Category of Noncommutative CW Complexes

Cate gory of Noncommutati v e CW Comple xe s ∗ † Do Ngoc Diep No vember 3, 2018 Abstract W e expos e the notion of nonc ommutati ve CW (NCCW ) comple xes, de- fine nonco mmutativ e (NC) mapping cylinder and NC mapping cone, and pro ve the noncommutati ve Approximatio n T heorem. The long exact homo - top y seq uences ass ociated with arbit rary morphisms are also deduce d. K e y W or ds : C*-alg ebra, nonco mm utativ e CW comple x, n oncommutative mapping cylind er , non commutative mapp ing cone . 1 Introduction Classical algebraic topology was fruitfully de veloped on the category of topologi- cal spaces with CW complex structure, see e.g. [W]. O ur goal is to show t hat with the same su ccess, theory can be developed in the frame work of noncommut ativ e topology . In noncom mutative geometry the notion of to pological spaces is changed by the notion of C*-algebras, motiv ating the spectra of C*-algebras as som e noncom- mutative spaces. In the works [ELP] and [P], it w as introduced the notion of non- commutative CW (NCCW) complex and proved some elementary properties of NCCW complexes. W e conti nue this line in proving some basic noncommutative results. In this paper we aim to explore the same prop erties of NC CW complexes, as the ones of CW com plexes from algebraic topology . In particular , we prove ∗ V ersion from November 3, 2018. † The work was supported in part by Vietnam National Project f or Research in Funda- mental Sciences and was c ompleted during the stay in Ju ne - July , 2007 a t the Abdus Sla m ICTP , T rieste , Italy 1 some NC Cellular App roximation Th eorem and the existence o f ho motopy exact sequences associated with morphisms . In the work [D3] we i ntroduced the notion of NC Serre fibrations (NCSF) and studied cyclic theories for the (co)hom ology of these NCCW complexes. In [D4] we studi ed t he Leray-Serre spectral sequ ences related with cyclic theories: periodic c yclic homology and KK-theory . In [DKT1] and [DKT2] we computed some noncom mutative Chern characters. Some deep study should be related with the Busby in v ariant, studied in [D1], [D2]. Let us describe in mo re detail th e content of the paper . In Section 2 we ex- pose the pu llback and pu shout d iagrams of G. Pedersen [P] on categories of C*- algebras. In Section 3 we i ntroduce NCCW complexes fol lowing S. Eilers, T .A. Loring and G. K. Pedersen, etc. W e prove in Section 4 a non commutative Cellu- lar Approximation Theorem. W e prove in Section 5 some long exact homotopy sequences associated with morphisms of C*-algebras. 2 Construc tions in categ ories of C*-algebras In this section we e xpose the pullback and pushout constructions o f S. Eilers, T .A. Loring and G. K. Pedersen [ELP] and of G. Pedersen [P] on categories of C*- algebras, and after that we define mapping cylinders and mapping cones associatd with arbitrary morphisms . Let introduce some general no tations. By I = [0 , 1] denote the closed interval from 0 to 1 on the real line of real numbers. It is easy to construct a hom eomor- phism I n ≈ B n between the n -cube and the