CCHARACTERISTIC CLASSES FOR TC STRUCTURES
MAURICIO CEPEDA DAVILA
Abstract.
In this article we study the construction of characteristic classesfor principal G -bundles equipped with an additional structure called transi-tionally commutative structure (TC structure). These structures classify, upto homotopy, possible trivializations of a principal G -bundle, such that theinduced cocycle have functions that commute in the intersections of their do-mains. We focus mainly on the cases where the structural group G equalsSU(n), U(n) or Sp(n). Our approach is an algebraic-geometric constructionthat relies on the so called power maps defined on the space B com G , the clas-sifying space for commutativity in the group G. Introduction
Suppose that G is a Lie group and consider the set of n -tuples with commutingelements which can be identified with Hom ( Z n , G ) . Adem, Cohen and Torres-Giese (see [ACT]) showed that { Hom ( Z n , G ) } n ≥ can be endowed with a simplicialstructure whose geometric realization is denoted by B com G . Additionally, consider aprinciple G -bundle over a compact Hausdorff space M , with a classifying function g : M → BG and trivializations with associated cocycle { ρ ij } . Suppose thenthat the cocycles commute with each other in the intersection of their domains, i.e. ρ ij · ρ jk = ρ jk ρ ij . Adem and Gomez showed in [AG] that the previous commutativitycondition holds if only if there is a lifting, up tho homotopy of the classifying map g ; that is a commutative diagram up to homotopy as the one shown below, B com G ι (cid:15) (cid:15) M f (cid:59) (cid:59) g (cid:47) (cid:47) BG.
The existence of such lifting is what we call a transitionally commutative(TC) structure on a principal G -bundle. Where we say that two TC structures f , f : M → B com G are equivalent if the functions are homotopic. TC structuresare meant to classify the different ways a principle bundle can have commutativecocycles, up to homotopy.The interest in studying the spaces Hom ( Z n , G ) arises from the study of modulispaces of flat bundles, which are important for Quantum field theories such asthe Yang-Mills and Chern- Simons theories. In particular, when the base space isthe torus (cid:0) S (cid:1) n and the structural group is a compact Lie group G , the moduli The author was sponsored by the Colombian Ministry of Sciences (previously Colciencias)under the public sponsorship act 647 of 2014 for national doctoral programs. Where the resultspresented here are part of his PhD thesis, supervised by José Manuel Gómez Guerra at theNational University of Colombia at Medellín. a r X i v : . [ m a t h . A T ] J a n HARACTERISTIC CLASSES FOR TC STRUCTURES 2 spaces of flat bundles can be identified with
Hom ( Z n , G ) /G , where G acts underconjugation.Mathematically speaking, the theory of commuting tuples is interesting in itsown right. For example, Adem and Gomez defined in [AG] the commutative K-theory of a finite CW-complex X to be K com X := Gr (Vect com ( X )) , where Gr denotes the Grothendieck construction and Vect com ( X ) is the set of equivalenceclasses of vector bundles over X with commuting cocycles. Later on Adem, Gómez,Lind and Tillman introduce the notion of q -nilpotent K -theory of a CW-complex X for any q ≥ , which extends the notion of commutative K -theory defined byAdem and Gomez, and show that it is represented by Z × B ( q, U ) , were B ( q, U ) isthe q -th term of a filtration of the infinite loop space BU . (See [AGLT].)In this article we define and develop characteristic classes for TC structures andmainly, we develop an algebraic-geometric method to use Chern-Weil theory tocompute them. To start we will see that there is a one to one correspondencebetween characteristic classes and elements of H * ( B com G, R ) . We then use thedescription of H * ( B com G, R ) given in [AG] for which we exhibit a set of algebraicgenerators. Then we show how we can use Chern-Weil theory to compute thecharacteristic classes associated to those generators. We do this for G equal toeither U ( n ) , SU ( n ) or Sp( n ) for the following reasons.In general the spaces Hom ( Z n , G ) are not path connected, so the simplicialconstruction can be reduced to consider the path connected components contain-ing the identity tuple (1 , , . . . , . These connected components are denoted by Hom ( Z n , G ) , and the geometric realization of them is denoted by B com G . Ademand Gomez showed in Proposition 7.1 of [AG] that the cohomology with real coef-ficients of B com G is isomorphic to(1.1) ( H ∗ ( BT, R ) ⊗ H ∗ ( BT, R )) W /J, where T ⊂ G is a maximal tori, W is the Weil group acting diagonally, and J isthe ideal generated by elements of the form p ( x ) ⊗ with p ( x ) an W -invariantpolynomial of positive degree.Additionally, Adem and Cohen showed in Corollary 2.4 of [AC] that Hom ( Z n , G ) is path connected when G is either U ( n ) , SU ( n ) or Sp( n ) . For them then Expres-sion 1.1 describes the cohomology of all B com G . To obtain the generators of thiscohomology, we first consider the natural inclusion Hom ( Z n , G ) ⊆ G n . The inclu-sion induces a simplicial map between the simplicial structure of { Hom ( Z n , G ) } n ≥ and the bar construction for the classifying space of G , BG . This in turn gives riseto a map ι : H * ( BG, R ) → H * ( B com G, R ) . Secondly, we need to consider the assignments
Hom ( Z n , G ) → Hom ( Z n , G )( g , . . . , g n ) (cid:55)→ (cid:0) g k , . . . , g kn (cid:1) . These assignments give rise to simplicial maps, allowing us to obtain the powermaps Φ k : H ∗ ( B com G, R ) → H ∗ ( B com G, R ) when k ∈ Z . By using characterizations of the cohomology rings as well as theeffect of these maps on them, we then use the particularities of the action of theWeil group for U ( n ) , SU ( n ) and Sp( n ) to show that HARACTERISTIC CLASSES FOR TC STRUCTURES 3
Theorem.
For G equal to U ( n ) , SU ( n ) or Sp ( n ) then H ∗ ( B com G ) is generatedas an algebra by (cid:8) Φ k (Im ι ) | k ∈ Z \ { } (cid:9) . In order to compute the TC characteristic class associated to a generator of theform Φ k ( ι ( s )) ∈ H ∗ ( B com G, R ) , s ∈ H ∗ ( BG, R ) and k ∈ Z \ { } , we developanother construction. For a TC structure f : M → B com G over a principal G -bundle E → M , we construct a family of principal G -bundles E k → M , called the k -th associated bundles. Then we prove that if Ω k is the curvature of E k we havethe equality f ∗ (cid:0) Φ k ( ι ( s )) (cid:1) = s (Ω k ) ∈ H ∗ ( M, R ) , where s (Ω k ) is the characteristic class of E k → M associated to s , which is com-puted using Chern-Weil theory. This let us obtain our main result: Theorem. (Chern-Weil theory for TC structures) Consider ε ∈ [ M, B com G ] anequivalence class with an underlying smooth vector bundle E → M , and structuregroup U ( n ) , SU ( n ) or Sp ( n ) . Also let Ω k be the curvature of E k , the k -th asso-ciated bundle of E . Then every TC characteristic class can be obtained as a linearcombinations of products of the form s (Ω k ) · s (Ω k ) · · · s m (Ω k m ) ∈ H ∗ ( M, R ) , where s i ∈ H ∗ ( BG ) and k i ∈ Z . Each s i (Ω k ) is the characteristic class of thevector bundle E k → M computed using its curvature. The outline of this article is as follows: in Section 2 we explain the constructionof B com G and define TC structures, TC characteristic classes, power maps and k -th associated bundles. We also show how these concepts relate to each other.In Section 3 we show the effect of the power maps in the cohomology with realcoefficients of B com G . In Section 4 we obtain the generators for H ∗ ( B com G, R ) for G equal to U ( n ) , SU ( n ) or Sp ( n ) . In Section 5 we develop the Chern-Weiltheory for TC characteristic classes. Finally in Section 6 we show an example of acomputation of a TC characteristic class using a TC structure developed by Ramrasand Villareal (see [RV]). 2. TC structures and B com G In this section we introduce all the basic concepts we are using that are relatedto commuting tuples in a Lie Group.2.1.
Simplicial construction for B com G : We first start with a basic descriptionof the simplicial structure used to define the space B com G . This will allow us toobtain the generators for its cohomology with real coefficients.Let us define a simplicial space whose n -th level is given by Hom ( Z n , G ) , whichis the subspace of G n consisting of all commuting n -tuples. This is ( g , . . . , g n ) such that g i g j = g j g i for every ≤ i, j ≤ n . Its face maps δ i : Hom ( Z n , G ) → Hom (cid:0) Z n − , G (cid:1) are given by δ i ( g , . . . , g n ) := ( g , . . . , g n ) i = 0 , ( g , . . . , , g i − , g i g i +1 , g i +2 , . . . , g n ) 1 ≤ i ≤ n − , ( g , . . . , g n − ) i = n, HARACTERISTIC CLASSES FOR TC STRUCTURES 4 and the degeneracy maps s i : Hom ( Z n , G ) → Hom (cid:0) Z n +1 , G (cid:1) are given by s i ( g , . . . , g n ) = ( g , . . . , , g i , , g i +1 , . . . , g n ) . It is routine to see that these maps satisfy the simplicial identities.
Definition 1.
The space B com G is defined as the fat realization of the simplicialspace { Hom ( Z n , G ) } n ≥ , that is B com G := (cid:107) Hom ( Z • , G ) (cid:107) . It is also important to mention that for this construction the fat realization is ho-motopy equivalent to the geometrical realization as the simplicial space
Hom ( Z • , G ) is proper. (See the appendix of [AG].)In general, Hom ( Z m , G ) is not connected. The path connected component of Hom ( Z m , G ) containing the element (1 , , . . . , is denoted by Hom ( Z m , G ) . Wecan restrict the face and degeneracy maps to obtain a simplicial space Hom ( Z • , G ) whose fat realization is denoted by B com G . However Adem and Cohen showed inCorollary 2.4 of [AC] that Hom ( Z m , G ) is path connected when G is either U ( n ) , SU ( n ) or Sp ( n ) . So for these groups we have the equality B com G = B com G. Power Maps:
For each k ∈ Z we define maps Φ km : Hom ( Z m , G ) → Hom ( Z m , G )( g , . . . , g m ) (cid:55)→ (cid:0) g k , . . . , g km (cid:1) . These maps are well defined since the power of commuting elements is still com-mutative. Commutativity is needed here in order for them to induce simplicialmaps. By this we mean maps commuting with the face and degeneracy maps.More precisely we need the equality ( g i g i +1 ) k = g ki g ki +1 to hold. Thus, only for commuting tuples we guarantee the existence of the k -thpower map Φ k : B com G → B com G . In the general bar construction for G , the powermaps do not necessarily induce simplicial maps.2.3. TC structures:
Consider a principal G -bundle π : E → M over a compactHausdorff space M . This implies that M has an open cover U := { U i } mi =1 andtrivializations ϕ i : π − ( U i ) → U i × G. By considering the second component of thecomposition ϕ j ◦ ϕ − i : ( U i ∩ U j ) × G → ( U i ∩ U j ) × G we obtain the cocycles ρ ij : U i ∩ U j → G , which satisfy that ϕ j ◦ ϕ − i ( x, g ) = ( x, ρ ij ( x ) · g ) , for every x ∈ U i ∩ U j and g ∈ G . Assume this cover is a good cover and considerthe simplicial construction of the nerve of the cover: N ( U ) n = (cid:71) ( U i ∩ U i · · · ∩ U i n ) . It is worth recalling that up to equivalence the cocycles characterize a principle bundle.
