defined as elemen ts of the r ed uced K-theory an d Cher n fun ctions as complex-v alued functions on the homotop y classes of paths, allo wing one to reco v er the fir st C hern class b y ev aluating a suitable p olynomial. When K is a base for the top ology of a space M , our results describ e lo cally constant ve ctor bu ndles o v er M , the first Ch ern class is an elemen t of the singular cohomology H 1 ( M , T ) and the Chern functions are defined on the homotop y group oid of M . Our results imply that when the p oset K is a base for the top ology of a manifold, eac h finite-dimensional Banac h net bun dle o v er K h as we ll-defined secondary charac teristic classes (see [1], [7, § 4.19],[2, § 1(g)]). It would b e interesting to find a description of suc h classes in terms of t he prop erties of K such as its n et co homology . Finally , we w ould like to men tion th at our a pp roac h to th e homotop y g roup ma y b e regarded as a particular case of the fu ndamen tal w ork of Quillen on higher algebraic K- theory ( [11]), fo cused on generic categories instead of p osets. Nev erthless our approac h has the adv antag e of b eing expr essed in a language d eriv ed fr om algebraic quant um field theory , whic h motiv ated our wo rk. An app endix is dev oted to describing the simplicial sets asso ciate d with a p oset, explaining the related notion of homotop y and pro ving that the homotop y group of a pro duct of symmetric simplicial set s is equal to th e pro du ct of th e homotop y grou p s. 2 Bac kground and Notations. F or the basic notions on the sim p licial set of a p oset K and its homotop y theory , includ ing the definition of path, w e refer th e reader to [1 4, 12, 13]. Ho w ev er the basic definitions and termin olog y can also b e foun d in the App endix. The fu ndamen tal co v ering of K is giv en by the family of sets V a := { o ∈ K : a ≤ o } , a ∈ Σ 0 ( K ) , whic h provide a b ase for th e Alexandroff top ology of K . W e d en ote the asso ciated sp ace b y τ K (see [13, § 2.3]). 3 In order to simplify the exp osition, in the p resen t pap er we shal l always assume that the p oset K is p athwise c onne cte d . This implies that the isomorphism class of the homotop y g roup π 1 ( K, a ) do es not d ep en d on the c hoice of a ∈ Σ 0 ( K ). In the same wa y , eac h sp ace M is assum ed to b e arcwise connected, and its homotop y group is denoted by π 1 ( M ). A pivot al resu lt that will b e used in the pr esen t pap er is the follo wing: when M is Hausd orff, and K is a base of arcwise and simply connected op en sub sets of M ordered u nder inclusion, t here is an isomorp hism π 1 ( M ) ≃ π 1 ( K, a ), a ∈ Σ 0 ( K ) (see [14, Thm.2.1 8]). W e denote the identit y map o f a set S b y id S . A m ultiplicativ e semigroup of p olynomials with coefficients in a ring R is defined by 1 + hR [[ h ]] := { 1 + r X k =1 x k h k , r ∈ N , x k ∈ R } . 3 Ab elian (co)homology . The relation b et wee n homotop y and Ab elian (co)homolo gy of a p oset can b e d educed b y the Quillen’s pap er [11]. Ho wev er, we prefer to analyze this r elati on in terms of a simplicial set asso ciate d with the p oset, an approac h closer to the l anguage used in the present p ap er than that used by Quillen. The main aim of this section is to giv e some v ersions of the classical Hurewicz theorems. W e refer the reader to the App end ix A for the d efi nition of the simp licia l s et asso ci- ated to the p oset and for the co rresp onding homotop y . 3.1 Net cohomology . The net cohomology of the p oset K with v alues in an Ab elian group A , written add itiv ely , is the cohomolog y of the simplicial set e Σ ∗ ( K ) with v alues in A (see App endix A). T o b e precise , one can define the set C n ( K, A ) of n –cochains of K with v alues in A as the set of functions v : e Σ n ( K ) → A . C n ( K, A ) inh erits from A the structure of an Abelian group: ( v + w )( x ) := v ( x ) + w ( x ) , ( − v )( x ) := − v ( x ) , x ∈ e Σ n ( K ) , for any v , w ∈ C n ( K, A ). T h e cob ound ary op erator d defined by d v ( x ) = n X k =0 ( − 1) k v ( ∂ k x ) , x ∈ e Σ n ( K ) , is a mapping d : C n ( K, A ) → C n +1 ( K, A ) s atisfying the equation dd v = ι , v ∈ C n ( K, A ), where ι is the trivial co c hain. Cle arly d is a group morp hism. An n –co c hain z is said to b e an n – c o cycle if b elongs t o the k ernel of d. W e denote the group of n –cocycles by Z n ( K, A ). S ince dC n − 1 ( K, A ) is a su bgroup B n ( K, A ) of C n ( K, A ), w e define the net c ohomolo gy with c o efficients in A H n ( K, A ) := Z n ( K, A ) / B n ( K, A ) . 4 The fu nctors H n ( − , − ), n ∈ N , are contra v arian t w.r.t the p oset and co v arian t w.r.t. to the grou p ; moreo v er, net cohomology has long exact sequences (see [10, Lemma 1.1]). Remark 3.1. Let K b e connected, a ∈ Σ 0 ( K ) and Hom( π 1 ( K, a ) , A ) denote the set of morphisms fr om π 1 ( K, a ) to A . Using [12, Prop.3.8] and the f act that A is Ab elian, there is an isomorph ism H 1 ( K, A ) ≃ Hom ( π 1 ( K, a ) , A ). This allo ws one to compu te several net cohomology groups. 3.2 Net homology . In th e presen t s ecti on, w e introd uce the not ion of the net homology of a p oset. Let A b e an Ab elian group with identit y 0 ∈ A . F or ev ery n ∈ N , w e let C n ( K, A ) denote the f r ee Ab elian group generated b y formal linear combinatio ns of elemen ts of e Σ n ( K ) w ith co efficien ts in A , and define the boun dary morphism as the A -linear ma p b n : C n ( K, A ) → C n − 1 ( K, A ) , b n x := n X k =0 ( − 1) k ∂ k x . (3.1) It is clear that b n − 1 b n = 0; we define Z n ( K, A ) := ker b n , B n ( K, A ) := b n +1 (C n +1 ( K, A )), and the net homolo gy gr oup H n ( K, A ) := Z n ( K, A ) / B n ( K, A ) (for n = 0, w e d efine C 0 ( K, A ) := A , B 0 ( K, A ) := 0 and b 0 := 0, so that H 0 ( K, A ) = A ). In the sequel, w e w ill use the notation b ≡ b n , so that b n − 1 b n ≡ b 2 = 0. Moreo v er, w e will use the same n otat ion for elemen ts of Z n ( K, A ) and H n ( K, A ), iden tifying cycles with the corresp onding net homolog y classes. When A = Z , C n ( K, Z ) redu ces to the Ab elian group generate d b y the elemen ts of e Σ n ( K ). Element s of Z 1 ( K, Z ) sat isfy the relation X i k i ( ∂ 0 b i − ∂ 1 b i ) = 0 , X i k i b i ∈ Z 1 ( K, Z ) , (3.2) so that, b ∈ e Σ 1 ( K ) ∩ Z 1 ( K, Z ) if, and only if, ∂ 0 b = ∂ 1 b . W e establish other useful relations. Given a ∈ Σ 0 ( K ) consider the d egenerate 1– simplex σ 0 a and note that σ 0 a = b( σ 0 σ 0 a ). So that, σ 0 a = 0 ∈ H 1 ( K, Z ) . (3.3) If b ∈ e Σ 1 ( K ) and b is defin ed by ∂ 0 b := ∂ 1 b , ∂ 1 b := ∂ 0 b , | b | := | b | , then defining c := ( | b | ; b, σ 0 ∂ 0 b, b ) we obtain b c = b − σ 0 ∂ 0 b + b , implying (by (3.3)) b + b = 0 ∈ H 1 ( K, Z ) . (3.4) No w, fo r ev ery n ∈ N w e consider the map C n ( K, A ) × C n ( K, A ) → A (3.5) 5 obtained extending th e ev aluation ( v , x ) 7→ v ( x ), v ∈ C n ( K, A ), x ∈ e Σ n ( K ), by linearit y . Some e lement ary computations s h o w that (3 .5) induces a b ilinear map H n ( K, A ) × H n ( K, A ) → A . (3.6) W e now analyze the connection b et wee n homotop y and n et homology , p ro viding a v ersion of a classical result ([5, Thm.2.A.1]). T o this end , we make some p reliminary remarks on π 1 ( K, a ), a ∈ Σ 0 ( K ), and the associated Ab elianized group π 1 ( K, a ) ab . W e r eca ll that π 1 ( K, a ) ab is defin ed as the quotien t of π 1 ( K, a ) by the commuta tor subgroup, that is the normal sub group generated b y elements of the form p 1 ∗ p 2 ∗ p − 1 1 ∗ p − 1 2 , p 1 , p 2 ∈ π 1 ( K, a ); by construction, π 1 ( K, a ) ab is the universal Abelian group such that eac h group morp h ism π 1 ( K, a ) → A , with A Ab elian, factorizes thr ough a morp hism π 1 ( K, a ) ab → A . F or eac h p ∈ π 1 ( K, a ), w e denote the corresp onding class in π 1 ( K, a ) ab b y [ p ], so that [ p ∗ p ′ ] = [ p ] ∗ [ p ′ ] = [ p ′ ] ∗ [ p ] = [ p ′ ∗ p ], p, p ′ ∈ K ( a ). Remark 3.2. Let c ∈ e Σ 2 ( K ). T hen th e p ath p c := ∂ 1 c ∗ ∂ 0 c ∗ ∂ 2 c is homotopic t o the constan t path. F or e ac h a ∈ Σ 0 ( K ), w e denote the set of paths starting and ending in a b y K ( a ). Remark 3.3. Let p ′ ∈ K ( a ′ ), a ′ ∈ Σ 0 ( K ), a ′ 6 = a ; then there is a path γ a,a ′ , starting in a ′ and end ing in a , so that w e can d efine p := γ a,a ′ ∗ p ∗ γ a,a ′ ∈ K ( a ). Remark 3.4. Let p := b n ∗ · · · ∗ b 1 ∈ K ( a ), a ∈ Σ 0 ( K ). Th en for ev ery ind ex i = 0 , . . . , n , w e define the p ath p i := b i ∗ · · · ∗ b 1 ∗ b n ∗ · · · ∗ b i +1 ∈ K ( ∂ 0 b i ) . W e sa y that p i is obtained from p by a shuffle . If γ a,∂ 0 b i is a path as in th e previous remark, then w e defi ne ˆ p i := γ a,∂ 0 b i ∗ p i ∗ γ a,∂ 0 b i ∈ K ( a ) . In th is w a y , w e fi nd (indep endently of the c hoice of γ a,∂ 0 b i ) [ ˆ p i ] = [ γ a,∂ 0 b i ∗ b i ∗ · · · ∗ b 1 ∗ b n ∗ · · · ∗ b i +1 ∗ γ a,∂ 0 b i ] = [ γ a,∂ 0 b i ∗ b i ∗ · · · ∗ b 1 ] ∗ [ b n ∗ · · · ∗ b i +1 ∗ γ a,∂ 0 b i ] = [ b n ∗ · · · ∗ b i +1 ∗ γ a,∂ 0 b i ] ∗ [ γ a,∂ 0 b i ∗ b i ∗ · · · ∗ b 1 ] = [ p ] . Of course, if ∂ 0 b i = a , th en we can pick γ a,∂ 0 b i = σ 0 a an d [ p ] = [ p i ]. Not e that if, for some i , p i is homotopic to a constan t path, then [ ˆ p i ] = [ γ a,∂ 0 b i ∗ γ a,∂ 0 b i ] and [ p ] is the iden tit y of π 1 ( K, a ) ab . Lemma 3.5. L et a ∈ K and p ∈ K ( a ) b e a p ath of the form p = . . . ∗ b m ∗ . . . ∗ b 1 . . . ∗ b m ∗ . . . ∗ b 1 , m = 2 , . . . . (3.7) Then [ p ] = [ σ 0 a ] ∈ π 1 ( K, a ) ab . 