n dimensional cl osed ball. Denote also the interior of th e cub e I n by I n 0 = (0 , 1 ) n = ◦ z}|{ I n . It is easy to show that the boundary ∂ I n = I n \ I n 0 is hom otpic to the ( n − 1 ) -dimensional sphere S n − 1 . Denote the space of all the continuous functions on I n with v alues in a C*-algebra A by I n = C ([0 , 1] n , A ) , and by analogy b y I n 0 A := C 0 ((0 , 1) n , A ) the space of all continuous functions with compact support with values in A , and finally , by S n A = C ( S n , A ) the space of all continuous maps from S n to A . Definition 2.1 (Pullback diagram) A com mutative diagram of C*-algebras and *-homomorphi sms X γ − − − → B   y δ   y β A α − − − → C (2 . 1) 2 is a pullback, if k er γ ∩ k er δ = 0 and if Y ψ − − − → B   y ϕ   y β A α − − − → C (2 . 2) is another commutative d iagram, then there e xists a uni que morphism σ : Y → X such that ϕ = δ ◦ σ and ψ = γ ◦ σ , i.e. we have the so ca lled pullb ac k diagram Y X B A C ❅ ❅ ❅ ❘ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ✲ ✲ ❄ ❄ ψ σ ϕ γ α δ β (2.3) Definition 2.2 (Pushout diagram) A commut ativ e diagram of C*-algebras and *-homomorphi sms C β − − − → B   y α   y γ A δ − − − → X (2 . 4) is a pushout, if X is generated by γ ( B ) ∪ δ ( A ) and if C − − − → β B   y α   y ψ A ϕ − − − → Y (2 . 5) is another commutative d iagram, then there e xists a uni que morphism σ : X → Y such that ϕ = σ ◦ γ and ψ = σ ◦ δ , i.e. we have the so ca lled pushou t diagr am 3 Y X B A C ❅ ❅ ❅ ❘ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ✲ ✲ ❄ ❄ ψ σ ϕ γ α δ β (2.6) Definition 2.3 (NC cone) F or C*-algebras the NC cone of A i s defined as the tensor product with C 0 ((0 , 1]) , i.e. Cone( A ) := C 0 ((0 , 1]) ⊗ A. (2 . 7) Definition 2.4 (NC suspension) F or C*-algebras the NC su spension of A is de- fined as the tensor product with C 0 ((0 , 1)) , i.e. S ( A ) := C 0 ((0 , 1)) ⊗ A. (2 . 8) Remark 2.5 If A admits a NCCW complex structu r e, the same have the cone Cone( A ) of A and the su spension S ( A ) o f A . Definition 2.6 (NC mapping cylinder) Consider a map f : A → B between C*-algebras. In the algebra C ( I ) ⊗ A ⊕ B consid er the closed two-sided ideal h{ 1 } ⊗ a − f ( a ) , ∀ a ∈ A i , generated by elements of type { 1 } ⊗ a − f ( a ) , ∀ a ∈ A . The quotient algebra Cyl( f ) = Cyl( f : A → B ) := ( C ( I ) ⊗ A ⊕ B ) / h{ 1 } ⊗ a − f ( a ) , ∀ a ∈ A i (2 . 9) is called the NC mapping cylinder and denote it by Cyl( f : A → B ) . Remark 2.7 It is easy to show that A is includ ed in Cy l ( f : A → B ) as C { 0 } ⊗ A ⊂ Cyl( f : A → B ) and B is includ ed in also B ⊂ Cyl( f : A → B ) . Definition 2.8 (NC mapping cone) In the algebra C ((0 , 1]) ⊗ A ⊕ B consider the closed two-sided ideal h { 1 } ⊗ a − f ( a ) , ∀ a ∈ A i , generated b y elements o f t ype { 1 } ⊗ a − f ( a ) , ∀ a ∈ A . W e define the mapping cone as the quotient algebra Cone( f ) = Cone( f : A → B ) := ( C 0 ((0 , 1]) ⊗ A ⊕ B ) / h{ 1 }⊗ a − f ( a ) , ∀ a ∈ A i . (2 . 