HARACTERISTIC CLASSES FOR TC STRUCTURES 5
Take N ( U ) := (cid:107)N ( U ) • (cid:107) . Since U is a good cover, M and N ( U ) are homotopyequivalent (See [Hatcher], Corollary 4G.3). This guarantees a biyection [ N ( U ) , Y ] ∼ = [ M, Y ] for any space Y .Recall that if we consider the bar construction of BG we have in every level theset of tuples, G l . Then we have a simplicial function g n : N ( U ) n → G n given by g n ( x ) := (cid:0) ρ i i ( x ) , ρ i i ( x ) , . . . , ρ i l − i l ( x ) (cid:1) . This induces a function g : N ( U ) → BG , which, up to homotopy, defines theclassifying function g : M → BG .Now suppose that for the principal G -bundle π : E → M there is a trivializationinducing cocycles that commute with each other. That is that for x ∈ U i ∩ U j ∩ U k we have ρ ik ( x ) ρ kj ( x ) = ρ kj ( x ) ρ ik ( x ) . Then we can define f n : N ( U ) n → Hom ( Z n , G ) given by f n ( x ) := (cid:0) ρ i i ( x ) , ρ i i ( x ) , . . . , ρ i l − i l ( x ) (cid:1) . We have a commuting diagram N ( U ) n f n (cid:47) (cid:47) g n (cid:39) (cid:39) Hom ( Z n , G ) (cid:15) (cid:15) G n where the vertical is the inclusion. This in turn leads to a diagram commuting upto homotopy M f (cid:47) (cid:47) g (cid:35) (cid:35) B com G (cid:15) (cid:15) BG, where the vertical map is the inclusion.Adem and Gomez proved in Theorem 2.2 of [AG] that if there is a lifting up tohomotopy of the classifying function of a principal G -bundle, then there exists atrivialization with commuting cocycles for that bundle. That is, that the existenceof a homotopy lifting for the classifying function is a necessary and sufficient condi-tion for the existence of commuting cocycles for the principal G -bundle. This allowus to define the following. Definition 2.
Given a space M and a principal G -bundle with classifying function f : M → BG , a TC structure over M is a function g : M → B com G such that M f (cid:47) (cid:47) g (cid:35) (cid:35) B com G (cid:15) (cid:15) BG, commutes up to homotopy. We say that two TC structures f , f : M → B com G are equivalent if the functions are homotopic. HARACTERISTIC CLASSES FOR TC STRUCTURES 6
At this point is important to remark that given a principal bundle there canbe several different TC structures over it. That is, there can exist functions g : M → BG and f , f : M → B com G such that there are homotopies ι ◦ f ∼ = g and ι ◦ f ∼ = g but where f is not homotopic to f . At the end of this article we exhibitan example with a homotopy trivial g : S → BSU (2) with a non homotopy triviallifting G : S → B com SU (2) .The assignment Top → [ − , B com G ] is a contravariant functor: given a continuousfunction h : M → N we can consider its pullback h ∗ : [ N, B com G ] → [ M, B com G ][ f ] (cid:55)→ [ f ◦ h ] , where by [ f ] we mean the homotopy class of the function f . From this we define Definition 3.
A characteristic class for TC structures or
TC characteristic class is a natural transformation η : [ − , B com G ] → H ∗ ( − , R ) . Here [ − , B com G ] is thefunctor assigning to a space the set of homotopy classes of functions from the spaceto B com G and H ∗ ( − , R ) is the functor of cohomology with real coefficients. Proposition 4.
There is a one to one correspondence between the TC characteristicclasses and elements of H ∗ ( B com G, R ) .Proof. Consider a TC characteristic class η , and define c η := η ( B com G ) ([Id B com G ]) ∈ H ∗ ( B com G, R ) where Id B com G is the identity on B com G . We want to see that the assignment θ : η (cid:55)→ c η is a one to one and onto.First, consider a continuous function f : M → B com G . By naturallity of η wehave a commuting diagram [ B com G, B com G ] η ( B com G ) (cid:47) (cid:47) f ∗∗ (cid:15) (cid:15) H ∗ ( B com G, R ) f ∗ (cid:15) (cid:15) [ M, B com G ] η ( M ) (cid:47) (cid:47) H ∗ ( M, R ) where we use f ∗∗ to distinguish the pullback of the functor [ − , B com G ] from thepullback from cohomology. The commutativity of the previous diagram meansthat f ∗ ( c η ) = η ( M ) ( f ∗∗ ([Id B com G ])) . But since f ∗∗ ([Id B com G ]) = [Id B com G ◦ f ] = [ f ] , we can conclude that(2.1) f ∗ ( c η ) = η ( M ) ([ f ]) ∈ H ∗ ( M, R ) . Now, to see that θ : η (cid:55)→ c η is surjective, take c ∈ H ∗ ( B com G. R ) and define η c ( M ) : [ M, B com G ] → H ∗ ( M, R )[ f ] (cid:55)→ f ∗ ( c ) . HARACTERISTIC CLASSES FOR TC STRUCTURES 7
This can be seen to be a well defined natural transformation thanks to the propertiesof cohomology, so η c is a TC characteristic class. Now, by definition and Equation2.1 it follows that c η c = η c ( B com G ) ([Id B com G ]) = Id ∗ B com G ( c ) = c, which implies that θ : η (cid:55)→ c η sends η c into c .On the other hand to prove inyectivity, take c η and consider η c η as defined before.For any f : M → B com G it follows that η c η ( M ) ([ f ]) = f ∗ ( c η ) = η ( M ) ([ f ]) , where the first equality is true by definition, and the sencond thanks to Equation2.1. The equality η c η ( M ) ([ f ]) = η ( M ) ([ f ]) means that η c η = η , so if ω is anotherTC characteristic class such that c η = c ω , then n = η c η = η c ω = ω. That is, θ : η (cid:55)→ c η is inyective. (cid:3) The k -th associated bundles: Once again let { U α } α ∈ J be an open cover ofa space M such that there are trivializations with a cocycle { ρ ij : U i ∩ U j → G } ,such that if x ∈ U i ∩ U j ∩ U l then ρ il ( x ) ρ lj ( x ) = ρ lj ( x ) ρ lk ( x ) . These transition functions satisfy the cocycle condition as well, that is, ρ ij ( x ) = ρ il ( x ) ρ lj ( x ) . In particular these two properties imply that for k ∈ Z we have ρ ij ( x ) k = ( ρ il ( x ) ρ lj ( x )) k = ρ il ( x ) k ρ lj ( x ) k . This tell us that the collection of functions ρ kij : U i ∩ U j → G defined as ρ kij ( x ) := ρ ij ( x ) k also satisfy the cocycle condition. Then for each k ∈ Z we can construct a principalbundle p k : E k → M with trivializations over { U i } i ∈ I with cocycle (cid:8) ρ kij (cid:9) . We callit the k -th associated bundle of E . Here E k is obtained as the quotient space (cid:32)(cid:71) i ∈ I U i × G (cid:33) (cid:30) ∼ , where for x ∈ U i and y ∈ U j , ( x, g ) ∼ ( y, h ) if only if x = y and ρ kij ( x ) · g = h . Proposition 5. (Classifying functions for k - th associated bundles) If f : M → B com G defines a TC structure over a principal G -bundle, and f k : M → B com G is the corresponding lifting over the k -th associated bundle, then the following mapdiagram commutes (2.2) M f (cid:47) (cid:47) f k (cid:35) (cid:35) B com G Φ k (cid:15) (cid:15) B com G. Where Φ k : B com G → B com G are the power maps. HARACTERISTIC CLASSES FOR TC STRUCTURES 8
Proof.
As it was explained before, to obtain the classifying functions for the k -th associated bundle p ( k ) : E k → M we need to consider a simplicial map f kl : N ( U ) l → Hom (cid:0) Z l , G (cid:1) . The components of this function are given by the transitionfunctions: if x ∈ U i ∩ U i ∩ · · · ∩ U i l +1 , we take f kl ( x ) = (cid:16) ρ ki i ( x ) , . . . , ρ ki l − i l ( x ) (cid:17) = (cid:16) ρ i i ( x ) k , . . . , ρ i l − i l ( x ) k (cid:17) . This can be rewritten using the power functions as f kl = Φ kl ◦ f l . The desired result is obtained after passing to the geometric realization. (cid:3) Power maps and cohomology of B com G In this section we go through the reasoning behind the computation of H ∗ ( B com G , R ) made in [AG] to obtain the effect of the power maps on cohomology. Thus we tracksuch effect in the main steps of the computation. To make notation simpler, weassume B com G = B com G , which is true when G is either U ( n ) , SU ( n ) or Sp ( n ) ,as mentioned before. We also fix a maximal torus T ⊆ G with Weyl group W andwe write H ∗ ( Y ) to refer to the cohomology of Y with real coefficients.In Section 7 of [AG] it is proved that for a maximal torus T of G with Weylgroup W we have H ∗ ( B com G ) ∼ = ( H ∗ ( BT ) ⊗ H ∗ ( BT )) W /J, where J is the ideal generated by the see { f ( x ) ⊗ ∈ H ∗ ( BT ) ⊗ H ∗ ( BT ) | f of positive degreepolynomial and n · f (x) = f (x) for all n ∈ W } and W acts on H ∗ ( BT ) ⊗ H ∗ ( BT ) diagonally.In order to reach the description of the induced power maps Φ k on cohomology,we need to consider some auxiliary maps that are used in [AG]. In this process wewill see what is their relationship with the power maps. First, since all the tuplesof T m have commuting elements, we can consider the power maps for the torus ψ k : H ∗ ( BT ) → H ∗ ( BT ) . This is the map induced in the m -level the by thepower maps, Φ km : Hom ( Z m , T ) → Hom ( Z m , T )( g , . . . , g m ) (cid:55)→ (cid:0) g k , . . . , g km (cid:1) . Also consider ϕ m : G × T m → Hom ( Z m , G )( g, t , . . . , t n ) (cid:55)→ (cid:0) gt g − , . . . , gt m g − (cid:1) . Because
Hom ( Z m , G ) is path connected, an m -tuple ( g , . . . , g m ) has commutingelements if and only if there is a maximal tori containing all g i (see Lemma 4.2 of[Baird]). Then, since every maximal tori is conjugated to T , the previous map is HARACTERISTIC CLASSES FOR TC STRUCTURES 9 surjective. We also have an action of the normalizer of T in G , N G ( T ) , on G × T m ,where for η ∈ N G ( T ) we have η · ( g, t , . . . , t m ) = (cid:0) gη − , ηt η − , . . . , ηt m η − (cid:1) . On the other hand, consider the flag variety
G/T . It is easy to verify that the maps ϕ m factor through the product G/T × T m giving us a commutative diagram G × T m ϕ m (cid:47) (cid:47) (cid:15) (cid:15) Hom ( Z m , G ) G/T × T m (cid:55) (cid:55) , such that the diagonal map is also surjective. We call it ϕ m as well. This family ofmaps give rise to a simplicial map ϕ • : G/T • × T • → Hom ( Z • , G ) . Here
G/T • is the trivial simplicial space with G/T on every level, and T • is thesimplicial space obtained by the bar construction for the classifying space appliedto T .Furthermore using representatives of the Weyl group [ η ] ∈ W = N G ( T ) /T , wehave a well defined action on G/T × T m given by [ η ] · ([ g ] , t , . . . , t m ) = (cid:0)(cid:2) gη − (cid:3) , ηt η − , . . . , ηt m η − (cid:1) . It is easy to see that this action makes ϕ m W -invariant. Also we can constructa simplicial space, G/T × W T • , having the space of orbits G/T × W T m on the m -th level. Where the simplicial structure is inhered form G/T • × T • , giving usa simplicial map π • : G/T • × T • → G/T × W T • where on each level we have thenatural quotient map. Then we have a commuting diagram G/T • × T • ϕ • (cid:47) (cid:47) π • (cid:15) (cid:15) Hom ( Z • , G ) ,G/T × W T • ¯ ϕ • (cid:54) (cid:54) where ¯ ϕ m : G/T × W T m → Hom ( Z m , G ) is the induced map. Finally, we havemaps P km : G/T × T m → G/T × T m ([ g ] , t , . . . , t m ) (cid:55)→ (cid:0) [ g ] , t k , . . . , t km (cid:1) . By direct computation it can be seen that these maps are compatible with thesimplicial structure. They are also W -equivariant, that is [ η ] · P km ( g, t , . . . , t m ) = P km ([ η ] · ( g, t , . . . , t m )) , This is true since, (cid:0) ηtη − (cid:1) k = ηt k η − for t ∈ T .From this point we are showing that the arguments given in [AG] are natural.This allow us to include the power maps in their conclusions. HARACTERISTIC CLASSES FOR TC STRUCTURES 10
Proposition 6.