6 Pr o of. p is of the form p = p 2 ∗ ( b 1 ∗ p 1 ∗ b 1 ), wh ere ( b 1 ∗ p 1 ∗ b 1 ) , p 2 ∈ K ( a ). S upp ose that p 1 con tains the 1–simplex b m and p 2 b m . Shuffle b 1 ∗ p 1 ∗ b 1 to giv e a path p ′ 1 ending with b m ; then, b y Remark 3.4, [ b 1 ∗ p 1 ∗ b 1 ] = [ γ a,∂ 0 b m ∗ p ′ 1 ∗ γ a,∂ 0 b m ]. S h uffle p 2 to g et a path p ′ 2 b eginning with b m . Then [ p 2 ] = [ γ a,∂ 0 b m ∗ p ′ 2 ∗ γ a,∂ 0 b m ]. It follo ws that [ p ] = [ γ a,∂ 0 b m ∗ p ′ 2 ∗ p ′ 1 ∗ γ a,∂ 0 b m ] . No w, p ′ 2 ∗ p ′ 1 con tains b m ∗ b m and b 1 ∗ b 1 , and these can b e remov ed without changing the homotop y c lass; w e then ha v e a path of the same t yp e w ith few er 1–simplices. Su pp ose on the other hand that there is no b m con tained in p 1 suc h that p 2 con tains b m . Then b oth p 1 and p 2 are of the form (3 .7 ) and the result follo ws b y induction. Theorem 3.6. L et ( K, ≤ ) b e a p athwise c onne cte d p oset, and a ∈ Σ 0 ( K ) . Then ther e is an isomorp hism H 1 ( K, Z ) ≃ π 1 ( K, a ) ab . Pr o of. Step 1 . Let a ∈ Σ 0 ( K ), and p := b n ∗ · · · ∗ b 1 a generic pat h in K ( a ). With this notation, we defin e the group morph ism T : K ( a ) → C 1 ( K, Z ) , T p := n X i =0 b i . (3.8) Since b T p = n X i =0 ( ∂ 0 b i − ∂ 1 b i ) = a + n − 1 X i =1 ( ∂ 0 b i − ∂ 1 b i +1 ) − a = 0 , w e conclude that T actually tak es v alues in Z 1 ( K, Z ). Step 2 . W e ve rify that (3.8) factorizes thr ough a ma p T : π 1 ( K, a ) → H 1 ( K, Z ) (3.9) (note the sligh t abuse of the notation T ). T o this end, we consider c ∈ e Σ 2 ( K ), p := b n ∗ · · · ∗ b 1 ∈ K ( a ), and an elemen tary deformation of p p erformed replacing a pair b i = ∂ 0 c , b i +1 = ∂ 2 c , i ∈ { 0 , . . . , n } , b y ∂ 1 c . If we denote the d eformed path by p c , th en T p c = n X i =0 b i − 2 X k =0 ( − 1) k ∂ k c = T p − b 2 c . In the same w a y , an elemen tary deformation p erformed replacing b i = ∂ 1 c b y ∂ 0 c ∗ ∂ 2 c giv es rise to the op eratio n T p c = n X i =0 b i + 2 X k =0 ( − 1) k ∂ k c = T p + b 2 c . This implies t hat (3.9) is w ell-defined. Step 3 . W e v erif y that (9) is surjectiv e. T o t his end, note that a generic el ement of Z 1 ( K, Z) can b e wr itten up to homology in the form x = m X j =1 b j , 7 no signs b eing needed since a b j can b e rep laced by b j if necessary . W e show that there is a bijection f on 1 , 2 , . . . , m and 0 = m 0 < m 1 < · · · < m ℓ = m and p aths p k = q k ∗ b f ( m k ) ∗ b f ( m k − 1) ∗ · · · ∗ b f ( m k − 1 +1) ∗ q k ∈ K ( a ) for k = 1 , 2 , . . . , ℓ . Th is suffices since setting p := p ℓ ∗ · · · ∗ p 1 , T p = x up to homology . W e set f (1) = 1 and since x ∈ Z 1 ( K, Z) we ma y pick a b j with ∂ 0 b j = ∂ 1 b 1 and set f (2) = j . W e con tin ue in this w a y , i.e. pic k a b k with ∂ 0 b k = ∂ 1 b j and set f (3) = k . After say r steps th is pro cess terminates w hen t wo conditions are f ulfilled, ∂ 1 b f ( r ) = ∂ 0 b 1 and there is n o b s with ∂ 0 b s = ∂ 1 b f ( r ) . W e s et m 1 := r and p ic k a path q 1 with ∂ 0 q = ∂ 0 b 1 and ∂ 1 q 1 = a . The sum o v er the remaining 1–simplices is still in Z 1 ( K, Z) and the argument ma y b e rep eated to reac h th e d esired conclusion. Note that by Lemma 3.5, T p = T p ′ implies [ p ] = [ p ′ ]. Step 4 . Let p := b n ∗ · · · ∗ b 1 ∈ ker T , i.e . x := P i b i ∈ B 1 ( K, Z). Then there are 2–cycles c 1 , . . . , c m suc h that x = X j ∈ J ( ∂ 0 c j − ∂ 1 c j + ∂ 2 c j ) − X k ∈ I ( ∂ 0 c k − ∂ 1 c k + ∂ 2 c k ) , where I ∪ J = { 1 , . . . , m } . Thus, X i b i + X j ( ∂ 1 c j + ∂ 1 c j ) + X k ( ∂ 0 c k + ∂ 0 c k + ∂ 2 c k + ∂ 2 c k ) = X j ( ∂ 0 c j + ∂ 1 c j + ∂ 2 c j ) + X k ( ∂ 0 c ′ k + ∂ c ′ k + ∂ 2 c ′ k ) , where c ′ k is the t w o simp lex got by exc hanging the vertice s 0 and 1 of c k . Ad ding on P j T ( γ j ∗ γ j ) + P k T ( γ k ∗ γ k ), where γ j is a path from ∂ 01 c j to a and γ k a path from ∂ 01 c k to a , to eac h side, w e get an elemen t y of B 1 ( K, Z). F r om the form of the left hand side, we see that there is a path p 1 homotopic to p w ith T p 1 = y and, f rom the form of the righ t h and side, there is a path p 2 homotopic to σ 0 a with T p 2 = y . Hence [ p ] = 0. Corollary 3.7. L et ( K, ≤ ) b e a p athwise c onne cte d p oset. F or every Ab e lian gr oup A , ther e is an isomorphism H 1 ( K, A ) ≃ Hom(H 1 ( K, Z ) , A ) . Pr o of. By Rem.3.1 , we h a ve an isomorphism H 1 ( K, A ) ≃ Hom( π 1 ( K, a ) , A ). Since A is Ab elian, eac h morphism φ ∈ Hom( π 1 ( K, a ) , A ) factorizes through an elemen t of Hom(H 1 ( K, Z ) , A ). No w, let M b e a Hausd orff space and let H 1 ( M , A ), H 1 ( M , A ) denote resp ectiv ely the singular homolo gy an d the s in gular cohomology of M w ith coefficien ts in A . W e fi x a p oset M ≺ pro viding a base of arcwise and simply connected op en subsets of M . Theorem 3.8. L e t M b e a Hausdorff, lo c al ly ar cwise and sim ply c onne cte d sp ac e. Then ther e is an isomorp hism H 1 ( M , Z ) ≃ H 1 ( M ≺ , Z ) . F or every Ab elian gr oup A ther e is an isomorph ism H 1 ( M ≺ , A ) ≃ Hom(H 1 ( M , Z ) , A ) , and this implies the isomorph ism H 1 ( M ≺ , A ) ≃ H 1 ( M , A ) . (3.10) 8 Pr o of. Let a ∈ Σ 0 ( M ≺ ). By [14, T hm.2.18], w e h a ve isomorphisms H 1 ( M , Z ) ≃ π 1 ( M ) ab ≃ π 1 ( M ≺ , a ) ab and, by the classical Hurewicz theorem there is an isomorphism H 1 ( M , Z ) ≃ π 1 ( M ) ab . The isomorp hism (3.10) f ollo ws since, by the un iv ersal co efficien t theorem, H 1 ( M , A ) ≃ Hom(H 1 ( M , Z ) , A ) (see [5, § 3.1]). When M is a manifold it mak es sense to consider the de Rham cohomology H 1 dR ( M , R ), and we hav e Corollary 3.9. L et M b e a c onne cte d manifold. Then ther e is an isomorph ism H 1 ( M ≺ , R ) ≃ H 1 dR ( M , R ) . 4 Homotop y and net cohomology of net bundles. 4.1 Net structures. A quasinet bund le with b ase K is giv en b y a 4-ple X := ( X, π , J, K ), wh ere X is a set called the total sp ac e , π : X → K is a su rjectiv e m ap with fibr es X a := π − 1 ( a ), a ∈ Σ 0 ( K ), and J is a family of injectiv e maps J b : X ∂ 1 b → X ∂ 0 b , b ∈ Σ 1 ( K ), called the net structur e of X , satisfying the c o cycle r elations J ∂ 0 c J ∂ 2 c = J ∂ 1 c , c ∈ Σ 2 ( K ), and J σ 0 a = id X a , a ∈ Σ 0 ( K ). When eac h J b is a bij ective map, we sa y that X is a net b und le . The fi bres of a net bun dle X are all isomorphic (see [13]); a distinguish ed fi bre X a of X w ill b e called the sta ndar d fibr e of X and emphasized with the notatio n F . Analogous d efinitions apply w hen considering categories with m ore structure than the catego ry of sets; in that case, the n et s tr ucture s hall b e a f amily of monomorphisms (or isomorphisms) in the appropriate cat egory . Of particular interest for our pu rp oses will b e H ilb ert (quasi)net bund les, g r oup (quasi)net bu nd les, C*-algebr a (quasi)net bund les and (quasi)net bund les of top olo gic al sp ac es . The first class of quasinet b undles will b e stud ied in § 5 . Quasinet bundles of C* - algebras motiv ated our w ork. They a rise as follo ws. Example 4.1. In the algebraic app roac h to quan tum field theory (see [4], for exam- ple), the set of qu antum observ ables is presente d as a net of C*-algebr as ; b y this we mean a map A , assigning a C* -algebra A ( o ) to elemen ts o of a distinguished p oset K of con tractible op en subsets of spacetime, and interpreted as the algebra of quant um observ ables lo calized in the op en set o ∈ K . Th e map A preserves the order, in the s en se that A ( o 1 ) ⊆ A ( o 2 ) wheneve r o 1 ⊆ o 2 . T his stru cture ma y b e replaced by a quasinet bund le of C* -a lgebras defined as fo llo ws: w e c onsider the set ˆ A := { ( o, A ( o )) , o ∈ K } endo we d with the pro jection π : ˆ A → K on to the first co mp onent and net s tr ucture J b , b ∈ Σ 1 ( K ), defined b y the inclusion A ( ∂ 1 b ) ⊆ A ( ∂ 0 b ). 