10) 4 Remark 2.9 It is easy to show that B is included in Cone( f : A → B ) . Pr opositi on 2.10 Both the ma pping cylinder and mapping cone satisfy t he pull- back dia grams Cone( ϕ ) pr 1 − − − → C 0 (0 , 1] ⊗ A pr 2   y   y ϕ ◦ ev(1) B id − − − → B Cyl( ϕ ) pr 1 − − − → C [0 , 1] ⊗ A pr 2   y   y ϕ ◦ ev(1) B id − − − → B , wher e ev (1) is the map of evaluation at the point 1 ∈ [0 , 1 ] . Remark 2.11 The pullback diagrams in Pr opositio n 2.10 can be used as t he ini- tial definition of mapping cyliner and ma pping cone. The p r evious definitions ar e ther efor e the e xistence of th ose universal objects. Remark 2.12 It is r easonabl e t o h ave that in the case of C*-algebras of contin- uous functions A = C ( X ) , B = C ( Y ) the C*-algebras of continuous functions over the cone and t he suspension of topol ogical spaces ar e in general dif fer ent fr om the cone and the suspensi on of C*-algebras, we have just defined; the sam e is true tha t the mappin g cylinder and the mapping cone of morphisms of C*-algebras ar e differ ent fr om the C *-algebra of continuous f unctions on the mappi ng cylinder and the mapping cone of spaces. 3 The category NCCW In this s ection we introduce NCCW complexes following J. Cuntz and fol lowing S. Eilers, T . A . Loring and G. K. Pedersen, [ELP] etc. Definition 3.1 A dimension 0 NC CW complex is defined, follo wing [P] a s a finite sum of C* algebras of finite linear dimens ion, i.e. a sum of finite dim ensional matrix algebras, A 0 = M k M n ( k ) . (3 . 1) In dimensi on n, an NCCW complex is d efined as a sequence { A 0 , A 1 , . . . , A n } of C*-algebras A k obtained e ach from the pre vious one by the pullback construction 0 − − − → I k 0 F k − − − → A k π − − − → A k − 1 − − − → 0      y ρ k   y σ k 0 − − − → I k 0 F k − − − → I k F k ∂ − − − → S k − 1 F k − − − → 0 , (3 . 2) 5 where F k is som e C*-algebra of finite linear dimensio n, ∂ the restriction mor- phism, σ k the connecting morphism, ρ k the projection on the fi rst coordinates and π the projection on the second coordinates in the presentation A k = I k F k M S k − 1 F k A k − 1 (3 . 3) Pr opositi on 3.2 If the algebras A and B admit a NCCW comp lex structur e , then the same has the NC mapping cylinder Cyl( f : A → B ) . P R O O F . Let us remember from [P] that the interval I admit s a structu re of an NCCW complex. Next, tensor product of two NCCW complex [P] is also an NCCW complex and finally the quotient of an NCCW complex is also an NCCW complex, loc. cit..  Pr opositi on 3.3 If the algebras A and B admit a NCCW comp lex structur e , then the same has the NC mapping cone Cone( f : A → B ) . P R O O F . The same argument as in Proof of Proposition 3.2.  4 A ppr oxi mation Theor em W e prove in thi s sectio n a noncomm utative analog of the w ell-known Cellular Approximation Theorem. First we introduce th e so called noncom mutative ho- motopy e xtensio n property (NC HEP). Definition 4.1 (NC HEP) F or a given ( f , ϕ t ) and a C*-algebra C , we say that ˜ h = ˜ ϕ t is a s olution of the extension pr obl em i f we ha ve the commutati ve ho mo- topy e xtension diagram C Cyl( i : A ֒ → B ) C [0 , 1 ] ⊗ B ❄ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✾ ✛ ( f , ϕ t ) ˜ ϕ t Definition 4.2 (NC NDR) W e say t hat t he pair of algebras ( B , A ) is a NCNDR pair , if t her e are continuous mo rphisms u : C [0 , 1] → B and ϕ : B → C [0 , 1 ] ⊗ B ∼ = C ( I , B ) such that 1. u − 1 ( A ) = 0 ; 6 2. If ϕ ( b ) = ( x ( t ) , b ′ ) and x ( t ) = 0 ∈ C ( I ) then b ′ = b , ∀ b ∈ B ; 3. ϕ ( a ) = ( x ( t ) , a ) , ∀ a ∈ A, x ( t ) ∈ C ( I ) ; 4. ϕ ( b ) = ( x ( t ) , b ′ ) and if x ( t ) = 1 ∈ C ( I ) th en b ′ ∈ A for al l b ∈ B such th at u ( b ) 6 = 1 . The following proposition is easily to prove . Pr opositi on 4.3 The assert ion that NC HEP has solutio n for eve ry ϕ t and C is equivalent to the the pr operty that ( B , A ) is a NC NDR pair . P R O O F . If ( B , A ) has NC H EP , we can for e very C , construct ˜ ϕ : B → C [0 , 1] ⊗ B satisfying the NC HEP diagram. Choose C = B and f = id we hav e the function ϕ and then choos e ( D , C ) = ( C [0 , 1] , 0) in the definiti on of NC NDR pair we hav e the function u . Con versely , if ( B , A ) is a NC NDR pair , we can define h = ϕ : B → C [0 , 1] ⊗ B , (4 . 1) the composition of which with f : C → B satisfy the NC HEP diagram.  Theor em 4.4 (Extension) Suppos e that B = I n F n ⊕ S n − 1 F n A and ( C , D ) is a NC NDR pair . Every r elat ive morphism of pairs of C *-algebras f : ( D , C ) → (Cyl( i : A ֒ → B ) , C { 1 } ⊗ A ) (4 . 2) can be up-to homotopy ex tended t o a r elative morphism of pairs of C *-algebras F : ( D , C ) → ( C ( I ) ⊗ B , C { 1 } ⊗ B ) . (4 . 3) P R O O F . The property that ( D , C ) is a NC NDR pair , there is a natural extension f 1 : ( D , C ) → ( C ( I ) ⊗ B , C { 1 } ⊗ A ) . (4 . 4) Composing f 1 with the map, e valuating the v alue at 1 g iv e a morphism ev(1) ◦ f 1 : ( D , C ) → ( C { 1 } ⊗ B , C { 1 } ⊗ A ) . (4 . 5) Therefore, there exists a natural extension f 2 from the pair ( D , C ) . to t he pair ( C { 0 } ⊗ C ( I ) ⊗ B + C ( I ) ⊗ C { 1 } ⊗ B , Cyl( C { 1 } ⊗ A ֒ → C { 1 } ⊗ B )) . 7 Once again, there is a natural e xtens ion f 3 from the pair to the pair ( C { 0 }⊗ C ( I ) ⊗ B + C { 1 }⊗ C ( I ) ⊗ B + C ( I ) ⊗ C ( I ) ⊗ A, C y l ( C ( I ) ⊗ A ֒ → C ( I ) ⊗ B )) = = ( C { 0 } ⊗ Cyl( C ( I ) ⊗ A ֒ → C ( I ) ⊗ B ) + C { 1 } ⊗ Cyl( C ( I ) ⊗ A ֒ → C ( I ) ⊗ B ) , , Cyl( C ( I ) ⊗ A ֒ → C ( I ) ⊗ B )) . And finally , there is a natural extension f 4 from the pair ( D , C ) to the pair C ( I ) ⊗ C ( I ) ⊗ B , Cyl ( C ( I ) ⊗ A ֒ → C ( I ) ⊗ B )) . W e define the desired extension F : ( D , C ) → ( C ( I ) ⊗ B , C { 1 } ⊗ B ) as F ( t, x ) := f 4 ( t, 0 , x ) . (4 . 6)  Theor em 4.5 Let { A 0 , A 1 , . . . , A n } and { B 0 , B 1 , . . . , B m } be two NCCW com- plexe s and f : A = A n → B m = B an a lgebraic homomorphism (map). Then f is homotopic to a cellular NCCW complex map h : A → B . P R O O F . W e constract a sequence of maps g p : A p → C ( I ) ⊗ B p , (4 . 7) with 4 well-known properties: 1. g p ( x ) = (0 , f ( x )) , ∀ x ∈ A p 2. If g p ( b ) = ( x ( t ) , f ( b )) and if x ( t ) = 0 ∈ C ( I ) , then f ( b ) = b, ∀ b ∈ B . 3. ev(1) ◦ g p = g p − 1 , 4. g p ( A p ) ⊂ C { 1 } ⊗ B p . Indeed, following the definition of an NCCW complex structure, we ha ve A 0 = F 0 ⊗ A, F 0 = M j 0 M n ( j 0 ) ( 4 . 