Suppose G is a compact connected Lie group such that Hom ( Z m , G ) is path connected for every non negative integer m . Then for the cohomology withreal coefficients we have a commutative diagram (3.1) H ∗ (Hom ( Z m , G )) ϕ ∗ m (cid:47) (cid:47) ( Φ km ) ∗ (cid:15) (cid:15) H ∗ ( G/T × T m ) W ( P km ) ∗ (cid:15) (cid:15) H ∗ (Hom ( Z m , G )) ϕ ∗ m (cid:47) (cid:47) H ∗ ( G/T × T m ) W . where the horizontal maps are isomorphisms.Proof. Under this setting, Theorem 3.3 of [Baird] is applied to conclude that wehave the following natural isomorphisms(3.2) H ∗ (Hom ( Z m , G )) ( ¯ ϕ m ) ∗ ∼ = H ∗ ( G/T × W T m ) π ∗ ∼ = H ∗ ( G/T × T m ) W . Now let us see how the power maps are related to this constructions so far. Wehave maps P km : G/T × T m → G/T × T m ([ g ] , t , . . . , t m ) (cid:55)→ (cid:0) [ g ] , t k , . . . , t km (cid:1) . By direct computation it can be seen that these maps are compatible with thesimplicial structure. They are also W -equivariant, that is [ η ] · P km ( g, t , . . . , t m ) = P km ([ η ] · ( g, t , . . . , t m )) , This is true since, (cid:0) ηtη − (cid:1) k = ηt k η − for t ∈ T . Thus, they induced a well definemap ¯ P km : G/T × W T m → G/T × W T m , and we get the following commutingdiagram H ∗ ( G/T × W T m ) π ∗ (cid:47) (cid:47) ( ¯ P km ) ∗ (cid:15) (cid:15) H ∗ ( G/T × T m ) P k (cid:15) (cid:15) H ∗ ( G/T × W T m ) π ∗ (cid:47) (cid:47) H ∗ ( G/T × T m ) . We know that the homomorphism π ∗ : H ∗ ( G/T × W T m ) → H ∗ ( G/T × T m ) actually has image equal to H ∗ ( G/T × T m ) W , since H ∗ ( G/T × W T m ) π ∗ ∼ = H ∗ ( G/T × T m ) W . Thus, we actually have the diagram H ∗ ( G/T × W T m ) π ∗ (cid:47) (cid:47) ( ¯ P km ) ∗ (cid:15) (cid:15) H ∗ ( G/T × T m ) WP k (cid:15) (cid:15) H ∗ ( G/T × W T m ) π ∗ (cid:47) (cid:47) H ∗ ( G/T × T m ) W . HARACTERISTIC CLASSES FOR TC STRUCTURES 11 where the horizontal maps are isomorphism. This implies that (cid:0) P km (cid:1) ∗ preserves W -invariance: (cid:0) P km (cid:1) ∗ (cid:16) H ∗ ( G/T × T m ) W (cid:17) ⊆ H ∗ ( G/T × T m ) W . Also, by direct computation from the definitions and the fact that (cid:0) gtg − (cid:1) k = gt k g − , it follows that ϕ m ◦ P km = Φ km ◦ ϕ m holds. And since (cid:0) P km (cid:1) ∗ preserves W -invariance, we obtain the following commuting diagram H ∗ (Hom ( Z m , G )) ϕ ∗ m (cid:47) (cid:47) ( Φ km ) ∗ (cid:15) (cid:15) H ∗ ( G/T × T m ) W ( P km ) ∗ (cid:15) (cid:15) H ∗ (Hom ( Z m , G )) ϕ ∗ m (cid:47) (cid:47) H ∗ ( G/T × T m ) W . Here the horizontal maps are isomorphism as they can be factored by the isomor-phisms ¯ ϕ ∗ m : H ∗ (Hom ( Z m , G )) → H ∗ ( G/T × W T m ) and π ∗ : H ∗ ( G/T × W T m ) → H ∗ ( G/T × T m ) W . (cid:3) Next, thanks to the naturality in Theorems 5.15 and 1.19 of [Dupont] it can beconcluded that
Proposition 7.
Let X • and Y • be two simplicial spaces with a simplicial map f : X • → Y • . Suppose also that there is a finite group K with an action on everylevel X q compatible with the simplicial structure, such that there is an isomorphism H p ( C ∗ ( X q )) K ∼ = H p ( C ∗ ( Y q )) induce on every level by the maps of f . Then thereis natural isomorphism (cid:107) f (cid:107) ∗ : H ∗ ( (cid:107) Y (cid:107) ) → H ∗ ( (cid:107) X (cid:107) ) K , where (cid:107) X (cid:107) and (cid:107) Y (cid:107) are the fat realizations. Proposition 8.
Suppose G is a compact connected Lie group such that Hom ( Z m , G ) is path connected for every non negative integer m . Then for the cohomology withreal coefficients we have a commutative diagram (3.3) H ∗ ( B com G ) π ∗ (cid:47) (cid:47) Φ k (cid:15) (cid:15) H ∗ ( (cid:107) ( G/T ) • × BT • (cid:107) ) WP k (cid:15) (cid:15) H ∗ ( B com G ) π ∗ (cid:47) (cid:47) H ∗ ( (cid:107) ( G/T ) • × BT • (cid:107) ) W , where the horizontal maps are isomorphisms. Here we are abusing notation by usingthe same names for the power map and its induce map on cohomology.Proof. Because of Proposition 6 the conditions of Proposition 7 can be appliedto conclude that π ∗ : H ∗ ( B com G ) → H ∗ ( (cid:107) ( G/T ) • × BT • (cid:107) ) W is an isomorphism.Then Diagram 3.1 implies that Diagram 3.3 commutes. (cid:3) HARACTERISTIC CLASSES FOR TC STRUCTURES 12
Proposition 9.
Suppose G is a compact connected Lie group such that Hom ( Z m , G ) is path connected for every non negative integer m . Then for the cohomology withreal coefficients we have a commutative diagram H ∗ ( B com G ) (cid:47) (cid:47) Φ k (cid:15) (cid:15) ( H ∗ ( BT ) ⊗ H ∗ ( BT )) W /J Id ⊗ ψ k (cid:15) (cid:15) H ∗ ( B com G ) (cid:47) (cid:47) ( H ∗ ( BT ) ⊗ H ∗ ( BT )) W /J, where the horizontal maps are the same isomorphism given above, and Φ k are thepower maps on cohomology.Proof. In the Diagram 3.3 we can consider the naturality on the Kunneth formulasand the fact that the realization of the simplicial product are naturally isomorphicto the product of the realizations of each of the simplicial spaces involved (seeTheorem 14.3 of [May]). Then we obtain the commuting diagram(3.4) H ∗ ( B com G ) (cid:47) (cid:47) Φ k (cid:15) (cid:15) ( H ∗ ( G/T ) ⊗ H ∗ ( BT )) W Id ⊗ ψ k (cid:15) (cid:15) H ∗ ( B com G ) (cid:47) (cid:47) ( H ∗ ( G/T ) ⊗ H ∗ ( BT )) W , where the horizontal maps are still isomorphisms.Next in the proof of Proposition 7.1 of [AG] they replace H ∗ ( G/T ) giving naturalarguments, which allow us to obtain our conclusion. (cid:3) The last result is important since it tell us that in order to obtain the effect ofpower maps on cohomology of B com G , we need to understand their effect whenthe Lie group is a torus, T = (cid:0) S (cid:1) n . Theorem 10.
Consider the k -th power map ψ k : T → T ( t , . . . , t n ) (cid:55)→ (cid:0) t k , . . . , t kn (cid:1) . Then by identifying H ∗ ( BT ) ∼ = R [ x , . . . , x n ] , the induced k -th power map is char-acterized by R [ x , . . . , x n ] → R [ x , . . . , x n ] x i (cid:55)→ kx i . Proof.
On a circle the k -th power of its elements induces the multiplication by k on the fundamental group: if S = { z ∈ C | | z | = 1 } then the k -th power map is given by η : S → S z (cid:55)→ z k When
Hom ( Z m , G ) is path connected for every m . HARACTERISTIC CLASSES FOR TC STRUCTURES 13 which is know to be a map of degree k . This means that if identify π (cid:0) S (cid:1) ∼ = Z then the k -th power maps induces multiplication by k on the fundamental group.Consider the projections p i : (cid:0) S (cid:1) n → S ( z , . . . , z n ) (cid:55)→ z i . It is well known that the map q : π (cid:0)(cid:0) S (cid:1) n (cid:1) → π (cid:0) S (cid:1) n given by q ([ α ]) := ([ p ◦ α ] , . . . , [ p n ◦ α ]) is an isomoprihsm, since S is path connected. Since the power map ψ k : T → T considers the k -th power component wise, it follows that we have a commutativediagram π ( T ) q (cid:47) (cid:47) ψ k ∗ (cid:15) (cid:15) π (cid:0) S (cid:1) n (cid:81) η ∗ (cid:15) (cid:15) π ( T ) q (cid:47) (cid:47) π (cid:0) S (cid:1) n . Since the horizontal maps are isomorphisms, we have the characterization ψ k ∗ : π ( T ) → π ( T ) α (cid:55)→ kα where we see α = ( α , . . . , α n ) ∈ Z n , and kα = ( kα , . . . , kα n ) .Now let us consider the fiber sequence of the classifying space of the torus T → ET → BT.
This induces an exact sequence on homotopy groups · · · → π m ( ET ) → π m ( BT ) → π m − ( T ) → · · · π ( ET ) → π ( BT ) →→ π ( T ) → π ( ET ) → π ( BT ) → but since ET is contractible, we get an isomorphism π m ( BT ) → π m − ( T ) . Inparticular we get π m ( BT ) = (cid:40) Z n m = 2 , . Since the exact sequence is natural, we get that the power map on BT inducesthe multiplication by k on the second homotopy group. Furthermore since BT issimply connected, by Hurewicz´s theorem we get that H ( BT, Z ) ∼ = π ( BT ) , andonce again because of naturality the effect on the second homology is multiplicationby k .We now apply the universal coefficients theorem to get that H ( BT ) ∼ = Hom ( H ( BT, Z ) , R ) ∼ = R n . Naturality allow us to conclude that the effect of the k -th power map is once againmultiplication by k . Finally it is known that the real cohomology of BT is thepolynomial ring R [ x , . . . , x n ] where x i ∈ H ( BT ) for ≤ i ≤ n (see [Dupont],Proposition 8.11). Since we know that the effect of the k power map is multiplicationby k on the x i , this determines the effect on the whole cohomology ring. (cid:3) As corollary of Proposition 9 and Theorem 10 we obtain the following:
HARACTERISTIC CLASSES FOR TC STRUCTURES 14
Theorem 11.