9 No w let ˆ X := ( ˆ X , ˆ π , ˆ J , K ) b e a quasinet bund le. A map T : X → X is said to b e a morphism if ˆ π T = π (this implies that T restricts to maps T a : X a → ˆ X a , a ∈ Σ 0 ( K )) and ˆ J b T ∂ 1 b = T ∂ 0 b J b , b ∈ Σ 1 ( K ) . In this case, w e use th e notation T ∈ ( X , ˆ X ); if eac h T a , a ∈ Σ 0 ( K ), is a bijectiv e map, then we say that T is an isomorphism . In this wa y , the set of quasinet bun dles b ecomes a catego ry . The fu ll sub category of n et bund les with fi b re F will b e denoted b y B ( K, F ). Ev ery net bund le X := ( X , π , J, K ) defines a net bund le X ◦ := ( X ◦ , π ◦ , J ◦ , K ◦ ) , (4.1) where K ◦ is the dual p oset (see App en dix A.2), X ◦ := X , π ◦ := π , and the net str u cture J ◦ is defined by J ◦ b := J − 1 ( ∂ 1 b,∂ 0 b ) , b ∈ Σ 1 ( K ◦ ). This p ro vides an iso morph ism B ( K, F ) ≃ B ( K ◦ , F ), whic h has b een d escrib ed in c ohomological te rms in [13, § 7 ]. No w, let G d en ote the group of automorphisms of F . By [12, Pr op.3.8] , [13, Thm.6.7], for eac h a ∈ Σ 0 ( K ) there are isomorp hisms ˙ B ( K, F ) ≃ H 1 ( K, G ) ≃ ˙ Hom( π 1 ( K, a ) , G ) , (4.2) where ˙ B ( K, F ) is the set of isomorphism classes of net b undles with fibre F and ˙ Hom( π 1 ( K, a ) , G ) d enotes the set of equiv alence classes of m orphisms fr om the h omo- top y group π 1 ( K, a ) to G , t wo morphisms b eing equiv alen t if they differ by an inner automorphism o f G . The notion of morphism can b e generalized to allo w c hanges of the base K ; to this end, we introduce the notion o f pul lb ack . Let η : K ′ → K b e a morphism of p osets. Giv en a quasinet bundle X ov er K , we define a quasinet bun dle X η , by considering the set X η := X × K K ′ :=  ( x, a ′ ) ∈ X × Σ 0 ( K ′ ) : π ( x ) = η ( a ′ )  endo we d w ith the obvio us pro jection π η : X η → K ′ and th e net stru cture J η b ′ ( x, ∂ 1 b ′ ) := ( J η ( b ′ ) x, ∂ 0 b ′ ), x ∈ X ∂ 1 η ( b ′ ) . By restricting the n otion of pullbac k to the su b categ ory of net b undles with fibre F , w e obtain a functor η ∗ : B ( K , F ) → B ( K ′ , F ) , η ∗ X := X η . W e are no w able to define the n otion of a ge ner alize d morphism fr om a quasinet b undle ˆ X := ( ˆ X , ˆ π , ˆ J , ˆ K ) to a quasinet bu ndle X := ( X , π , J, K ) as a pair ( η , T ), wh ere η : ˆ K → K is a morphism of p osets, and T ∈ ( X η , ˆ X ) is a morp hism in the usual sense. W e also recall that a lo c al se ction of a qu asinet bund le ( X, π , J, K ) is given by a map σ from a sub p oset V ⊆ K into X , such that π σ = id V and J b σ ( ∂ 1 b ) = σ ( ∂ 0 b ), b ∈ Σ 1 ( V ). If V = K , then σ is said to b e glob al . W e denote th e set of lo cal sections of X ov er V by S ( V ; X ) (for V = K , w e use the anal ogous notat ion for the set o f g lobal sections). 10 4.2 P oset net bundles. In the pr esen t section, w e consider the notion of p oset net bundle. The in teresting applications are when the fibre is the p oset arising from the top ology of a space. In particular, in the follo wing w e relat e the homotop y of a p oset net bu ndle to that of the underlying p oset. A p oset net bund le is a net bun dle ( X, η , J, K ) where eac h fi bre X a , a ∈ Σ 0 ( K ), is endo we d with an orderin g ≤ a and eac h map J b , b ∈ Σ 1 ( K ), is an isomorp h ism of p osets. A stand ard argument for net bundles allo w s one to conclude that eac h p oset net bund le adm its a fixed p oset ( F , ≤ ) as standard fibre, end o wed with p oset isomorphisms V a : F → X a , a ∈ Σ 0 ( K ). Lemma 4.2. L et ( X, η , J, K ) b e a p oset net bund le. Then ther e is a c anonic al or dering ≺ on the total sp ac e X making η a p oset epimorph ism. In this way, morp hisms of p oset net bund les give rise to p oset morphisms of the underlying total sp ac es. Pr o of. Define x 1 ≺ x 0 ⇔ a 1 := η ( x 1 ) ≤ η ( x 0 ) =: a 0 and J ( a 0 ,a 1 ) ( x 1 ) ≤ a 0 x 0 . The p revious lemma illustrates the main adv an tage of considerin g p oset n et bund les: in fact, the u s ual n et bu ndles ( X , p, J, K ) corresp ond to partial orderin gs wh ere eac h fibre X a has the “discrete” order relation x ≤ x ′ ⇔ x = x ′ , x, x ′ ∈ X a (see [13, § 4.1]), whereas a p oset n et bun dle may b e end o w ed with a more in teresting partial ordering. By th e previous lemma it mak es sense to consid er the maps η ∗ : H 1 ( K, G ) → H 1 ( X, G ) , (4 .3) η ∗ : π 1 ( X, x ) → π 1 ( K, a ) , x ∈ X a , (4.4) for eve ry p oset net bundle ( X, η , J, K ) and group G . Theorem 4.3. L et ( X , η , J, K ) b e a p oset net bu nd le with p athwise c onne c te d fibr e F . Then for e ach a ∈ Σ 0 ( K ) ther e is a morphism j a : π 1 ( X a , x ) → π 1 ( X, x ) with π 1 ( X a , x ) ≃ π 1 ( F , a ′ ) , x ∈ X a , a ′ ∈ Σ 0 ( F ) , and an exact se qu enc e π 1 ( X a , x ) j a → π 1 ( X, x ) η ∗ → π 1 ( K, a ) → 0 . (4.5) In p articular, if ( F , ≤ ) is simply c onne cte d, then η ∗ is an isomo rphism Pr o of. The morp hism j a is ind uced by the ord er -p reserving inclusion j a : ( X a , ≤ a ) → ( X, ≺ ). Clearly , η ∗ j a [ p ] = [ σ 0 a ] , [ p ] ∈ π 1 ( X a , x ) , where σ 0 a , a ∈ Σ 0 ( K ), d enotes, as usual, the degenerate 1-simplex. In other w ords, j a ( π 1 ( X a , x )) ⊆ k er η ∗ . Let p := b n ∗ · · · ∗ b 1 b e a closed p ath in Σ 1 ( K ). W e fi x | v 1 | ∈ X | b 1 | and d efine r ecursiv ely      | v k +1 | := J | b k +1 | ,∂ 0 b k J − 1 | b k | ,∂ 0 b k | v k | ∂ i v k +1 := J − 1 | b k +1 | ,∂ i b k +1 | v k +1 | , i = 0 , 1 v k := ( | v k | ; ∂ 0 v k , ∂ 1 v k ) 11 (recalling that ∂ 0 b k = ∂ 1 b k +1 , k = 1 , . . . , n − 1). I n this wa y , w e obtain a path ˜ p 0 := e b n ∗ · · · ∗ e b 1 in Σ 1 ( X ) suc h that η ∗ ˜ p 0 = p . S ince ˜ p 0 ma y b e not closed, we consider a p ath ˜ p 1 in Σ 1 ( X ∂ 1 b 1 ≃ F ) starting in ∂ 0 v n and ending in ∂ 1 v 1 , and define ˜ p := ˜ p 0 ∗ ˜ p 1 . By construction, η ∗ ( ˜ p ) coincides with the c losed path b n ∗ . . . ∗ b 1 ∗ σ 0 ∂ 1 b 1 ∗ . . . ∗ σ 0 ∂ 1 b 1 where σ 0 ∂ 1 b 1 is the deg enerate 1-simplex in Σ 1 ( K ). The closed path constructed ab ov e is clea rly homotopic to p . F or the con v erse, let ˜ p := v n ∗ · · · ∗ v 1 b e a closed path in X with ˜ p ∈ k er η ∗ . W e pro v e t hat ˜ p is homotopic to a p ath in X a for some a ∈ Σ 0 ( K ). T o this end , w e consider the path in K η ∗ ( ˜ p ) := b n ∗ b n − 1 ∗ · · · ∗ b 1 , b k := η ∗ v k , k = n , . . . , 1 . Since ˜ p ∈ ker η ∗ , we ma y find 2-simplices c 1 , . . . , c m ∈ Σ 2 ( K ) pro viding elementa ry deformations o f η ∗ ( ˜ p ) to the constant path σ 0 a , a := ∂ 1 b n , σ 0 a := ( a ; a, a ) . In particular we m a y ha ve, for example, an elemen tary deformation con tracting b 2 ∗ b 1 to ∂ 1 c 1 . Now, v 2 ∗ v 1 is a path in η − 1 ( V | c 1 | ); using the definition of lo cal chart ([13, § 4.3]) and Lemm a 4.2, w e conclude that there is an isomorph ism of p osets η − 1 ( V | c 1 | ) ≃ V | c 1 | × F . The image of v 2 ∗ v 1 under the ab o ve isomorph ism can b e written, according to § A.1, as a pair ( p α , p µ ), where p α = ∂ 1 c 1 is a path in V | c 1 | and p µ is a path in F . Applying (A.1), w e co nclude that v 2 ∗ v 1 is homotopic to a path v ′ m ∗ . . . ∗ v ′ 1 suc h that η ∗ ( v ′ m ∗ . . . ∗ v ′ 1 ) = ∂ 1 c 1 ∗ σ 0 a ∗ . . . ∗ σ 0 a . Th us, ˜ p is h omotopic to a p ath of th e t yp e ˜ p 1 := v n ∗ v n − 1 ∗ · · · ∗ v 3 ∗ v ′ m ∗ . . . ∗ v ′ 1 , Note th at if there is a c 2 ∈ e Σ 2 ( K ) con tr acting b 3 ∗ ∂ 1 c 1 , then v 3 ∗ v ′ m ∗ . . . ∗ v ′ 1 is a p ath in η − 1 ( V | c 2 | ) ≃ V | c 2 | × F , so that we can again apply the ab ov e argument. Moreo ve r, the same p r ocedu re applies to ampliations of η ∗ ( ˜ p ). Iterating th e ab o ve op erations for eac h deformation of η ∗ ( ˜ p ), w e conclud e that ˜ p is h omotopic to a p ath ˜ p m suc h that η ∗ ( ˜ p m ) = σ 0 a ∗ . . . ∗ σ 0 a , i.e. ˜ p m is a path in X a . This pro v es k er η ∗ = j a ( π 1 ( X a , x )) . Corollary 4.4. L et ( X, η , J, K ) b e a p oset net bund le with simply c onne cte d fibr e ( F , ≤ ) . Then the maps (4.3), (4.4) define gr oup i somorphisms. Pr o of. It su ffices to note that π 1 ( F , a ′ ) = 0 and apply (4.2), (4.5 ). 