8) 8 a finite system of quantum points, i.e. a com mutative diagram 0 − − − → I 1 0 F 1 − − − → A 1 − − − → A 0 − − − → 0      y ρ 1   y σ 1 0 − − − → I 1 0 F 1 − − − → I 1 F 1 − − − → α 1 S 0 F 1 − − − → 0 (4 . 9) in which the second square is a pullback diagram, F 1 = M j 1 M n ( j 1 ) = M j 1 Mat n ( j 1 ) , (4 . 10) and we can present A 1 as A 1 ≈ I 1 F 1 M S 0 F 1 A 0 . (4 . 11 ) Follo wing the compessible th eorems [W] and t he previous Extension Th eorem 4.4, the function g 0 can be naturally extended to a function g 1 with properties 1. - 4. and now we ha ve again follo win g the definition of an NCCW complex, 0 − − − → I 2 0 F 2 − − − → A 2 − − − → A 1 − − − → 0      y ρ 2   y σ 2 0 − − − → I 2 0 F 2 − − − → I 2 F 2 − − − → α 2 S 1 F 2 − − − → 0 (4 . 12) F 2 = M j 2 M n ( j 2 ) = M j 2 Mat n ( j 2 ) , (4 . 13) and we can present A 2 as A 2 ≈ I 2 F 2 M S 1 F 2 A 1 . (4 . 14 ) Follo wing the compessible th eorems [W] and t he previous Extension Th eorem 4.4, the function g 1 can be naturally extended to a function g 2 with properties 1. - 4. The procedure is continued for all p . Once these functions g p were defined, the function g : A → C ( I ) ⊗ B which is continu ous and g is a homo topy of f to h , where h ( x ) := ev (1) ◦ g ( x ) . (4 . 15) Because of 4. th e function h : A → B is a cellular NCCW complex map.  9 5 Homotop y of NCC W complexes W e prove in this section the standard long exact homotopy sequences. Let us first recall the definition of homotopi c morphisms. Definition 5.1 A homotopy between two morphis ms ϕ, ψ : A → B is a morphism Φ : A → C ( I ) ⊗ B , such that Φ(0 , . ) = ϕ and Φ(1 , . ) = ψ . Pr opositi on 5.2 Ther e is a natural homotopy Cyl( ϕ : A → B ) ≃ B and Cone( ϕ : A → B ) ≃ B / A , if the last one B / A is defined. Theor em 5.3 F or every morphism ϕ ; A → B , t her e is a n atural long e xact ho- motopy sequence . . . − − − → S 2 ( A ) − − − → S (Cone( ϕ : A → B )) − − − → S (Cyl( ϕ : A → B )) − − − → S ( A ) − − − → Cone( ϕ : A → B ) − − − → Cyl( ϕ : A → B ) − − − → A ϕ − − − → B (5 . 1) P R O O F . Put A 0 = B , A 1 = A and ϕ 0 = ϕ we have A 0 ϕ 0 = ϕ − − − → A 1 . (5 . 2) Because of Proposition 5.2 we ha ve A 0 = B ϕ 0 ← − − − A 1 = A ϕ 1 ← − − − A 2 = Cyl ( ϕ ) ϕ 2 ← − − − A 3 = Cone( ϕ ) . (5 . 3) Because of the exact sequence S ( A ) ← − − − Cyl( ϕ ) ← − − − Cone( ϕ ) , we ha ve A 0 = B ϕ 0 ← − − − A 1 = A ϕ 1 ← − − − A 2 = Cyl( ϕ ) ϕ 2 ← − − − A 3 = Cone( ϕ ) ϕ 3 ← − − − ϕ 3 ← − − − A 4 = S ( A ) = C 0 ((0 , 1)) ⊗ A. (5 . 4) Because the t ensor product C 0 ((0 , 1)) ⊗ . is a left exact functor and because of (5.3) , we ha ve A 0 = B ϕ = ϕ 0 ← − − − A 1 = A ϕ 1 ← − − − A 2 = Cyl( ϕ ) ϕ 2 ← − − − A 3 = Cone( ϕ ) ϕ 3 ← − − − ϕ 3 ← − − − S ( A ) ϕ 4 ← − − − S (Cyl( ϕ )) ϕ 5 ← − − − A 5 = S (Cone( ϕ )) ϕ 5 ← − − − ϕ 5 ← − − − A 6 = S 2 ( A ) ϕ 6 ← − − − . . . , (5 . 5) etc.  10 Acknowledgm ents The work was supported in part by V ietnam National Project for Research in Fun- damental Sciences and was compl eted during the stay in June and July , 2007 of the aut hor , in Abdus Salam ICTP , Trieste, Italy . 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