Identify the real cohomology ring of an n -torus with R [ x , . . . , x n ] ,and suppose that G is a Lie group such that Hom ( Z m , G ) = Hom ( Z m , G ) for every m , then H * ( B com G ) ∼ = ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) W /J. Here J is the ideal generated by the invariant polynomials of positive degree onthe x i under the action of the Weyl group, W . Further, the power maps Φ k : H * ( B com G, F ) → H * ( B com G, F ) are induced by the homomorphism characterizedby sending x i (cid:55)→ x i and y i (cid:55)→ ky i for every ≤ i ≤ n . Generators of H ∗ ( B com G, R ) for G = U ( n ) , Sp ( n ) and SU ( n ) Here we are going to examine the cases of the Lie groups G = U ( n ) , Sp ( n ) and SU ( n ) . From this we see that possible differences between the different cases depend entirelyon the Weyl group and its action on the cohomology of BT . For this we first needto establish some facts and definitions.If B com G = B com G we know that H * ( B com G, R ) ∼ = ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) W /J. While in general it is known that for a compact and connected Lie group GH * ( BG, R ) ∼ = H * ( BT, R ) W ∼ = P [ t ] W , where W is its Weil group. Its action is induced by adjunction. That is, if t is the Liealgebra of T , P [ t ] is the polynomial algebra of t . An element [ n ] ∈ W ∼ = N G ( T ) /T has a well defined action given by adjunction, ad ( n ) : t → t . This in turn induces anaction of W on P [ t ] . Even further, if n is the dimension of t , P [ t ] can be identifiedwith R [ z , . . . , z n ] , and under such identification, we have an action of W on thelatter.There is a natural inclusion B com G (cid:44) → BG , inducing a map ι : H * ( BG, R ) → H * ( B com G, R ) . In terms of the previous identifications, ι is induced by the homomorphism ([Gritschacher],Corollary A.2.) R [ z , . . . , z n ] → R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] z i (cid:55)→ x i + y i . Additionally we saw in the previous section that the power maps, Φ k : H * ( B com G, R ) → H * ( B com G, R ) are induced by the map characterized by sending x i (cid:55)→ x i and y i (cid:55)→ ky i for every ≤ i ≤ n . HARACTERISTIC CLASSES FOR TC STRUCTURES 15
Definition 12.
We call the subalgebra generated by (cid:8) Φ k (Im ι ) | k ∈ Z \ { } (cid:9) ⊂ H * ( B com G, R ) by S := (cid:10) Φ k (Im ι ) | k ∈ Z \ { } (cid:11) . On this section we use the previous maps to see that if G = U ( n ) , Sp ( n ) and SU ( n ) then S is all of H ∗ ( B com G, R ) . We do this by dealing with the explicit descriptionsof their actions and the specific Weyl groups on each case.Before dealing with each individual case, it is worth proving the following Lemma 13.
The subalgebra S is closed under the power maps.Proof. This is true since Φ k is a R -homomorphism of algebras as well because ofthe equality Φ k ◦ Φ l = Φ kl . This implies that for q j ∈ R [ z , . . . , z n ] , α j ∈ R Φ k (cid:32) s (cid:88) l =1 α j Φ k j ◦ ι ( q j ) (cid:33) = s (cid:88) l =1 α j Φ kk j ◦ ι ( q j ) ∈ S . (cid:3) Next we explore individually the particular cases of G equal to U ( n ) , SU ( n ) and Sp ( n ) to show that S = H ∗ ( B com G, R ) . Generators of H ∗ ( B com U ( n ) , R ) : For this case recall that the Weil groupof U ( n ) is isomorphic to the symmetric group S n . By the previous section we knowthat H ∗ ( B com U ( n ) , R ) = ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n /J where S n acts diagonally on the tensor product, permuting the variables of eachfactor. J is the ideal generated by the symmetric polynomials of positive degree onthe x i . It is also known that H ∗ ( BU ( n ) , R ) = ( R [ x , . . . , x n ]) S n , where the action is once again by permuting variables. H ∗ ( BU ( n ) , R ) is generatedby the power polynomials p m := z m + z m + · · · + z mn , which are clearly invariant under the action of S n . These polynomials have theircounterparts on two variables polynomials in the form of P a,b ( n ) := x a y b + x a y b + · · · + x an y bn , where ≤ a + b ≤ n . These generate the algebra ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n (See [Vaccarino], Theorem 1). Thus to prove that S is all of H ∗ ( B com U ( n ) , R ) itis enough to see that the multisymmetric polynomials (modulo J ) are in fact in S .To see it, we first need a couple of lemmas. Lemma 14.
For every n ∈ N and ≤ a + b ≤ n with a, b ≥ we have Φ k ( P a,b ( n )) = k b P a,b ( n ) . HARACTERISTIC CLASSES FOR TC STRUCTURES 16
Proof.
Since Φ k is a homomorphism of algebras, we have Φ k ( P a,b ( n )) = Φ k (cid:0) x a y b + x a y b + · · · + x an y bn (cid:1) = n (cid:88) i =1 Φ k (cid:0) x ai y bi (cid:1) . But we have Φ k (cid:0) x ai y bi (cid:1) = Φ k ( x i ) a Φ k ( y i ) b = k b x ai y bi . Where the last equality is true since we already saw that Φ k ( x i ) = x i and Φ k ( x i ) = ky i for every ≤ i ≤ n . (cid:3) To prove the goal of this subsection, we illustrate explicitly the cases n = 2 and n = 3 . • Suppose first that n = 2 .We want to show that the following multisymmetric polynomials areindeed in S : – P , (2) = y + y , – P , (2) = x y + x y and – P , (2) = y + y .We ignore P , (2) = x + x since this is zero modulo J . For this firstobserve that ι ( z + z ) = ( x + y ) + ( x + y ) = ( x + x ) + ( y + y ) = P , (2) + P , (2) clearly belongs to S . Since P , (2) = 0 mod J we are done. For P , (2) and P , (2) notice that the total degree (the sum of the power of each term) is2, thus we have to consider ι ( p ) : ι (cid:0) z + z (cid:1) = ( x + y ) + ( x + y ) = (cid:0) x + x (cid:1) + 2 ( x y + x y ) + (cid:0) y + y (cid:1) . This can be rewritten as ι (cid:0) z + z (cid:1) = P , (2) + 2 P , (2) + P , (2) . Then we consider Φ − (cid:0) ι (cid:0) z + z (cid:1)(cid:1) = (cid:0) x + x (cid:1) − x y + x y ) + (cid:0) y + y (cid:1) giving us that ι (cid:0) z + z (cid:1) + Φ − (cid:0) ι (cid:0) z + z (cid:1)(cid:1) = 2 (cid:0) y + y (cid:1) mod J meaning that P , (2) ∈ S , since S is a subalgebra closed under power maps.Finally modulo J we get P , (2) = ι (cid:0) z + z (cid:1) − P , (2)2 ∈ S which finishes the proof for n = 2 . • Suppose now that n = 3 .The arguments used in the case n = 2 can be used to obtain the firsttwo of the next equalities, where once again they are taken to be modulo J : HARACTERISTIC CLASSES FOR TC STRUCTURES 17 (1) P , (3) = ι ( z + z + z ) .(2) P , (3) = (cid:0) ι (cid:0) z + z + z (cid:1) + Φ − (cid:0) ι (cid:0) z + z + z (cid:1)(cid:1)(cid:1) .(3) P , (3) = (cid:0) ι (cid:0) z + z + z (cid:1) − P , (cid:1) .We are left to obtain P a,b (3) such that a + b = 3 . For this we can reorderto see that ι (cid:0) z + z + z (cid:1) = ( x + y ) + ( x + y ) + ( x + y ) = (cid:0) x + x + x (cid:1) + 3 (cid:0) x y + x y + x y (cid:1) + 3 (cid:0) x y + x y + x y (cid:1) + (cid:0) y + y + y (cid:1) which amounts to ι (cid:0) z + z + z (cid:1) = 3 P , + 3 P , + P , mod J. We use the power maps to get that Φ − (cid:0) ι (cid:0) z + z + z (cid:1)(cid:1) = − P , + 3 P , − P , mod J. By adding the last two equalities we get P , mod J = 16 (cid:0) Φ − (cid:0) ι (cid:0) z + z + z (cid:1)(cid:1) + ι (cid:0) z + z + z (cid:1)(cid:1) ∈ S . Thus we have ι (cid:0) z + z + z (cid:1) − P , mod J ∈ S , and by closure underpower maps we obtain P , mod J = Φ (cid:0) ι (cid:0) z + z + z (cid:1) − P , (cid:1) − (cid:0) ι (cid:0) z + z + z (cid:1) − P , (cid:1) ∈ S from we conclude that P , mod J ∈ S . We finally have P , = 13 (cid:0) ι (cid:0) z + z + z (cid:1) − P , − P , (cid:1) mod J which finishes the case n = 3 .In the previous two examples we see that for non negative numbers a and b , weproved that P a,b ( n ) belongs to S using induction on the value a + b . This was donein such a way that the induction process did not depend on n . These argumentscan be generalized more methodically to obtain. Theorem 15.
The algebra H ∗ ( B com U ( n ) ; R ) is equal to the subalgebra S := (cid:10) Φ k (Im ι ) | k ∈ Z \ { } (cid:11) . Proof.