12 The previous theorem h as a we ll-kno wn coun terpart, inv olving top ologica l spaces; in this case, (4.5) is a long exact s equence inv olving higher homotop y groups. O ne migh t hop e to generalize Thm .4.3 and get a lo ng exact sequence b y in tro ducing higher homotop y g roup s for p osets. No w let X := ( X, p, J, K ) b e a net bund le of top ologica l spaces, s o that eac h J b , b ∈ Σ 1 ( K ), is a homeomorph ism. W e fix a p oset X ≺ ,a of op en subsets U ⊂ X a , U 6 = ∅ , ord ered un der inclusion and forming a base for X a . W e r equire that eac h J b is a p oset isomorph ism from X ≺ ,∂ 1 b to X ≺ ,∂ 0 b . Then defining X ≺ := ˙ ∪ a X ≺ ,a and letting p ≺ : X ≺ → K b e the ob vious p ro jection, we conclude t hat X ≺ := ( X ≺ , p ≺ , J, K ) (4.6) is a p oset net bund le, called the p oset net bund le asso ciate d with X . By [14, T h m.2.8], the top ological homotop y grou p π 1 ( X a ) is isomorphic to π 1 ( X ≺ ,a , U ), a ∈ Σ 0 ( K ), U ∈ Σ 0 ( X ≺ ,a ); thus, by T h m.4.3 w e conclude the f ollo wing: Corollary 4.5. L e t X b e a net bund le of top olo gic al sp ac es with fibr e a Hausdorff, lo c al ly ar cwise and simply c onne cte d sp ac e M . F or e ach a ∈ Σ 0 ( K ) , ther e is an exact se quenc e π 1 ( M ) j a → π 1 ( X ≺ , U ) η ∗ → π 1 ( K, a ) → 0 . (4.7) Example 4.6. F or the notion of principal net bun dle, w e refer th e r eader to [1 3 ]. Let P := ( P , π , J, K , R ) b e a p rincipal net bun dle with a connected Lie group G as standard fibre. T hen (pro vided th e net structure is defined by means of isomorphisms of Lie groups), P is also endo w ed with the structur e of a top ological net bund le and the pr evious corollary ap p lies. The global space of a net b undle of top ological spaces X := ( X, p, J, K ) can b e top ologi zed in th e follo wing w a y: pic k for eac h a ∈ Σ 0 ( K ) a b ase U a for the top ology of X a . If U ∈ U a , we defin e the cylinder with b ase U to b e T a,U := { x ∈ J ( o,a ) ( U ) , o ∈ V a } . (4.8) Clearly p ( T a,U ) = V a . The family { T a,U } is a base for a top ology on X and we denote the asso ciated top ologica l space by τ X . Th is top ology is indep endent of the c hoice of bases. Note t hat there is a con tinuous b ijectio n T a,U → V a × U , x 7→ ( p ( x ) , J ( a,p ( x ) x ) . The p ro jection p is therefore con tinuous as a map f rom τ X to τ K . Lemma 4.7. L et X := ( X , p, J, K ) , ˆ X := ( ˆ X , ˆ p, ˆ J , K ) b e net bund les of top olo gic al sp ac es and f ∈ ( X , ˆ X ) a morphism. Then f : τ X → τ ˆ X is c ontinuous. Pr o of. It s uffices to sh o w that if ˆ V is op en in ˆ X a then f − 1 ( T a, ˆ V ) = T a,f − 1 ( ˆ V ) . If x ∈ J ( o,a ) ( f − 1 ( ˆ V )) then f ( x ) ∈ f J ( o,a ) ( f − 1 ( ˆ V )) ⊂ ˆ J ( o,a ) ( ˆ V ). Hence T a,f − 1 ( ˆ V ) ⊂ f − 1 ( T a, ˆ V ). If f ( x ) ∈ ˆ J ( o,a ) ( ˆ V ) then x ∈ f − 1 ˆ J o,a ( ˆ V ) = J o,a f − 1 ( ˆ V ) so f − 1 ( T a, ˆ V ) ⊂ T a,f − 1 ( ˆ V ) . 13 When X is trivial, the previous Lemm a implies that there is a homeomorphism τ X ≃ τ K × M , (4.9) where τ K is th e space K with the Alexandr off top olog y (see [13, § 2.3]). The pr evious elemen tary remark giv es an idea of th e lo cal b ehaviour of τ X for a generic X , and implies that the bundle p : τ X → τ K is lo cally t rivial when X is lo cally trivial. If s ∈ S ( V a ; X ) and s ( a ) ∈ U , then s ( o ) ∈ T a,U for eac h o ∈ V a and s − 1 ( T a,U ) = V a ; w e co nclude that eac h s ∈ S ( V a ; X ) defines a local sect ion s : V a → τ X . W e n o w consider the asso ciated net b undle of top ological spaces X ◦ , see (4.1), and the asso ciated poset X ◦ , ≺ , s ee (4.6) and Lemma 4.2. Theorem 4.8. L et X := ( X, p, J, K ) b e a net bund le of top olo gic al sp ac e s with standa r d fibr e a sp ac e M . Then the map p : τ X → τ K defines a fibr e bund le with fibr e M . Mor e over, ther e is an isomorphism π 1 ( X ◦ , ≺ , U ) ≃ π 1 ( τ X ) , U ∈ Σ 0 ( X ◦ , ≺ ) , (4.10) and for every gr oup G ther e ar e isomorp hisms H 1 ( X ◦ , ≺ , G ) ≃ ˙ Hom( π 1 ( τ X ) , G ) . (4.11) Pr o of. In order to pro v e th e theorem it just remains to c hec k the details in (4.10). T o this end, let us consider the p oset X cy l with elemen ts the sets (4.8), ordered un der inclusion. Since X cy l is a base for τ X , w e hav e an isomorphism π 1 ( X cy l , T ) ≃ π 1 ( τ X ), T ∈ Σ 0 ( X cy l ). Th us, in ord er to prov e (4.10), it suffices to constru ct a p oset isomorphism from X ◦ , ≺ to X cy l . By the definition of X ◦ , X ◦ , ≺ coincides w ith X ≺ as a set, thus elemen ts of X ◦ , ≺ are giv en by op en sets U ∈ X ≺ ,a , a ∈ Σ 0 ( K ◦ ). The order r elation for X ◦ , ≺ is giv en by U ≺ U ′ ⇔ a ≥ a ′ and J ◦ ( a ′ ,a ) ( U ) ⊆ U ′ , i.e. U ⊆ J ( a,a ′ ) ( U ′ ) (see Lemma 4.2). W e consider the biject ive map X ≺ → X cy l , U ∈ X ≺ ,a 7→ T a,U . (4.12) By construction, U ≺ U ′ implies V a ⊆ V a ′ and U ⊆ J ( a,a ′ ) ( U ′ ), thus T a,U ⊆ T a ′ ,U ′ , and the th eorem is p ro v ed. Let M b e a sp ace and P → M a locally trivial bun dle. W e say that P is lo c al ly c onstant whenever it has a set of lo cally constan t transition maps (that is, the tr ansition maps are constan t on the connected c omp onents of their d omains). Corollary 4 .9. L et G b e a gr oup. F or e v ery princip al net G -bund le P := ( P , p, J, R, K ) , the fibr e bund le p : τ P → τ K is lo c al ly c onstant. Pr o of. F or ev ery a ∈ Σ 0 ( K ), V a is a n open n eigh b ourho o d of a trivializing τ P , so ther e are lo cal c harts θ a : V a × G → p − 1 ( V a ), a ∈ Σ 0 ( K ), giving rise to lo cally constan t transition m aps (see [13, Lemma 5.6]). 14 5 K-theory of a p oset. 5.1 Net bundles of B anac h spaces. W e n ow sp ecialize our d iscussion to the case of quasin et bundles of Banac h spaces. These ob j ects hav e alrea dy b een studied in a com binatorial se tting ([9]). In the p resen t pap er, w e emphasize the top ological vie wp oint. Banac h qu asinet bun dles are quasinet bu ndles ha ving Banac h spaces as fibr es, and net stru cture giv en b y b oun ded, inj ective , linear op erators. If E := ( E , π , J, K ), ˆ E := ( ˆ E , ˆ π , ˆ J , ˆ K ) are Banac h qu asinet bund les, then we denote the set of b ound ed morph isms from E in to ˆ E by ( E , ˆ E ). If T ∈ ( E , ˆ E ), then eac h T a : E a → ˆ E a , a ∈ Σ 0 ( K ), is a b ounded linear op erator satisfying the rela tions T ∂ 0 b J b = ˆ J b T ∂ 1 b , b ∈ Σ 1 ( K ) . (5.1) If E , ˆ E are Banac h net bund les, then ( K b eing path wise connected) (5.1) implies that k T a k d oes not dep end on the c hoice o f a ∈ Σ 0 ( K ). The Banac h quasinet bun dle E is said to b e finite dimensional if the dim en sion of the fib res E a , a ∈ Σ 0 ( K ), has an up p er b oun d d ∈ N . S ince eac h J b , b ∈ Σ 1 ( K ), b et wee n v ector spaces with th e same fi nite dimension is also sur jectiv e, w e ha ve Lemma 5.1. L et E b e a finite-dimensional Banach quasinet bund le. Then E is a Banach net bund le if and only if the r ank function d ( a ) := dim( E a ) , a ∈ Σ 0 ( K ) , is c onstant. A Banac h quasinet bundle E is said t o b e a Hi lb ert quasinet bund le if eac h fi bre E a , a ∈ Σ 0 ( K ), is a Hil b ert space, and eac h J b , b ∈ Σ 1 ( K ), preserv es the scala r pro duct. Some remarks on the classification of Banac h net bu ndles follo w. If we wan t to apply (4.2) to Banac h net bundles, then we ha v e to pick the group G to tak e accoun t of the structure of interest. F or examp le, we may consider the group of inv ertible op erators of a Banac h space, or the unitary group U of a Hilb ert space H if w e are inte rested in isomorphisms preserving the Hermitian str u cture. In particular, in the finite-dimensional case, G is the co mplex linear group GL ( d ), d ∈ N , or the unitary group U ( d ). Let us denote t he set of isomorphism classes of Hilb ert net bu ndles with fi b re H b y ˙ H net ( K, H ), and th e ˇ Cec h cohomolog y of the dual p oset K ◦ b y H 1 c ( K ◦ , U ). App lying [13, T hm.8.1], w e find Prop osition 5.2. F or e ach p oset K and a ∈ Σ 0 ( K ) , ther e ar e isomorph isms ˙ H net ( K, H ) ≃ H 1 ( K, U ) ≃ H 1 c ( K ◦ , U ) ≃ ˙ Hom( π 1 ( K, a ) , U ) . Unlik e ordinary v ector b undles, a finite-dimensional Banac h net b u ndle cannot in general b e endo wed with an Hermitian stru cture, in fact the equiv alence relation induced b y inn er automorphisms lea v es the d eterminan t map d et χ ( p ), χ ∈ Hom( π 1 ( K, a ) , GL ( d )), p ∈ π 1 ( K, a ), inv arian t. Th us, if χ do es not take v alues in U ( d ), the same is tru e of ev ery χ ′ ∈ Hom( π 1 ( K, a ) , GL ( d )) equiv alen t t o χ . 15 W e denote the category of Hilb ert quasinet b undles o ver K b y q H net ( K ), and the f ull sub category of Hilb ert net bundles by H net ( K ). F or finite d imensional Hilb ert quasinet bund les, we use the analogo us notations q V net ( K ) and V net ( K ). Some algebraic stru ctures are naturally defin ed on q H net ( K ): ( 1) Th e dir ect s um E ⊕ ˆ E ; (2) The adjoin t ∗ : ( E , ˆ E ) → ( ˆ E , E ); (3) T h e tensor pro duct E ⊗ ˆ E ; (4) T he symmetry θ ∈ ( E ⊗ ˆ E , ˆ E ⊗ E ), θ a ( v ⊗ ˆ v ) := ˆ v ⊗ v , v ∈ E a , ˆ v ∈ ˆ E a ; (5) The conjugate net bund le E := ( E , π , J , K ); (6 ) The an tisymmetric tensor p o wers λ r E , r ∈ N . There is a Banac h quasinet bun dle of morph isms B ( E , ˆ E ) asso ciat ed w ith Hilb ert quasinet b undles E , ˆ E . It is defined as follo ws: f or eac h a ∈ K , consider the vecto r space ( E a , ˆ E a ) of linear op erators from E a in to ˆ E a and the disjoint union B ( E , E ′ ) := ˙ ∪ a ( E a , ˆ E a ). Then define t he net structure I b ( t ) := ˆ J b tJ − 1 b , t ∈ ( E ∂ 1 b , ˆ E ∂ 1 b ), b ∈ e Σ 1 ( K ). In the finite-dimensional c ase, w e clearly hav e an isomorphism o f Ba nac h quasinet bun- dles B ( E , ˆ E ) → E ⊗ ˆ E . (5.2) The ab ov e isomorphism ind uces an Hermitian s tr ucture on B ( E , ˆ E ), making it in to a Hilb ert quasinet bundle. Lemma 5.3. Every morphism T ∈ ( E , ˆ E ) defines a se ction of B ( E , ˆ E ) . Pr o of. If T ∈ ( E , ˆ E ), then we ma y regard T as a map T : Σ 0 ( K ) → B ( E , ˆ E ), T ( a ) := T a . By the definition of morph ism, we find that T satisfies the compatibilit y proper ty w.r.t. the n et structure of B ( E , ˆ E ), i.e. I b ◦ T ( ∂ 1 b ) = T ( ∂ 0 b ). The existence of the adjoin t and norm on q H net ( K ) imply that the space ( E , E ) is a unital C* -algebra. W e can n o w pro v e the follo wing: Lemma 5.4. The c ate gory q H net ( K ) has sub obje cts. Pr o of. Consider E ∈ q H net ( K ), E := ( E , π , J, K ) and a pro jection P ∈ ( E , E ). W e define P E := ( P E , π ′ , J ′ , K ), w h ere P E := ˙ ∪ a P a E a , π ′ is the map from P E onto K , and J ′ b v := J b v , v ∈ P ∂ 1 b E ∂ 1 b , b ∈ Σ 1 ( K ), a ∈ Σ 0 ( K ). Since J b P ∂ 1 b = P ∂ 0 b J b , w e conclude that J ′ b P ∂ 1 b E ∂ 1 b ⊆ P ∂ 0 b E ∂ 0 b , so that J ′ is a well -defin ed net stru cture, and P E is a Hilb ert quasinet bundle. Hilb ert quasinet b undles asso ciate d with p ro jections as in the previous lemma are called dir e ct summands . A Hilb ert net bu ndle E is said to b e irr e ducible if it do es not admit direct su mmands differen t from 0 and E . W e sum m arize the results of the present section in th e f ollo wing prop osition. Prop osition 5.5. The c ate gory q H net ( K ) with arr ows b ounde d morphisms is a sym- metric tensor C*-c ate gory with sub obje cts, dir e ct sums, and identity obje c t give n by the trivial Hilb ert net bund le ι := ( K × C , π , j, K ) . (When K is p athwise c onne cte d) ι is simple, i.e. ( ι, ι ) = C . In the n ext lemma, w e establish an equiv alence b et w een th e p r esence of n o w here zero global s ections and trivial direct summand s . 16 Lemma 5.6. L et E := ( E , π , J, K ) b e a Hilb ert quasinet bu nd le. Ther e is a nowher e zer o se ction σ ∈ S ( K ; E ) if and only if E has a trivial dir e ct summand of r ank one. Thus, an irr e ducible Hilb ert net bund le is nontrivial if and only i f it la cks nowher e zer o se ctions. Pr o of. If E has a trivial direct su m mand, then it is clear that there is a n o w here zero section σ : K → E . C on v ersely , s ince the net stru cture J inv olve s isometric maps, if there is suc h a section σ then u p to n ormalizat ion we ma y assume that k σ ( o ) k = 1, o ∈ K . Given the trivial net bundle ι := ( K × C , p, j, K ), w e defin e the map I σ ( o, z ) := z σ ( o ) o ∈ K , z ∈ C . It is clear that I σ is injectiv e. Since σ ( o ) ∈ E o , o ∈ K , I σ preserve s the fi bres. W e n o w ve rify that I σ preserve s the n et structure, i.e. I σ ◦ j = J ◦ I σ : for ev ery b ∈ Σ 1 ( K ), w e compute I σ ◦ j b ( ∂ 1 b, z ) = I σ ( ∂ 0 b, z ) = z σ ( ∂ 0 b ) = z J b ◦ σ ( ∂ 1 b ) = J b ◦ I σ ( ∂ 1 b, z ) , where we used the fact that J b ◦ σ ( ∂ 1 b ) = σ ( ∂ 0 b ). This p ro v es th at I σ is a n et bu n dle morphism. Corollary 5.7. L et d ∈ N and E := ( E , π , J, K ) b e a r ank d H ilb ert net bund le. If ther e ar e d line arly indep endent se ctions of E , then E is t rivial. The ab o v e co rollary sho ws that the e xistence o f sufficien t glob al sectio ns means that the Hilb ert net bun d le is trivial, in contrast to the u s ual top ological setting. The reason is t hat, a section of a Hilb ert n et bun dle not z ero o v er some o ∈ K is no w h ere zero. Of course, this is not true for ordinary v ector bundles. Th e Chern functions in trod uced in Sec.5.3 measur e the ob s truction to the existence of global sect ions. 5.2 Basic Prop erties of K -theory . In th e sequel, we alwa ys assume that our Hilb ert net bun d les are finite-dimensional. Let E b e a Hilb ert n et b undle o v er K . W e denote the isomorp hism class of E by { E } . Direct sum and tensor p rod uct induce a natural semiring structur e on the set ˙ V net ( K ) of isomorphism classes of fi n ite-dimensional Hilb ert net bu ndles. W e defi ne K 0 ( K ) to b e the Grothendiec k rin g asso ciated w ith ˙ V net ( K ), and d en ote the semir ing morph ism assigning to eac h isomorphism c lass { E } the asso ciated elemen t of K 0 ( K ) b y ˙ V net ( K ) → K 0 ( K ) , { E } 7→ [ E ] . (5.3) By definition, every element of K 0 ( K ) can b e wr itten as [ E ] − [ F ], with { E } , { F } ∈ V net ( K ). In analogy with the us ual K-theory , we c haracterize Hilb ert net bund les E , E ′ → K w ith [ E ] = [ E ′ ]: as this result d ep ends only on the definition of Grothendiec k rin g, we omit the pro of. Lemma 5.8. L et E , E ′ b e Hilb ert net bund les over K . Then [ E ] = [ E ′ ] ∈ K 0 ( K ) if and only if ther e exists a Hilb e rt net bund le F su ch tha t E ⊕ F is isomorph ic to E ′ ⊕ F . The n ext t w o results ind icate a d rastic difference b et w een ordinary K-theory and n et K-theory . In f act, we are going to prov e that eve ry Hilb ert net bu ndle with a complement is t rivial (in ordinary K-t heory , ev ery v ector bundle has a complemen t, see [6, I.6 .5]). 17 Prop osition 5.9. L et T d b e the trivial r ank d Hilb ert net bund le. Then every dir e ct summand of T d is trivial. Pr o of. Let E ′ b e a Hilb ert net subbu n dle of T d , and P ∈ ( T d , T d ) the pro jection asso ciated with E ′ . Then P is a section of the b undle B ( T d , T d ) (see Lemma 5.3). No w, B ( T d , T d ) is isomorphic to th e trivial net bundle K × M ( d ) (w h ere M ( d ) denotes the matrix algebra of order d ). Thus, every s ecti on of B ( T d , T d ) is constan t. In particular, P is a c onstant section whose range is a fixed vect or subs pace V of C d , and E ′ is isomorp h ic to the trivial bund le K × V . Corollary 5.10. L et E , ˆ E b e Hilb ert net bund les such that E ⊕ ˆ E ≃ T n for some n ∈ N . Then E , ˆ E ar e trivial. Hilb ert n et bun dles E , F (not n ecessarily of the same rank) are said to b e stably e qui v alent if E ⊕ T j is isomorphic to F ⊕ T k for s ome j, k ∈ N . It follo ws from the pr evious corollary that if a Hilb ert net bund le is st ably equiv alen t to a trivial net bun d le, then it is trivial. The set V s net ( K ) of stable equiv alence classes is an Ab elian semiring w.r.t. th e op eration of direct sum a nd te nsor pro duct. By definition, there is an e pimorp hism ˙ V net ( K ) → V s net ( K ) (5.4) mapping eac h { E } ∈ ˙ V net ( K ) into its stable equiv alence class. The kernel of (5.4 ) is giv en b y the set of isomorp hism classes o f trivial Hilb ert n et b undles, hence la b elled b y the non-negativ e in tegers. F or the notion of represen tation ring, w e recommend [1 5] to the r eader; the fo llo wing result is a direct c onsequence of [14, Thm.2.8 ]. Theorem 5.11. L e t a ∈ Σ 0 ( K ) . Then the c ate gory V net ( K ) is e quivalent to the c ate gory of unitary, finite-dimensional r epr ese ntations of π 1 ( K, a ) , so that K 0 ( K ) is i somorphic to the r epr esentation ring of π 1 ( K, a ) . Corollary 5.12. L et E := ( E , π , J, K ) b e a Hilb ert net bund le. Then ther e is a unique de c omp osition E = ⊕ r i =1 n i V i , w her e e ach V i is irr e ducible, n i ∈ N , and n i V i denotes the dir e ct sum of n i c opies of V i . Pr o of. In fact, the ab o v e deco mp osition corresp onds to the decomposition of the r epre- sen tation χ : π 1 ( K, a ) → U ( d ) asso ciated with E in to irr educibles. Note that a Hilb ert net bund le is irreducible if and only if the asso ciated represen tation of π 1 ( K, a ) is irre- ducible. Corollary 5.13. L et ( K, ≤ ) b e a p oset such that π 1 ( K, a ) is Ab elian. Then every Hilb ert net bund le over K is a dir e ct sum of line net bund les. The ring K 0 ( K ) is g e ner ate d by H 1 ( K, T ) as a Z -mo dule. Let us no w consider the r ank function ρ : ˙ V net ( K ) → N , assigning to the (isomorphism class of a) Hilb ert net bundle the corresp ond ing rank. It is clear that ρ is a semiring epimorphism, so that it mak es sense to consider the asso ciated extension ρ : K 0 ( K ) → Z . 