For this proof we will be working modulo J . Also, for an arbitrary n considera fixed m ∈ { , , . . . , n } . Now take p m := z m + z m + · · · + z mn . An easy reordering gives us ι ( p m ) = (cid:32) n (cid:88) i =1 ( x i + y i ) m (cid:33) = n (cid:88) i =1 m (cid:88) j =0 (cid:18) mj (cid:19) x m − ji y ji = m (cid:88) j =0 (cid:18) mj (cid:19) P m − j,j ( n ) = m (cid:88) j =1 (cid:18) mj (cid:19) P m − j,j ( n ) , (4.1) HARACTERISTIC CLASSES FOR TC STRUCTURES 18 where the last equality holds because we are working modulo J . From this pointwe will use the power maps Φ k to obtain the various P m − j , j ( n ) . First we use thefollowing recursion to get first P ,m ( n ) from 4.1: Let A := ι ( p m ) , A := Φ ( A ) − A = m (cid:88) j =2 (cid:0) j − (cid:1) (cid:18) mj (cid:19) P m − j,j ( n ) and A := Φ ( A ) − A = m (cid:88) j =3 (cid:0) j − (cid:1) (cid:0) j − (cid:1) (cid:18) mj (cid:19) P m − j,j ( n ) . In general for ≤ k ≤ m − we define A k := Φ k +1 ( A k − ) − ( k + 1) k A k − . Notice that every A k has non zero coefficients only for P m − j,j ( n ) for k +1 ≤ j ≤ m .Since A ∈ S by definition and every A k is defined in terms of the power mapsand A k − , induction implies that A k ∈ S for every ≤ k ≤ m − . Some easycalculations allow us to obtain that P ,m ( n ) = (cid:32) m (cid:89) k =2 (cid:0) k m − k k − (cid:1)(cid:33) − A m − ∈ S . And thus we obtain that ι ( p m ) − P ,m ( n ) = m − (cid:88) j =1 (cid:18) mj (cid:19) P m − j,j ( n ) ∈ S . Then we can apply a new recursion to conclude that P ,m − ( n ) ∈ S . By continuingwith this backwards recursion we obtain that P a,b ( n ) ∈ S for all positive a, b suchthat a + b = m . Since we picked m ∈ { , , . . . , n } arbitrarily, this finishes theproof. (cid:3) Generators of H ∗ ( B com SU ( n ) ; R ) : To obtain that H ∗ ( B com SU ( n ) , R ) = (cid:10) Φ k (Im ι ) | k ∈ Z \ { } (cid:11) , we use a different presentation of H ∗ ( BT, R ) . A maximal torus of SU ( n ) is theset of diagonal matrices with entries in S ⊆ C , such that their product equals one.Under such presentation it is routine to show that H ∗ ( BT, R ) ∼ = ( R [ z , . . . , z n ] / (cid:104) z + · · · + z n (cid:105) ) where the Weyl group is then S n acting by permutation. This implies that H ∗ ( BSU ( n ) , R ) ∼ = ( R [ z , . . . , z n ] / (cid:104) z + · · · + z n (cid:105) ) S n , but since p := z + · · · + z n is already invariant, the previous ring is isomorphic to H ∗ ( BSU ( n ) , R ) ∼ = R [ z , . . . , z n ] S n / (cid:104) z + · · · + z n (cid:105) . Since R [ z , . . . , z n ] S n is itself a polynomial algebra on p i = z i + · · · + z in for ≤ i ≤ n ,(see [Humphrey], Chapter 3.5: Chevalley’s Theorem), we finally get that H ∗ ( BSU ( n ) , R ) ∼ = R [ p , . . . , p n ] / (cid:104) p (cid:105) ∼ = R [ p , . . . , p n ] . We will use this to conclude the following
HARACTERISTIC CLASSES FOR TC STRUCTURES 19
Theorem 16.
The real cohomology of B com SU ( n ) can be given by H * ( B com SU ( n ) , R ) ∼ = ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n / ˜ J, where ˜ J is the ideal generated by x i + · · · + x in , ≤ i ≤ n and y + · · · + y n .Proof. We saw in Theorem 9 that H * ( B com SU ( n ) , R ) ∼ = ( H ∗ ( BT ) ⊗ H ∗ ( BT )) S n /J, where J is the ideal generated by the S n -invariants on the first component. Now,for convenience, let us call R [ x ] := R [ x , . . . , x n ] , R [ y ] := R [ y , . . . , y n ] ,f ( x ) = x + · · · + x n and f ( y ) = y + · · · + y n The previous reasoning then gives us H * ( B com SU ( n ) , R ) ∼ = ( R [ x ] / (cid:104) f ( x ) (cid:105) ⊗ R [ y ] / (cid:104) f ( y ) (cid:105) ) S n /J. Notice that this is well defined since the S n -invariance of x + · · · + x and y + · · · + y allow us to have a well define action of S n on R := R [ x ] / (cid:104) f ( x ) (cid:105) ⊗ R [ y ] / (cid:104) f ( y ) (cid:105) . Consider first the map p : R [ x ] ⊗ R [ y ] → R, which is induced by the projection R [ x ] × R [ y ] → R [ x ] / (cid:104) f ( x ) (cid:105) × R [ y ] / (cid:104) f ( y ) (cid:105) . The map p is naturally S n -equivariant, thus it induces a map ˜ p : ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n → R S n . Also, since p is surjective, and the action is diagonal, we have that ˜ p is also onto. Wecan further consider the composition with the quotient by J to obtain a surjectivemap q : ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n → ( R ) S n /J. It is easy to see that the kernel of this map is what we called ˜ J , so the resultfollows. (cid:3) Even further, since the map R [ z , . . . , z n ] → R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] z i (cid:55)→ x i + y i . induces the map ι : H * ( BSU ( n ) , R ) → H * ( B com SU ( n ) , R ) , we still have thesame characterization under the identifications given above. That is, ι can be seenas the map ( R [ z , . . . , z n ] / (cid:104) z + · · · + z n (cid:105) ) S n → ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n / ˜ J induce by z i (cid:55)→ x i + y i . The k -th power map on ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n / ˜ J HARACTERISTIC CLASSES FOR TC STRUCTURES 20 is also still induce by the assignment x i (cid:55)→ x i and y i (cid:55)→ ky i . Thus, with slightchanges we can still apply the arguments given in the proof of Theorem 15, toobtain the main result. Theorem 17.
The algebra H ∗ ( B com SU ( n ) ; R ) is equal to the subalgebra S := (cid:10) Φ k (Im ι ) | k ∈ Z \ { } (cid:11) , where Φ k is the k -th power map. Generators of H ∗ ( B com Sp ( n ) ; R ) : In this section Z will denote the multi-plicative group {− , } .The Weyl group W of the simplectic group Sp ( n ) is isomorphic to the semidirectproduct Z n (cid:111) S n , where σ ∈ S n acts on ( a , . . . , a n ) ∈ Z n by σ · ( a , . . . , a n ) = (cid:0) a σ (1) , . . . , a σ ( n ) (cid:1) . Under these identifications, if f ∈ R [ x , . . . , x n ] ∼ = H ∗ ( T ) and g = (( a , . . . , a n ) , σ ) ∈ Z n (cid:111) S n we have g · f ( x , . . . , x n ) = f (cid:0) a x σ (1) , . . . , a n x σ ( n ) (cid:1) . Recall that H ∗ ( B com Sp ( n ) ; R ) ∼ = ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) W /J where W acts diagonally: for n ∈ W and p ( x ) ⊗ q ( y ) ∈ R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] we have n · ( p ( x ) ⊗ q ( y )) := ( n · p ( x )) ⊗ ( n · q ( y )) .J is the ideal generated by the symmetric polynomials on the variables x i . Forbrevity, let us call R := ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) W the signed multisym-metric polynomials .Once again we want to see that S := (cid:10) Φ k (Im ι ) | k ∈ Z \ { } (cid:11) is equal to all of H ∗ ( B com Sp ( n ) ; R ) . For this let us see first that the set { P a,b ( n ) | a, b ≥ a + b ∈ Z } generates all of the signed multisymmetric polynomials as an algebra. This willallow us to use the same arguments used in the case of U ( n ) to obtain that S = H ∗ ( B com Sp ( n ) ; R ) . We need the following lemmas, where the first has astraightforward proof. Lemma 18.
Let µ : R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] → R be the operator defined as µ ( f ) = 1 | W | (cid:88) g ∈ W g · f. This is a well defined R -linear map, where | W | is the cardinality of the Weyl group.We call this operator the symmetrization operator . Lemma 19. If f ∈ R and h ∈ R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] , then µ ( f ) = f and µ ( f h ) = f · µ ( h ) . HARACTERISTIC CLASSES FOR TC STRUCTURES 21
Proof.
Since f is invariant, we have that g · f = f for all g ∈ W , thus µ ( f ) = 1 | W | (cid:88) g ∈ W g · f = | W || W | f = f. Also, since by definition g · ( f h ) = ( g · f ) ( g · h ) for every g ∈ W and f, h ∈ R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] . In particular if f is invariant it follows that µ ( f h ) = 1 | W | (cid:88) g ∈ W g · ( f h ) = f | W | (cid:88) g ∈ W g · h = f · µ ( h ) . (cid:3) In order to prove our objective we need to analyze the summands (or monomials)of a signed multisymmetric polynomials first. For this consider sets of indices I = ( i , . . . , i n ) , J = ( j , . . . , j n ) ∈ N n (including zero as a natural number) and letus denote x I y J := x i x i · · · x i n n y j · · · y j n n . Definition 20.
We say a pair of multi indices ( I, J ) ∈ N n × N n is odd if there if ≤ k ≤ n such that i k + j k is odd. Such a pair is even if it is not odd. Lemma 21.
If a pair of multi indices ( I, J ) is odd, then µ (cid:0) x I y J (cid:1) = 0 .Proof. Let ( I, J ) = (( i , . . . , i n ) , ( j , . . . , j n )) and let’s assume i k + j k is odd. Let h k := , . . . , , − (cid:124)(cid:123)(cid:122)(cid:125) k − position , , . . . , , e ∈ W, where e is the identity permutation. Denote by H ⊆ W the subgroup generated by h k and the partition by right cosets { Hg , . . . , Hg m } of W. Since h k has order 2 W = { g , . . . , g m } ∪ { h k g , . . . , h k g m } and thus µ (cid:0) x I y J (cid:1) = 1 | W | m (cid:88) l =1 (cid:0) g l x I y J + h k (cid:0) g l x I y J (cid:1)(cid:1) . Notice that in general if g = (( a , . . . , a n ) , σ ) , then since i k + j k is odd we get h k (cid:0) g · x I y J (cid:1) = h k (cid:16) a i + j · · · a i n + j n n x i σ (1) · · · x i n σ ( n ) y j σ (1) · · · y j n σ ( n ) (cid:17) = ( − i k + j k a i + j · · · a i n + j n n x i σ (1) · · · x i n σ ( n ) y j σ (1) · · · y j n σ ( n ) = − a i + j · · · a i n + j n n x i σ (1) · · · x i n σ ( n ) y j σ (1) · · · y j n σ ( n ) = − g · x I y J . This implies that µ (cid:0) x I y J (cid:1) = 1 | W | m (cid:88) l =1 (cid:0) g l x I y J − g l x I y J (cid:1) = 0 . (cid:3) HARACTERISTIC CLASSES FOR TC STRUCTURES 22
Theorem 22.
If a polynomial is signed multisymmetric then its monomials haveall even multi indices.Proof.
An element f ∈ R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] can be uniquely written as f = c + m (cid:88) k =1 c k x I k y J k . Where c ∈ R and for k > , c k ∈ R \{ } , I k and J k are multi indices of n variables,not all of them zero. If f is signed multisymmetric, f = µ ( f ) = c + m (cid:88) k =1 c k µ (cid:0) x I k y J k (cid:1) . These two last expressions for f imply that(4.2) m (cid:88) k =1 c k x I k y J k = m (cid:88) k =1 c k µ (cid:0) x I k y J k (cid:1) . But by the previous lemma, we know that if ( I t , J t ) is odd for a given t , then µ (cid:0) x I t y J t (cid:1) = 0 . Since µ (cid:0) x I k y J k (cid:1) is itself a sum of monomials, the expression m (cid:88) k =1 c k µ (cid:0) x I k y J k (cid:1) must have only monomials with an even set of multi indices. Since all the coefficientsin m (cid:88) k =1 c k x I k y J k are non zero, the last equality and the uniqueness of the expression for non zerocoefficients of a polynomial, allow us to conclude that ( I k , J k ) is even for every ≤ k ≤ m . (cid:3) In particular this proof allows us to obtain
Corollary 23.
Every signed multisymmetric polynomial can be written in the form f = c + m (cid:88) k =1 c k µ (cid:0) x I k y J k (cid:1) , where ( I k , J k ) is even for every ≤ k ≤ m . If a multisymmetric polynomial has monomials with even multi indices, suchpolynomial is signed symmetric, meaning that is invariant under the action of ele-ments of the form (( a , . . . , a n ) , e ) ∈ W . In particular we can now conclude: Theorem 24.