18 The kernel of ρ is c alled the r e duc e d net K -theory of K and denote d b y e K 0 ( K ). I n this w a y w e hav e a direct su m decomp ositio n K 0 ( K ) = Z ⊕ e K 0 ( K ) . (5.5) so that e K 0 ( K ) emb eds in to K 0 ( K ). Th e reduced group e K 0 ( K ) enco des the nontrivial part of th e K-theory of K : if K is simply conn ecte d, then e K 0 ( K ) = 0. By the defi n ition of stable equ iv alence class and recalling (5.4 ,5.5), w e conclude that the canonical map ˙ V net ( K ) → K 0 ( K ) factorizes through V s net ( K ), in such a wa y that th e f ollo wing d iag ram comm utes: ˙ V net ( K ) / /   V s net ( K )   K 0 ( K ) e K 0 ( K ) o o (5.6) Net K-theory satisfies natural fun ctorial prop erties. If η : K ′ → K is a p oset morph ism , then the pullbac k induces a ring morphism η ∗ : K 0 ( K ) → K 0 ( K ′ ). If X := ( X, η , J, K ) is a p oset net b undle with fi bre F , then b y ( 4.5) w e fin d an exact sequence o f rings Z → K 0 ( K ) η ∗ → K 0 ( X ) → K 0 ( F ) . (5.7) Note that the images of Z w.r.t. the ab ov e m aps corresp ond to trivial repr esen tations of the homotop y g roup s π 1 ( K, a ), π 1 ( X, x ), π 1 ( F , a ′ ); so that, ( 5.7) restricts to a n exact sequence of reduced K -groups 0 → e K 0 ( K ) η ∗ → e K 0 ( X ) → e K 0 ( F ) . W e conclude that if K 0 ( F ) = Z (i.e. e K 0 ( F ) = 0 ), then e K 0 ( X ) and e K 0 ( K ) are isomo r- phic. Let u s now consider a rank d Hilb ert net bu ndle E := ( E , π , J, K ). F or ev ery fi bre E a , a ∈ Σ 0 ( K ), w e consider the asso ciated pro jectiv e space P E a . Eac h un itary J b , b ∈ Σ 1 ( K ), defin es a homeomorphism [ J ] b : P E ∂ 1 b → P E ∂ 0 b . Defining P E := ˙ ∪ a P E a and the ob vious p r o jection [ π ] : P E → K , w e obtai n a net bundle of t op ological spaces PE := ( P E , [ π ] , [ J ] , K ) , (5.8) called t he p r oje ctive net bund le associated with E . W e denote b y PE ≺ := ( P E ≺ , [ π ] ≺ , [ J ] , K ) the p oset net bundle asso ciate d with PE in the sense o f Sec.4.2. Theorem 5.14 (The Thom isomorphism) . L et ( E , π , J, K ) b e a r ank d Hilb ert net bun- d le. F or every gr oup G , the pul lb ack over P E ≺ induc es isomorphisms [ π ] ∗ ≺ : H 1 ( K, G ) → H 1 ( P E ≺ , G ) , [ π ] ∗ ≺ : K 0 ( K ) → K 0 ( P E ≺ ) . (5.9) 19 Pr o of. Since the pr o jectiv e space is pathwise and simp ly connected, we apply Cor.4.4, Cor.4.5, an d conclude that the map [ π ] ≺ : P E ≺ → K in duces an i somorph ism [ π ] ≺ , ∗ : π 1 (( P E ≺ , v ) → π 1 ( K, a ) , v ∈ P E ≺ ,a . (5.10) Since we find Z 1 ( K, G ) ≃ Hom( π 1 ( K, a ) , G ) , Z 1 ( P E ≺ , G ) ≃ Hom( π 1 (( P E ≺ , v ) , G ) , for ev ery group G , the theorem is pro v ed at the lev el of cohomo logy . T h e isomorphism at the lev el of K-theory follo ws b y Thm.5.11 and (5.10). In contrast to ordinary geometry , the Thom isomorphism really is an isomorphism and not a monomorph ism. But this resu lt is n ot as useful as its top ologic al counterpart, since there is not a well -defin ed noti on of canonical line net bundle with base P E ≺ . W e shall r eturn to th is p oin t in the sequel (Rem.5.16). F or eve ry top ologic al space Y , we denote the cate gory of v ector b undles o v er Y by V top ( Y ), and th e sub category (not full) of lo call y constant v ector bund les with arrows lo cally constan t morphisms by V lc ( Y ) (see Sec.6 or [8, Ch.I.2]). No w, a Hilb ert net bund le E := ( E , π , J, K ) is a n et bun dle of top olog ical spaces in a natural wa y , so there is a n associated v ector b u ndle π : τ E → τ K (see Thm.4.8). Theorem 5 .15. F or every Hilb ert net bund le E := ( E , π , J, K ) , the pr oje ction π defines a c ontinuous map π : τ E → τ K , and τ E b e c omes a lo c al ly c onstant ve ctor bund le over τ K . Thus, ther e is a functor τ ∗ : V net ( K ) → V top ( τ K ) , E 7→ τ E , (5 .11) pr oviding an isomorphism V net ( K ) ≃ V lc ( τ K ) . F or every net bund le of top olo gic al sp ac es X := ( X , p, Φ , K ) , the top olo gic al pul lb ack p ∗ ( τ E ) defines a functor p ∗ : V net ( K ) → V lc ( τ X ) . (5.12) Pr o of. π : τ E → τ K is a v ector b undle as a direct consequence of Th m.4.8. The functorialit y of the m ap (5.11) follo ws b y Lemm a 4.7. Ap plying Cor.4.9 to the U ( d )– co cycle asso ciated with E w e see th at τ E is locally co nstant. Finally , p ∗ tak es v alues in V lc ( τ X ) since the p ullbac k of a lo cally constant bu ndle is locally constan t. W e emph asize the fact that τ K is ju st a T 0 -space, th us w e cannot use the usual mac hinery of differenti al geometry , as in [8, C h.I] for example, to stud y p r op erties of the lo cally constant bun dle τ E . Remark 5.16. The fibr e bundle [ π ] : τ P E → τ K has simp ly conn ected fib res, thus the top ologi cal v ersion of Th m.4.3, and the f act that τ P E is lo cally constant, imp ly that we ha v e isomorphisms [ π ] ∗ : π 1 ( τ P E ) → π 1 ( τ K ) , [ π ] ∗ : V lc ( τ K ) → V lc ( τ P E ) . (5.13) 20 No w, let us consider the top olog ical pullbac k [ π ] ∗ ( τ E ) → τ P E . By definition, [ π ] ∗ ( τ E ) has as elemen ts pairs ( v , ξ ) ∈ E a × P E a , a ∈ τ K , and w e can defi ne the c anonical line bund le τ L → τ P E by τ L := { ( v , ξ ) ∈ [ π ] ∗ ( τ E ) : v ∈ ξ } . As in [6, Ch.IV.2], th e restriction of τ L ov er P E a , a ∈ τ K , is ju st the usual canonical line b undle L a → P E a . Since L a is not lo cally constant τ L is not lo cally constan t, so it do es not b elong to the image of [ π ] ∗ , and [ π ] ∗ ( τ E ) = τ L ⊕ E ′ is d irect sum of not lo cally c onstant bundles. W e co nclud e th at the splitting principle cannot b e app lied i n the category of Hilb ert n et bun dles. 5.3 Chern classes for Hilb ert net bundles. As we sa w in the previous section, the algebraic prop erties of the category of Hilb ert net bund les o ve r a p oset K d iffer drastically from those of vect or bundles ov er a top ologica l space. This fact is also reflected in the construction of Chern classes. As we saw in Theorem 5.15 and sha ll see in § 6, Hilb ert net bu ndles are strict y related to locally constan t v ector bun dles, wh ic h ha ve trivial Chern classes w hen the b ase space is a manifold. A differen t approac h to Chern classes ma y b e b ased on the picture of Hilb ert net bund les as unitary repr esen tatio ns of the homotop y group. F rom this p oin t of view, there are s ome r esults ([17, 16]) on Ch ern classes asso ciated with repr esentati ons of a d iscr ete group G (in our case, the homotop y group). But u nfortunately , such Chern classes d o not suit our purp ose, as they are expr essed in terms of the group cohomo logy of G , or, equiv alent ly , in terms o f the singular cohomol ogy of a suitable Eilen b erg-McLane space asso ciate d with G . In b oth the cases, an immediate in terpretation in terms of p r op erties of the initial p oset is lost; moreo ve r, these classes v anish w h en the ab ov e-mentio ned space is homoto pic to a manifold. In the follo wing we d efi ne analogues of Ch ern classes, in terms of homotop y-inv ariant complex fu nctions on the path group oid of the p oset. 5.3.1 T he first Chern class. Let d ∈ N and E b e a Hilb ert net b u ndle of r ank d . According to (4.2), E is c harac- terized by a cohomology class z ∈ H 1 ( K, U ( d )). By the functorialit y of H 1 ( K, · ), the determinan t map det : U ( d ) → T induces a map d et ∗ : H 1 ( K, U ( d )) → H 1 ( K, T ). W e define the first Chern class of E to b e c 1 ( E ) := det ∗ z ∈ H 1 ( K, T ) . By the elementa ry prop erties of determinants, w e find c 1 ( E ⊕ E ′ ) = c 1 ( E ) c 1 ( E ′ ), and obtain an epimorphism of Abelian groups c 1 : K 0 ( K ) → H 1 ( K, T ) . (5.14) 21 The fi rst Chern c lass is natural in th e sense t hat, if η : K ′ → K is a morphism, then η ∗ c 1 ( E ) = c 1 ( η ∗ E ) . (5.15) F rom a catego rical p oin t of view, the first C hern class enco des the cohomologic al ob- struction for E to b e a special ob ject: if c 1 ( E ) = 0, then the totally ant isymmetric line net b undle λ d E is trivial, and ev ery normalized section R ∈ S ( K ; λ d E ) is a solution of the equ atio n ([3, (3.19)]). No w, c 1 ( E ) can b e regarded as a morp hism from π 1 ( K, a ) in to T . By Thm .3.6 , we conclude th at c 1 ( E ) factorizes through a morphism ˆ c 1 ( E ) ∈ Hom(H 1 ( K, Z ) , T ) , (5.16) that w e call the Ab elianize d first Chern class . By Cor.3.7 H 1 ( K, T ) is isomorphic to Hom(H 1 ( K, Z ) , T ), thus it is essent ially equiv alen t to consider ˆ c 1 ( E ) instead of c 1 ( E ); but since H 1 ( K, Z ) is generally easier to compu te, it is u sually con v enient to us e (5.16). 