A multisymmetric polynomial is signed symmetric if only if all itsmonomials have even multi indices.
This result grant us the frame work to obtain generators for the algebra H ∗ ( B com Sp ( n ) ; R ) . Recall that multisymmetric are generated by the power polynomials P a,b := n (cid:88) i =1 x ai y bi . HARACTERISTIC CLASSES FOR TC STRUCTURES 23
On the other hand, due to the last result we know P a,b is signed multi symmetricif and only if a + b is even. Let’s see that they in fact generate all of the signedmultisymmetric polynomials. Theorem 25. ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) Z n (cid:111) S n is generated as an algebra bythe set G := (cid:40) P a,b := n (cid:88) i =1 x ai y bi | ≤ a, b and a + b ∈ Z (cid:41) . Proof.
By Corollary 23 is enough to show that for even multi indices ( I, J ) , µ (cid:0) x I y J (cid:1) ∈ gen G . To see this, note that any permutation of the set of indices have the same sym-metrization. This is, for k , . . . , k p ∈ { , . . . , n } all mutually different, p ≤ n , wehave µ (cid:16) x i k · · · x i p k p y j k · · · y j p k p (cid:17) = µ (cid:16) x i · · · x i p p y j · · · y j p p (cid:17) . So it is enough to show that µ (cid:16) x i · · · x i p p y j · · · y j p p (cid:17) ∈ gen G , where of course i k + j k is even for every ≤ k ≤ p . We do it using induction on p . The cases p = 1 is immediate, since in this case µ (cid:0) x I y J (cid:1) is a scalar multiple ofeven power polynomials of the form P a, , P ,b or P a,b .Next, assume we know µ (cid:16) x i · · · x i p p y j · · · y j p p (cid:17) ∈ gen G for ≤ p ≤ k . Byreordering we have µ (cid:16) x i y j (cid:17) µ (cid:16) x i · · · x i k +1 k +1 y j · · · y j k +1 k +1 (cid:17) = cµ (cid:16) x i · · · x i k +1 k +1 y j · · · y j k +1 k +1 (cid:17) + Θ , with Θ = k +1 (cid:88) r =2 c r µ (cid:16) x i · · · x i r + i r · · · x i k +1 k +1 y j · · · y j r + j r · · · y j k +1 k +1 (cid:17) , where c , . . . , c k +1 are integers, and c is a non zero integer. This implies that µ (cid:16) x i · · · x i k +1 k +1 y j · · · y j k +1 k +1 (cid:17) = 1 c (cid:16) µ (cid:16) x i y j (cid:17) µ (cid:16) x i · · · x i k k y j · · · y j k k (cid:17) − k +1 (cid:88) r =2 c r µ (cid:16) x i · · · x i r + i r · · · x i k k y j · · · y j r + j r · · · y j k k (cid:17)(cid:33) . By the induction hypothesis all of the terms in the right are in gen G , which impliesthat µ (cid:16) x i · · · x i k +1 k +1 y j · · · y j k +1 k +1 (cid:17) belongs to gen G . (cid:3) With the last result at hand we can imitate the reasoning in the proof of Theorem15 to obtain the main result of this part.
Theorem 26.
The algebra H ∗ ( B com Sp ( n ) ; R ) is equal to the subalgebra S := (cid:10) Φ k (Im ι ) | k ∈ Z \ { } (cid:11) . HARACTERISTIC CLASSES FOR TC STRUCTURES 24
Where Φ k are the power maps and ι : H ∗ ( B Sp ( n ) ; R ) → H ∗ ( B com Sp ( n ) ; R ) isthe map induced by the homomorphism R [ z , . . . , z n ] → R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ] z i (cid:55)→ x i + y i . Proof.
Take once again p m = z m + z m + · · · + z mn ∈ R [ z , . . . , z n ] for m even. We also work modulo J , the ideal generated by the x i . Recall that ι ( p m ) = m (cid:88) j =1 (cid:18) mj (cid:19) P m − j,j ( n ) . Since ( m − j ) + j = m , all of the power polynomials P m − j,j ( n ) are even. Nowwe use recursion to get first P ,m ( n ) from the last equality: for this we name A := ι ( p m ) , then we take A := Φ ( A ) − A = m (cid:88) j =2 (cid:0) j − (cid:1) (cid:18) mj (cid:19) P m − j,j ( n ) and A := Φ ( A ) − A = m (cid:88) j =3 (cid:0) j − (cid:1) (cid:0) j − (cid:1) (cid:18) mj (cid:19) P m − j,j ( n ) . In general for ≤ k ≤ m − we define A k := Φ k +1 ( A k − ) − ( k + 1) k A k − . Notice that every A k has non zero coefficients only for P m − j,j ( n ) for k +1 ≤ j ≤ m .Since A ∈ S by definition and every A k is defined in terms of the power maps and A k − , induction implies that A k ∈ S for every ≤ k ≤ m − . In particular wehave P ,m ( n ) = (cid:32) m (cid:89) k =2 (cid:0) k m − k k − (cid:1)(cid:33) − A m − ∈ S . We now can apply a similar procedure to ι ( p m ) − P ,m ( n ) = m − (cid:88) j =1 (cid:18) mj (cid:19) P m − j,j ( n ) ∈ S to conclude that if m = 2 k , and P a,b is such that a + b = m then P a,b ∈ S . (cid:3) Chern-Weil theory for TC structures
In this section we develop characteristic classes for TC structures. Our centralgoal is to obtain characteristic classes for TC structures using Chern-Weil theory.Specifically, we will develop this for TC structures over vector bundles whose struc-tural group is either U ( n ) or SU ( n ) . Thus, by G we will mean one of these groups.Recall that we have the k -th power maps Φ k : H ∗ ( B com G ) → H ∗ ( B com G ) , anda natural inclusion ι : H ∗ ( BG ) → H ∗ ( B com G ) . We already proved that Theorem 27.
For G equal to U ( n ) or SU ( n ) , then H ∗ ( B com G ) = gen (cid:8) Φ k ◦ ι ( c ) | c ∈ H ∗ ( BG ) , k ∈ Z \ { } (cid:9) . HARACTERISTIC CLASSES FOR TC STRUCTURES 25
This means that given a class in H ∗ ( B com G ) , it can be written as a sum of finiteproducts of elements of the form Φ k ◦ ι ( c ) , c ∈ H ∗ ( BG ) , k ∈ Z \ { } . Now wecontinue with a construction that allow us to use Chern-Weil theory to computecharacteristic classes associated to the previous classes.5.1. Chern-Weil theory for TC structures:
For the rest of this section let ε ∈ [ M, B com G ] be an equivalence class with an underlying smooth vector bundle E → M and structure group U ( n ) or SU ( n ) . For an element p ∈ H ∗ ( B com G, R ) we denote by p ( ε ) ∈ H ∗ ( M ) the value of the TC characteristic class on the TCequivalence class ε . Also, recall that via the Chern-Weil isomorphism, if g is theLie algebra of G , then H ∗ ( BG ) ∼ = I ( g ) . Here I ( g ) is the subalgebra of invariantpolynomials under conjugation of the polynomial algebra of g . Under this iden-tification, every characteristic class for vector bundles (having G as its structuregroup) can be identified with a polynomial c ∈ I ( g ) .Now recall that for a smooth vector bundle F → M with curvature Ω , the valueon F of the characteristic class associated to c is equal to c (Ω) ∈ H ∗ ( M ) . Underthese terms, we are now able to compute the TC characteristic classes associatedto the set of generators of H ∗ ( B com G ) , (cid:8) Φ k ◦ ι ( c ) | ≤ i ≤ n, k ∈ Z \ { } (cid:9) . Here,we take ι to be a map from I ( g ) to H ∗ ( B com G ) . Theorem 28.
Consider ε ∈ [ M, B com G ] an equivalence class with an underlyingsmooth vector bundle E → M , and structure group U ( n ) or SU ( n ) . Also let Ω k be the curvature of E k , the k -th associated bundle of E . Then for c ∈ I ( g ) and p = Φ k ◦ ι ( c ) ∈ H ∗ ( B com G ) , the TC characteristic class p ( ε ) has same class in H ∗ ( M ) as the characteristic class for vector bundles c (cid:0) E k (cid:1) . This implies that p ( ε ) = c (Ω k ) ∈ H ∗ ( M ) . Proof.
This is straight forward. First, by Theorem 5 we know that if f and f k thethe classifying functions for TC structures over E → M and E k → M , respectively,then there is the following commuting diagram H ∗ ( B com G ) f ∗ (cid:47) (cid:47) H ∗ ( M ) H ∗ ( B com G ) . f ∗ k (cid:55) (cid:55) Φ k (cid:79) (cid:79) This means that for c ∈ H ∗ ( BG ) we have the identity f ∗ (cid:0) Φ k ◦ ι ( c ) (cid:1) = f ∗ k ( ι ( c )) in H ∗ ( M ) .In turn, since the composition f ∗ k ◦ ι is a classifying function for the vector bundle E k → M , we can apply the Chern-Weil isomorphism. That is, we can consider thecurvature Ω k of E k to obtain that f ∗ k ( ι ( c )) = c (Ω k ) . The conclusion of the theorem then follows by transitivity. (cid:3)
Theorem 29. (Chern-Weil theory for TC structures) Consider ε ∈ [ M, B com G ] anequivalence class with an underlying smooth vector bundle E → M , and structuregroup U ( n ) or SU ( n ) . Also let Ω k be the curvature of E k , the k -th associated HARACTERISTIC CLASSES FOR TC STRUCTURES 26 bundle of E . Then every TC characteristic class can be obtained as a linear com-binations of products of the form s (Ω k ) · s (Ω k ) · · · s m (Ω k m ) ∈ H ∗ ( M ) , where s i ∈ H ∗ ( BG ) and k i ∈ Z . Each s i (Ω k ) is the characteristic class of thevector bundle E k → M computed using its curvature.Proof. Recall that if we set S as the subalgebra of H ∗ ( B com G ) generated by (cid:8) Φ k ◦ ι ( s ) | ≤ i ≤ n, k ∈ Z \ { } , s ∈ H ∗ ( BG ) (cid:9) then we have S = H ∗ ( B com G ) . Thus, every element of H ∗ ( B com G ) can be writtenas a linear combination of products of the form Φ k ( ι ( s )) · Φ k m ( ι ( s )) · · · Φ k m ( ι ( s m )) . Then we can apply the previous theorem to obtain Φ k i ( ι ( s i )) = s i (Ω k i ) . (cid:3) As suggested by the name of the theorem, we are now able to compute TCcharacteristic classes by using Chern-Weil theory. This is done in a three stepsprocess for a class in s ∈ H ∗ ( B com G ) and a TC structure ξ over a vector bundle E → M : first we need to decompose s in terms of the generators in (cid:8) Φ k ◦ ι ( c ) | ≤ i ≤ n, k ∈ Z \ { } (cid:9) . Secondly, for each of the generators Φ k ◦ ι ( c ) in the decomposition of s we use thecurvature of the k -th associated bundle, Ω k , to compute the characteristic classassociated to it, c (Ω k ) ∈ H ∗ ( M ) (this class is equal to the TC class given by (cid:0) Φ k ◦ ι ( c ) (cid:1) ( ξ ) ). Finally we replace the values of each (cid:0) Φ k ◦ ι ( c ) (cid:1) ( ξ ) to obtain s ( ξ ) ∈ H ∗ ( M ) .Recall from Chapter 3 that when G is equal to U ( n ) , then H ∗ ( B com G, R ) ∼ = ( R [ x , . . . , x n ] ⊗ R [ y , . . . , y n ]) S n /J where S n acts by permutation on their indexes and J is the ideal generated by theinvariant polynomials of positive degree on the x i . When G is SU ( n ) is the samedescription for H ∗ ( B com G, R ) except J is generated by the invariant polynomialsof positive degree on x i and the polynomial y + · · · y n .We also have the identifications H ∗ ( BU ( n ) , R ) ∼ = R [ z , . . . , z n ] S n and H ∗ ( BSU ( n ) , R ) ∼ = R [ z , . . . , z n ] S n / (cid:104) z + · · · + z n (cid:105) . Then we have that the polynomials p i = z i + · · · + z in ∈ R [ z , . . . , z n ] generated all of H ∗ ( BG, R ) , when G is U ( n ) or SU ( n ) . Even further for a, b ∈ N ∪ { } such that < a + b then P a,b ( n ) := n (cid:88) i =1 x a y b mod J. HARACTERISTIC CLASSES FOR TC STRUCTURES 27 generated all of H ∗ ( B com G, R ) as an algebra. We also saw in the proof of Theorem15 there every P a,b ( n ) can be obtain, via a recursive procedure, as a linear combi-nation of elements of the form Φ k ( ι ( p i )) . With that recursive procedure and theprevious theorem, we can compute the TC characteristic classes corresponding toeach P a,b ( n ) .Recall that another set of generators for H ∗ ( BG, R ) , when G is U ( n ) or SU ( n ) is given by the polynomials σ i , characterized by the equation det ( I − tX ) = 1 + tσ ( X ) + t σ ( X ) + · · · + t n σ n ( X ) . These generators are more commonly used instead of the p i , as σ i are used in thedefinition of Chern classes. Example 30.
We saw previously that for G = U (3) we have the equalities(1) y + y + y = ι ( z + z + z ) .(2) y + y + y = (cid:0) ι (cid:0) z + z + z (cid:1) + Φ − (cid:0) ι (cid:0) z + z + z (cid:1)(cid:1)(cid:1) .(3) x y + x y + x y = (cid:0) ι (cid:0) z + z + z (cid:1) − Φ − (cid:0) ι (cid:0) z + z + z (cid:1)(cid:1)(cid:1) .Now consider a TC strcutrure ξ with underlying vector bundle E → M , withcurvature Ω and Ω k is the curvature of the k -th associated bundle. Now since wehave that p = σ and that p = σ − σ we obtain that(1) ( y + y + y ) ( ξ ) = σ (Ω) .(2) (cid:0) y + y + y (cid:1) ( ξ ) = (cid:16) σ (Ω) + σ (Ω − ) (cid:17) − ( σ (Ω) + σ (Ω − )) .(3) And finally ( x y + x y + x y ) ( ξ ) = 14 (cid:16) σ (Ω) − σ (Ω − ) (cid:17) + 12 ( σ (Ω − ) − σ (Ω)) . For G = U ( n ) we know that H ∗ ( BG ) is a polynomial algebra generated bythe Chern classes c i , ≤ i ≤ n . Thus it follows that S is generated by the set (cid:8) Φ k ◦ ι ( c i ) | ≤ i ≤ n, k ∈ Z \ { } (cid:9) . Definition 31.
We call the classes of the form c ki := Φ k ◦ ι ( c ) ∈ H ∗ ( B com U ( n )) the TC Chern classes . Also, for a TC structure ε with underlying vector bundle E → M we call c ki ( ε ) := f ∗ (cid:0) c ki (cid:1) ∈ H ∗ ( M ) the TC ( i, k ) -Chern class. Here f : M → B com U ( n ) is the classifying function ofthe TC structure.From the previous theorem we have the immediate following corollary: Corollary 32.
Let E → M by the underlying bundle of a TC structure structure ε , and let Ω k be the curvature of the k -th associated bundle. Then c ki ( (cid:15) ) = c i (Ω k ) . It is immediate from our results that (cid:8) c ki | k ∈ Z , i ∈ N (cid:9) HARACTERISTIC CLASSES FOR TC STRUCTURES 28 generates all of B com U ( n ) as an algebra. That is, every class in H ∗ ( B com U ( n )) can be written in the form s = m (cid:88) j =1 α j C j where α j ∈ R and C j = m j (cid:89) t =1 c k j,t i j,t , where k j,t ∈ Z and i j,t ∈ N . Then it follows that if ξ is a TC structure withunderlying vector bundle E → M , with curvature its Ω and Ω k the curvature ofthe k -th associated bundle, then s ( ξ ) = m (cid:88) j =1 α j (cid:32) m j (cid:89) t =1 c i j,t (cid:0) Ω k j,t (cid:1)(cid:33) ∈ H ∗ ( M ) . At this point it is worth mentioning that H ∗ ( B com G ) is in general not a poly-nomial algebra. For example when G = U ( n ) , the TC Chern classes are notalgebraically independent. However, the relationships governing them are rathercomplicated. As such, their values in a given TC structure can vary significantly.We see an example of this in the next chapter. Remark . The concepts developed in this section can also be applied to vectorbundles on the quaternions. In this case, the structural group is the simplecticgroup
Sp ( n ) . The main ideas we needed to developed TC characteristic classesalso hold for this group. As we saw in the previous section we also have powermaps on cohomology, and H ∗ ( B com Sp ( n ) , R ) is also generated by as an alge-bra by (cid:8) Φ k ◦ ι ( c i ) | ≤ i ≤ n, k ∈ Z \ { } (cid:9) . Where again ι : H ∗ ( B Sp ( n ) , R ) → H ∗ ( B com Sp ( n ) , R ) is induced by the natural inclusion B com Sp ( n ) → B Sp ( n ) .Also, since Sp ( n ) is a compact group, the Chern-Weil homomorphism is in fact anisomorphism. Thus, most of the ideas we used through out this section apply tothis case as well. 6. Example
In this final section we exhibit explicit calculations of examples using Chern-Weiltheory to compute TC characteristic classes. In particular we show there is a TCstructure ξ over a 4 sphere such that c i ( ξ ) = 0 for every i ∈ N while c − ( ξ ) (cid:54) = 0 .This shows that a TC Chern class c ni does not necessarily determines another TCChern class c mi , if m (cid:54) = n . This confirms that the underlying vector bundle of a TCstructure does not determine completely the TC structure.For this we develop the computations that allow us to obtain Chern classes interms of clutching functions, with SU (2) as the structural group. This treatmentis based on the concepts presented in [Morita], Chapter 5.6.1. Connection for a vector bundle with a two sets cover with trivializa-tions:
Let π : E → M denote a smooth vector bundle over C of dimension n , with HARACTERISTIC CLASSES FOR TC STRUCTURES 29 M a closed manifold. Assume we can find an open cover { U , U } of M togetherwith trivializations ϕ i : π − ( U i ) → U i × C n e (cid:55)→ ( π ( e ) , h i ( e )) . Suppose these trivializations have structure group a Lie group of matrices G . Thisis, we have a function ρ : U ∩ U → G ⊆ GL n ( C ) characterized by ϕ ◦ ϕ − : U ∩ U × C n → U ∩ U × C n ( x, v ) (cid:55)→ ( x, ρ ( x ) v ) . These trivializations induce smooth sections s ij : U i → π − ( U i ) x (cid:55)→ ϕ − j ( x, e j ) , where e j is the j -th vector of the standard basis of C n , and i = 1 , . This settingimplies that for x ∈ U i the set { s i ( x ) , s i ( x ) , . . . , s in ( x ) } ⊂ π − ( x ) is a basis.Under these conditions, for a point x ∈ U ∩ U it follows that(6.1) s k ( x ) = n (cid:88) l =1 ρ lk ( x ) s l ( x ) where we take ρ = [ ρ lk ] nk,l =1 .Now let { f , f } be a partition of the unity subordinated to { U , U } , as well asthe trivial connections over each U i , ∇ i . We can now define the connection ∇ X s := f ∇ X s + f ∇ X s. This means that for a vector field X and a section s , we consider their restriction to U i in order to evaluate ∇ iX . That is, we need first to consider the decomposition of s in terms of the basis { s i , . . . , s in } , which means that there are smooth functions α ji : U i → C such that for x ∈ U i s | U i ( x ) = n (cid:88) j =1 α ij ( x ) s ij ( x ) . Then, applying the product rule and the definition, we have ∇ X s := (cid:88) i,j f i X (cid:0) α ij (cid:1) s ij . Recall that with n -linearly independent sections { s , . . . , s n } we have the localexpressions for both the connection and the curvature, R : X ( M ) × X ( M ) → Γ ( E ) .There exists 1-forms ω ij and two forms Ω ij such that we can write ∇ X s i = (cid:88) i ω ij ( X ) s j and R ( X, Y ) ( s i ) = (cid:88) j Ω ij ( X, Y ) s j which gives rise to the local connection and curvature matrices ω := [ ω ij ] and Ω := [Ω ij ] . HARACTERISTIC CLASSES FOR TC STRUCTURES 30
These local forms are related to the transition function in the following way. FromEquality 6.1 we get that ∇ X s k = n (cid:88) l =1 f X ( ρ lk ( x )) s l ( x ) . From differential geometry we know that for a function f : M → R , X ( f ) = df ( X ) holds, where d is the external derivation. Thus we get the expression ∇ X s k = n (cid:88) l =1 f d ( ρ lk ( x )) ( X ) s l ( x ) , which allow us to write ∇ s k = n (cid:88) l =1 f d ( ρ lk ( x )) s l ( x ) . By the properties of cocycles we also know that s l ( x ) = n (cid:88) t =1 ρ − tl ( x ) s t ( x ) , where by ρ − tl we mean the components of the matrix ρ − . Thus, we can write ∇ s k = n (cid:88) t =1 (cid:32) f (cid:32) n (cid:88) l =1 ρ − tl ( x ) d ( ρ lk ( x )) (cid:33)(cid:33) s t . By comparing this expression with the local form, we conclude that(6.2) ω = f ρ − dρ. Our next step is to obtain the local form of the curvature. For this we use thestructural equation (see [Morita] Theorem 5.21.) Ω i = dω i + ω i ∧ ω i . Consider the equality ρ − ρ = I . An application of the product rule allow us towrite: dI = d (cid:0) ρ − (cid:1) ρ + ρ − dρ. This in turn implies that d (cid:0) ρ − (cid:1) ρ = − ρ − dρ ⇒ d (cid:0) ρ − (cid:1) = − ρ − ( dρ ) ρ − . Since dd = 0 , we obtain d (cid:0) ρ − dρ (cid:1) = d (cid:0) ρ − (cid:1) ∧ d ( ρ ) , which allow us to concludethat dω = (cid:0) ( df ) ρ − dρ − f ρ − dρ ∧ ρ − dρ (cid:1) . On the other hand ω ∧ ω = (cid:0) f ρ − dρ (cid:1) ∧ (cid:0) f ρ − dρ (cid:1) = f ρ − dρ ∧ ρ − dρ, which finally gives us(6.3) Ω = (cid:0) ( df ) ρ − dρ − f ρ − dρ ∧ ρ − dρ (cid:1) + f ρ − dρ ∧ ρ − dρ. Observe that in a point x / ∈ U ∩ U , Ω is zero since the closure of the support of f is contained in U . Similarly, an analogue formula can be deduce for the localform of the curvature in U , and deduced that it is also zero outside U ∩ U . Thus,we can conclude that HARACTERISTIC CLASSES FOR TC STRUCTURES 31
Proposition 34.