5.3.2 C hern K-classes W e no w adapt a classical construction to net K-theory ([6, IV.2.17]). Let E := ( E , π , J, K ) b e a rank d Hilb ert net b undle. W e consider th e antisymmetric tensor p o w ers λ k E , k = 1 , . . . , d , and define k i ( E ) := i X k =0 ( − 1) k  d − k i − k  [ λ k E ] , i = 1 , . . . , d . (5.17) Some elementa ry computations inv olving the dimens ions of an tisymmetric tensor p ow er s imply that ρ (k i ( E )) = 0, thus k i ( E ) ∈ e K 0 ( K ). W e call the classes k i ( E ) the Chern K- classes . Keeping (5.14) in min d, w e find c 1 (k 1 ( E )) = − c 1 [ E ]. Applying the we ll-kno wn iden tit y λ i ( E ⊕ E ′ ) = ⊕ l + m = i λ l E ⊗ λ m E ′ (5.18) to E ′ = T k , we see that if E has rank d and admits a tr ivial, rank k d irect su mmand, then k i ( E ) = 0, k ≤ i ≤ d . W e define the to tal Chern K- class to b e k( E ) := 1 + d X i =1 k i ( E ) h i , E ∈ V net ( K ) . By defi n ition, k( E ) = k( E ⊕ T k ) for ev ery k ∈ N . This fi ts in well with the id ea that k( E ) should enco de the nontrivia l p rop erties of E , and implies that the classes k i are w ell-defined for elemen ts of V s net ( K ). The ab o v e considerations and the naturalit y of the pullbac k yield Prop osition 5.17. We have k i ( E ⊕ E ′ ) = d + d ′ X l + m = i k l ( E ) k m ( E ′ ) , E , E ′ ∈ V net ( K ) . (5.19) 22 Thus, the total Chern K-class facto rizes thr ough a morphism k : V s net ( K ) → 1 + h e K 0 ( K )[[ h ]] , k( E ⊕ E ′ ) = k( E ) k( E ′ ) , such that η ∗ k( E ) = k( η ∗ E ) for every p oset morphism η : K ′ → K . 5.3.3 C hern functions. Let Π 1 ( K ) denote the set of p aths of K . Since eac h 1–simplex is a path of length 1, e Σ 1 ( K ) is con tained in Π 1 ( K ). Giv en a set S , a map f : Π 1 ( K ) → S is said to b e homoto py-invariant if f ( p ) = f ( p ′ ) w henev er p is homotopic to p ′ . Let E := ( E , π , J, K ) b e a rank d Hilb ert net bund le with asso ciated U ( d )-cocycle z ∈ Z 1 ( K, U ( d )). By [12, Eq.32], z can b e extend ed to a U ( d )-v alued, homotop y-in v arian t map on Π 1 ( K ); in particular, if p is homotopic to a constan t path, then z ( p ) is the iden tit y 1 ∈ U ( d ). Using the trace map T r a nd the exterior p o w ers ∧ k , k = 1 , . . . , d , w e define th e maps χ k z ( p ) := T r ∧ k z ( p ) , p ∈ Π 1 ( K ) . Clearly , the restriction of χ k z to e Σ 1 ( K ) yields an elemen t of C 1 ( K, C ). O n the other hand, if w e restrict χ k z to K ( a ), a ∈ Σ 0 ( K ), then by homotop y in v ariance we find that χ k z can b e regarded as the c haracter of the repr esen tation of π 1 ( K, a ) asso ciated with λ k E , k = 1 , . . . , d . I n particular, sin ce the trace is the iden tit y map for rank one represent ations, w e find χ d z = c 1 ( E ) , d = ρ ( E ) . (5.20) Let us denote the ring of b ounded, homotop y-in v arian t m aps fr om Π 1 ( K ) to C , b y R 1 ( K, C ). By analogy w ith the previous section, we int ro duce the Chern functions c i E ∈ R 1 ( K, C ) , c i E := i X k =0 ( − 1) k  d − k i − k  χ k z , i = 1 , . . . , d . (5.21) F or i > d , w e set c i E := 0. If T d is the t rivial Hilb ert net bundle of r ank d , then (5.18) implies c i T d = 0, i = 1 , . . . , d . If L ∈ V net ( K ) is a line net bu ndle with T -co cycle ζ , then from (5.20) w e fin d χ 1 ζ = c 1 ( L ) ⇒ c 1 L = 1 − c 1 ( L ) . (5.22) Since the trace is additiv e on direct su ms and multiplica tiv e on tensor pro ducts, the argumen t of P r op.5.17 shows th at c i ( E ⊕ E ′ ) = X l + m = i c l E · c m E ′ , (5.23) so th at c d + k ( E ⊕ T k ) = 0 , c i ( E ⊕ T k ) = c i E , i, k = 1 , . . . . (5.24) The previous equalities yield the u s ual int erpr etat ion of the functions c i E as obstr uctions to the trivialit y of E . Moreo ver, (5.24) imp lies that e ac h c i E dep ends o nly on the class of E in V s net ( K ). F or the first Chern class, we ha ve the f ollo wing result. 23 Lemma 5.18. F or a Hi lb ert net bund le E of r ank d , we have c 1 ( E ) = 1 + d X i =1 ( − 1) i c i E . (5.25) Pr o of. Let us recall the obvio us iden tit y N X j =0 ( − 1) j  N j  = 0 , N ∈ N . (5.26) W e put χ 0 z := 1, so that, by the definition of c i E , the r.h.s. of (5.25) co incides with d X i =0 ( − 1) i i X k =0 ( − 1) k  d − k i − k  χ k z . (5.2 7) Let us put together the co efficien ts of eac h χ k z and set j := i − k , s o that in particular ( − 1) i + k = ( − 1) j +2 k = ( − 1) j . Then we find that the q u an tit y (5.27) coincides with d X k =0   d − k X j =0  d − k j  ( − 1) j   χ k z . Using (5.26), we conclude that the terms b et w een the br ac kets v anish es, except in the case d = k , which pro vides the co efficien t χ d z . Thus, the r.h.s. of (5.25) coincides with χ d z , w hic h is equal to c 1 ( E ) b y (5.20). Theorem 5.19. Defining the p olynomial c E ( h ) := 1 + d X i =1 c i E h i for e ach E ∈ V net ( K ) gives a morphism c : V s net ( K ) → 1 + h R 1 ( K, C )[[ h ]] , c ( E ⊕ E ′ ) = c E · c E ′ , (5.28) such that c 1 ( E ) = c E ( − 1) . Pr o of. After Prop.5.17, (5 .23) and (5.2 5 ), t he only non trivial assertio n th at w e ha v e to v erify is the naturalit y of (5.28). Let η : K ′ → K b e a morp hism and z ∈ Z 1 ( K, U ( d )) th e co cycle asso ciated with E . Then η ∗ z ∈ Z 1 ( K ′ , U ( d )) is the co cycle asso ciated with the pullbac k η ∗ E , an d clearly χ k η ∗ z = χ k z ◦ η 1 , wh ere η 1 : Π 1 ( K ′ ) → Π 1 ( K ) is th e map induced b y η . This shows that t he functions c i E are n atural, i.e. c i ( η ∗ E ) = η ∗ c i E := c i E ◦ η 1 . 24 It is ins tructiv e to give the details when E = ⊕ d i L i is the d irect sum of line net bund les. S ince c is a h omomorphism, w e ge t c E ( h ) = d Y i =1 c L i ( h ) = d Y i =1 [1 + ˆ c 1 L i h ] = d Y i =1 [1 + (1 − c 1 ( L i )) h ] , (5.29) so th at w e obtain c i E = X 1 ≤ k 1 <... 1 , so H odd ( S 1 , R / Q ) ≃ Z ⊕ R / Q and (6.4) tak es the form Z [ T ] → Z ⊕ R / Q , ccs ( p ) = X z ∈ S n z ! ⊕ X z ∈ S n z log z mod Q ! , ∀ p ∈ Z [ T ] , ha ving used the logarithm log : T → R / Z and the quotien t mo d Q : R / Z → R / Q . 29 The homotopy g r oup oid of M is giv en by th e categ ory with ob jects t he p oin ts o f M and set of arro ws the set ˜ π 1 ( M ) of homotop y classes of con tin uous cur v es in M . W e denote the ring of b oun ded, complex functions on ˜ π 1 ( M ) b y R 1 ( M , C ) and th e Ab elian semigroup of stable equiv alence classes of lo cally co nstant vect or bu ndles, defined as i n (5.4), b y V s lc ( M ). Theorem 6.4. F or e ach lo c al ly c onstant v e ctor bund le E → M of r ank d , ther e ar e functions c i E ∈ R 1 ( M , C ) , i = 1 , . . . , d , such that c i ( E ⊕ E ′ ) = P l + m = i c l E · c m E ′ . The p olynomial c E ( h ) := 1 + d X i c i E h i defines a morp hism c : V s lc ( M ) → 1 + h R 1 ( M , C )[[ h ]] , c ( E ⊕ E ′ ) = c E · c E ′ . When π 1 ( M ) is Ab elian, R 1 ( M , C ) c an b e r eplac e d by the ring of b ounde d c omplex func- tions on the singular hom olo gy H 1 ( M , Z ) . Pr o of. By [1 4, Thm.2.18], the homotop y group oid of M is isomorphic to the h omoto py group oid of M ≺ , s o that there is a ring isomorphism R 1 ( M ≺ , C ) ≃ R 1 ( M , C ). T h us, we apply Th m.3.8, Thm.5.19 an d Rem.5.20. A Simplicial sets. A simplicial set is a con tra v ariant fun ctor from the s im p licial category ∆ + to the category of sets. ∆ + is a sub category of the category of sets ha ving as ob jects n := { 0 , 1 , . . . , n − 1 } , n ∈ N and as mapp ings the order preserving mappin gs. A simplicial set has a w ell kno wn description in te rms of generators, the face and degeneracy maps, and relatio ns. W e use the standard notatio n ∂ i and σ j for the fac e and degeneracy maps, and denote the comp ositions ∂ i ∂ j , σ i σ j , resp ectiv ely , by ∂ ij , σ ij . A p ath in a simp licia l set is an expression of the form p := b n ∗ b n − 1 ∗ · · · ∗ b 1 , where the b i are 1–simplices and ∂ 0 b i = ∂ 1 b i +1 for i = 1 , 2 , . . . , n − 1. W e set ∂ 1 p := ∂ 1 b 1 , ∂ 0 p := ∂ 0 b n and ℓ ( p ) := n the length of p . Concatenation give s us an obvious asso ciativ e comp osition la w for paths and in this wa y w e get a category without u nits. Homotop y pro vides us with an equiv alence relation ∼ on this structure. This is the equ iv alence relation generated by a fi nite sequence s ( i ), with i = 1 , . . . , k sa y , of elemen tary deformations. An elementary deformation of a path consists in replacing a 1–simplex ∂ 1 c of th e path b y a pair ∂ 0 c, ∂ 2 c , wh ere c ∈ Σ 2 , or, con versely in replaci ng a consecutiv e pair ∂ 0 c, ∂ 2 c of 1– simplices of p b y a single 1–simplex ∂ 1 c . The former t yp e 30 of deformation is called an ampliation of th e path, the latter a c ontr action . Quotien ting b y t his equiv alence rela tion yields th e homoto py category of the simplicia l set. W e shall mainly use symmetric simplicial sets. These are con tra v arian t f unctors fr om ∆ s to the category of sets, where ∆ s is the full su b categ ory o f the category o f set s with the same ob jects as ∆ + . A symmetric simp licial set also h as a description in terms of generators and relations, wh ere the generators n o w in clud e the p ermutatio ns of adjacent v ertices, denoted τ i . In a symmetric sim p licial set we define th e rev erse of a 1–simplex b to b e th e 1–simplex b := τ 0 b and the reve rse of a path p = b n ∗ b n − 1 ∗ · · · ∗ b 1 is the path p := τ 0 b 1 ∗ τ 0 b 2 ∗ · · · ∗ τ 0 b n . The reve rse acts as an inv erse after taking equiv alence classes so the homotopy category b ecomes a homotopy group oid. Giv en a symmetric simp licia l set e Σ ∗ w e denote its homo topy group oid by π 1 ( e Σ ∗ ). A.1 Homotop y of pro ducts Consider a pair e Σ α ∗ and e Σ µ ∗ of symmetric simplicial sets. Let Π α 1 and Π µ 1 denote the corresp onding set of paths, and let ∼ α and ∼ µ denote the corresp onding homotop y equiv alence relations. No w, consider the p ro duct simplicial set e Σ α ∗ × e Σ µ ∗ . Let Π α × µ 1 b e the s et of paths of the p ro d uct simplicial set and denote the homotop y equiv alence relation by ∼ . A path p in this set is a pair ( p α , p µ ), where p α , p µ are paths in Π α 1 and Π µ 1 resp ectiv ely with ℓ ( p α ) = ℓ ( p µ ). Note that a homotop y in this set is a finite sequence s ( i ) = ( s α ( i ) , s µ ( i )), with i = 1 , . . . , m to sa y , wh ere s α ( i ) a nd s µ ( i ) a re either b oth ampliations or b oth con tractions of the paths p α and p µ for an y i = 1 , . . . , m . In particular note th at, if p ∼ q , then p α ∼ α q α and p µ ∼ µ q µ . Our aim is to show that the fun damen tal group oid of the pr od uct simplicial set is equal to the p ro duct of the fundamental group oids. T o this end, consider the set Π α 1 × Π µ 1 pro duct of paths. Note that the el ements p of this set are pairs ( p α , q µ ) of paths where, in general, ℓ ( p α ) 6 = ℓ ( p µ ). Finally , observe that the pro of of our claim follo ws once w e ha v e sho wn that the identit y map Π α × µ 1 ∋ p → p ∈ Π α 1 × Π µ 1 , is injective and surj ecti ve up to equiv alence. It is easily seen to b e surj ecti ve. In f act l et p ∈ Π α 1 × Π µ 1 . Assu me that ℓ ( p α ) = ℓ ( p µ ) + k . If a is the starting p oint of th e path p µ in Σ µ ∗ , th en w e consider the path p µ ∗ ( σ 0 a ) ∗ k , w here ( σ 0 a ) ∗ k is the k -fold comp osition of the degenerate 1–simplex σ 0 a . C learly p µ ∗ ( σ 0 a ) ∗ k ∼ µ p µ and ( p α , p µ ∗ ( σ 0 a ) ∗ k ) ∈ Π α × µ 1 . F or inject ivit y we m ust sho w that p α ∼ α q α , p µ ∼ µ q µ ⇒ p ∼ q . (A.1) If p ′ ∼ p and q ′ ∼ q then it obvio usly suffices to p ose the question for p ′ and q ′ instead. p ∗ σ 0 ∂ 1 p is homotopic to p so it suffices to sup p ose that ℓ ( p ) = ℓ ( q ). Pick sequences s α 0 and s µ 0 of elemen tary deformations leading from p α to q α and from p µ to q µ . Since ℓ ( p α ) = ℓ ( q α ) and ℓ ( p µ ) = ℓ ( q µ ), eac h sequence con tains as man y ampliations as con tractions but they ma y not h a v e the same length. If ℓ ( s α 0 ) < ℓ ( s µ 0 ), s ay , w e ma y adjoin pairs 31 consisting of an ampliation adding on σ 0 ∂ 1 p α and a cont raction removing it again. W e ma y therefore rep lace s α 0 and s µ 0 b y sequences s α 1 and s µ 1 of the same length. W e now pro ceed in ductiv ely: if after n − 1 steps we hav e sequences s α n and s µ n and if the n − th elemen ts of s α n and s µ n are b oth ampliations or b oth con tractions, set s α n +1 := s α n and s µ n +1 := s µ n . If this is not the case and the n − th element of s α n , sa y , is a con traction, s α n +1 is obtained by inserting σ 0 ∂ 1 p α as n − th member and s µ n +1 b y addin g σ 0 ∂ 1 p α as a final element. The pro cess terminates with sequences s α k and s µ k , sa y , having amp liations and con tractions in the same relativ e p ositions. Setting s ( i ) := ( s α k ( i ) , s µ k ( i )). s is then a sequence of deformations fr om p to q ∗ σ 0 a ∗ · · · ∗ σ 0 a ∼ q , wh ere a = σ 0 ∂ 1 ( p α , p µ ), completing t he proof. A.2 Simplicial sets of a p oset W e shall b e concerned h ere with t wo d ifferen t simp licial sets that can b e asso ciate d with a p oset and we giv e their d efinitions not just for a p oset b ut for an arbitrary category C . T he first denoted Σ ∗ ( C ) is just the usual nerve of the category . Thus the 0–simplices are just the ob jects o f C , the 1– simplices are the arr o ws of C a nd a 2–simplex c is made up of its three faces which are arro ws satisfying ∂ 0 c ∂ 2 c = ∂ 1 c . The exp licit form of higher simplices will n ot b e needed in th is pap er. The h omoto py catego ry of Σ ∗ ( C ) is canonically isomorphic to C itself. T h e second simplicial set ˜ Σ ∗ ( C ) is a symmetric simplicial set. It is constructed as follo ws. Consider the p oset P n of non-v oid su bsets of { 0 , 1 , . . . , n − 1 } ordered under inclus ion. An y mapping f from { 0 , 1 , . . . , m − 1 } to { 0 , 1 , . . . , n − 1 } ind uces an order pr eservin g mapp in g from P m to P n . Regarding th e P n as categorie s, we h a ve realized ∆ s as a sub category of th e category of catego ries. W e then get a symmetric simp licial set where an n –simplex of ˜ Σ ∗ ( C ) is a f unctor from P n to C . Giv en a p oset K with order relation ≤ . W e recall that K is upwar d directed w henev er for an y p air o, ˆ o ∈ K there is ˜ o suc h that o, ˆ o ≤ ˜ o . It is downwar d directed if th e dual p oset K ◦ is up w ard d irected. The dua l K ◦ of K is the p oset h a ving the same e lement s as K and o pp osite order r elat ion ≤ ◦ , i.e., a ≤ ◦ ˜ a i f, and only if, a ≥ ˜ a . When w e sp ecialize to a p oset K th e nerve Σ ∗ ( K ) and e Σ ∗ ( K ) admit the f ollo wing represent ation (see [13] for details). A 0–simplex of e Σ ∗ ( K ) is ju st an element of th e p oset. F or n ≥ 1, an n − simplex x is form ed by n + 1 ( n − 1) − simplices ∂ 0 x, . . . , ∂ n x , and by a 0–simplex | x | called the supp ort of x such that | ∂ 0 x | , . . . , | ∂ n x | ≤ | x | . The nerv e Σ ∗ ( K ) turns out to b e a sub simplicial set of ˜ Σ ∗ ( K ). T o see this, it is enough to define f 0 ( a ) := a on 0–simplices and, inductiv ely , | f n ( x ) | := ∂ 01 ··· ( n − 1) x and ∂ i f n ( x ) := f n − 1 ( ∂ i x ). So w e obtain a simplicial map f ∗ : Σ ∗ ( K ) → ˜ Σ ∗ ( K ). W e sometimes adopt the follo wing notation: ( o ; a, ˜ a ) is the 1–simplex of ˜ Σ 1 ( K ) whose sup p ort is o and whose 0– and 1–face are, r esp ectiv ely , a and ˜ a ; ( a, ˜ a ) is the 1–simplex of the nerve Σ 1 ( K ) wh ose 0– and 1–face are, resp ectiv ely , a and ˜ a . The p oset K is said to b e p ath wise connected whenev er the simp licia l set ˜ Σ ∗ ( K ) is path wise connected. 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