Let π : E → M be a smooth vector bundle with { U , U } an opencover of M , both having trivializations of E , ϕ and ϕ , respectively. Let { f , f } be a partition of unity associated to { U , U } , respectively. If ρ is the transitionfunction associated to ϕ ◦ ϕ − , then the curvature Ω k of the k -th associated bundleis given by (Ω k ) x = (cid:40)(cid:0) Ω k (cid:1) x x ∈ U ∩ U . x / ∈ U ∩ U . Where (6.4) Ω k = ( df ) ρ − k d (cid:0) ρ k (cid:1) + (cid:0) f − f (cid:1) ρ − k d (cid:0) ρ k (cid:1) ∧ ρ − k d (cid:0) ρ k (cid:1) is the local expression on U .Proof. Since the k -th associated vector bundle has the same cover associated to itsTC structure, with transition functions equal to ρ k , the previous discussion providesa proof of the theorem. (cid:3) It is worth mentioning that it is possible to deduce a similar formula to (6.3) fora arbitrary number of sets in an open cover, but we will not need this.6.2.
Second Chern class for clutching functions with values on SU (2) . Suppose that we have a vector bundle p : E → M in such a way that we can find anopen cover { U , U } of M together with a transition function ρ : U ∩ U → SU (2) .First, we are going to compute the determinant of the curvature form in terms ofthe components of the matrices in SU (2) , SU (2) := (cid:26)(cid:20) z − ¯ ww ¯ z (cid:21) | | z | + | w | = 1 (cid:27) . So let us take ρ = (cid:20) z − ¯ ww ¯ z (cid:21) , for which we want to compute the curvature Ω = ( df ) ρ − d ( ρ ) + (cid:0) f − f (cid:1) ρ − d ( ρ ) ∧ ρ − d ( ρ ) . Since z ¯ z + w ¯ w = 1 , we get by differentiating that zdz + ¯ wdw ) + ( zd ¯ z + wd ¯ w ) ⇒ zd ¯ z + wd ¯ w = − (¯ zdz + ¯ wdw ) and so we have τ := ρ − dρ = (cid:20) ¯ zdz + ¯ wdw ¯ wd ¯ z − ¯ zd ¯ w − wdz + zdw − (¯ zdz + ¯ wdw ) (cid:21) . Now take θ := ρ − dρ ∧ ρ − dρ . Using that τ = − τ and | z | + | w | = 1 we getthat θ = (cid:20) ( ¯ wd ¯ z − ¯ zd ¯ w ) ∧ ( − wdz + zdw ) 2 d ¯ z ∧ d ¯ w − dz ∧ dw − ( ¯ wd ¯ z − ¯ zd ¯ w ) ∧ ( − wdz + zdw ) (cid:21) . which is the same as expressing it as θ = (cid:20) τ ∧ τ θ θ − τ ∧ τ (cid:21) . Now, by making f := f and g := ( f − f we may express the curvature as Ω = df τ + gθ = (cid:20) df τ + gτ ∧ τ df τ + gθ df τ + gθ − ( df τ + gτ ∧ τ ) (cid:21) , HARACTERISTIC CLASSES FOR TC STRUCTURES 32 and its determinant is then given by det (Ω) = − ( df τ + gτ ∧ τ ) ∧ ( df τ + gτ ∧ τ ) − ( df τ + gθ ) ∧ ( df τ + gθ ) . In order to reduce this expression, we recall that the wedge product of a oneform with itself is zero. Also, one forms commute with two forms, so we get: det (Ω) = − g θ ∧ θ − gdf ∧ ( τ ∧ θ + τ ∧ θ + 2 τ ∧ τ ∧ τ ) . By recalling that τ ∧ τ = d ¯ z ∧ d ¯ w we get: τ ∧ τ ∧ τ = − ( wdzd ¯ zd ¯ w + zd ¯ zdwd ¯ w ) ,τ ∧ θ = 2 (¯ zdzdwd ¯ w + ¯ wdzd ¯ zdw ) and τ ∧ θ = − zd ¯ zdwd ¯ w + wdzd ¯ zd ¯ w ) . Now take A := τ ∧ θ + τ ∧ θ + 2 τ ∧ τ ∧ τ then A =2 [(¯ zdzdwd ¯ w + ¯ wdzd ¯ zdw ) − ( zd ¯ zdwd ¯ w + wdzd ¯ zd ¯ w ) − ( wdzd ¯ zd ¯ w + zd ¯ zdwd ¯ w )] which gives us A = 2 (¯ zdzdwd ¯ w + ¯ wdzd ¯ zdw − zd ¯ zdwd ¯ w + wdzd ¯ zd ¯ w )) which we can now replace to have(6.5) det (Ω) = 4 ( f − f dzd ¯ zdwd ¯ w − ( f − f df ∧ A. Now we are going to use this formula to find the second Chern class in terms of asmooth Clutching function ϕ : S → SU (2) (see [Hatcher II], Chapter 1). Considerthe sets S = (cid:8) x = ( x , . . . , x ) ∈ R | (cid:107) x (cid:107) = 1 (cid:9) ,D + = (cid:8) ( x , . . . , x ) ∈ S | x ≥ (cid:9) ,D − = (cid:8) ( x , . . . , x ) ∈ S | x ≤ (cid:9) and the open set V = (cid:8) ( x , . . . , x ) ∈ S |− / < x < / } . Also let U := D + ∪ V and U := D − ∪ V and identify S with the equator (cid:8) ( x , . . . , x ) ∈ S | x = 0 (cid:9) .Using "bump" functions we can obtain a partition of the unity f , f : S → [0 , such that they depend only on the "height" x and f i | U i \ V ≡ . Also the clutchingfunction ϕ : S → SU (2) can be composed with a smooth "perpendicular" retrac-tion of V to S , to obtain a transition function ρ : V → SU (2) independent of x .Under this conditions is clear that • df = ∂f ∂r dr , and HARACTERISTIC CLASSES FOR TC STRUCTURES 33 • If ρ = (cid:20) z − ¯ ww ¯ z (cid:21) any four form depending on z , ¯ z , w and ¯ w is zero, since these functionsdepend only on three variables.We are in position to apply the previous results to obtain that det (Ω) = 4 ( f − f dzd ¯ zdwd ¯ w − ( f − f df ∧ A. Where A = 2 (¯ zdzdwd ¯ w + ¯ wdzd ¯ zdw − zd ¯ zdwd ¯ w + wdzd ¯ zd ¯ w )) . However, by con-struction we have that dzd ¯ zdwd ¯ w = 0 and so (cid:90) S det (Ω) = (cid:18)(cid:90) − (cid:18) (1 − f ) f ∂f ∂r (cid:19) dr (cid:19) (cid:90) S A. First, notice that by construction it follows that (cid:90) − (cid:18) (1 − f ) f ∂f ∂r (cid:19) dr = − . Finally since the second Chern class in this case is the determinant of the curvaturetimes (cid:0) i π (cid:1) , we get Proposition 35.
The second Chern class associated to a clutching function ϕ : S → SU (2) is given by c = 124 π (cid:90) S A. Here A is a 3-form given by zdzdwd ¯ w + ¯ wdzd ¯ zdw − zd ¯ zdwd ¯ w + wdzd ¯ zd ¯ w )) and the functions z, w : S → SU (2) are determined by the clutching function, ϕ = (cid:20) z − ¯ ww ¯ z (cid:21) . A non trivial TC structure over a trivial vector bundle.
It is alreadyknown that there are trivial vector bundles with non trivial TC structures overthem. In this section we are going to use such a structure to show that:
Theorem 36.
There exists a TC structure ξ = (cid:8) E → S , { U , U , U } , ρ ij : U i ∩ U j → SU (2) (cid:9) such that E → S is a trivial bundle, and such that c − ( ξ ) = − , implying that theTC structure is non trivial. This in particular highlights how the TC characteristic classes depends on theTC structure and not on the equivalence class of their underlying bundle.Now, to prove this theorem we are based on the construction made by D. Ramrasand B. Villareal ([RV], Chapter 3). In what follows, we first define the vector bundleby defining an open cover on S and transition functions on them. This defines aTC structure ξ = (cid:8) E → S , { U , U , U } , ρ ij : U i ∩ U j → SU (2) (cid:9) . Then by considering the ( − -powers of these transition functions we also obtainthe ( − -th associated bundle, E − . HARACTERISTIC CLASSES FOR TC STRUCTURES 34
Figure 6.1.
Retraction r : D − → D Next we are going to use Lemma 3.1 of [RV] to show that both vector bundlesobtained can be described, up to isomorphism, by a given clutching functions.Then on one hand by showing that the clutching function associated to E is trivial,we conclude that E is trivial. On the other hand, we use the clutching functionassociated to E − together with the formulas of the previous sections, to concludethat c − ( ξ ) = − .We outline how their initial construction can be made in the smooth category,which allows us to reduce the problem of computing the Chern class by usingClutching functions.We are constructing a TC structure on a vector bundle defined over S in termsof a triple open cover { U , U , U } and transition functions between them. Thesetransition functions themselves will be described in terms of two functions ρ , ρ : D → SU (2) , where D is the 3-dimensional closed disk of radius 1.For this, take S = (cid:8) x = ( x , . . . , x ) ∈ R | (cid:107) x (cid:107) = 1 (cid:9) and for / > (cid:15) > consider the triple open cover U := (cid:8) ( x , . . . , x ) ∈ S | x > − (cid:15) (cid:9) ,U := (cid:8) ( x , . . . , x ) ∈ S | x < , x > − (cid:15) (cid:9) and U := (cid:8) ( x , . . . , x ) ∈ S | x < , x < (cid:15) (cid:9) . Also call D − = (cid:8) ( x , . . . , x ) ∈ S | x ≤ (cid:9) and identify the closed 3-dimensionaldisk with D = (cid:8) ( x , . . . , x ) ∈ S | x ≤ , x = 0 (cid:9) . There is a natural retraction r : D − → D leaving D fixed (See Figure 6.1). Thisis a smooth function almost everywhere.Take V = D ∩ U . Then we get that V = { ( x , . . . , x ) ∈ D | x > − / } . Now suppose that the functions ρ , ρ : D → SU (2) are smooth functions suchthat: • They are independent of the radius in D in V . • They are commutative in the closure of V .We define the transition functions ρ ij : U i ∩ U j → SU (2) by • ρ := ρ ◦ r. HARACTERISTIC CLASSES FOR TC STRUCTURES 35 • ρ := ρ ◦ r. • ρ := ( ρ ◦ r ) ( ρ ◦ r ) . Since r ( U ∩ U ∩ U ) ⊆ V by construction, the previous cocycles commute witheach other in their common domain U ∩ U ∩ U . This transition functions allowus to construct a smooth vector bundle E → S , and so we have constructed a TCstructure ξ = (cid:8) E → S , { U , U , U } , ρ ij : U i ∩ U j → SU (2) (cid:9) . Associated clutching functions:
Before dealing with the result we need, itis important to highlight the following. Suppose E → M and E → M are smoothvector bundles with classifying functions f i : M → BSU ( n ) , i = 1 , . If thereis a (non necessarily continous) homotopy between f and f , and there is class c ∈ H ∗ ( BSU ( n )) , it follows that f ∗ ( c ) = f ∗ ( c ) ∈ H ∗ ( M ) . Now consider the cur-vatures Ω and Ω for E and E , respectively. By the Chern-Weil isomorphism, weget that c (Ω ) = f ∗ ( c ) and c (Ω ) = f ∗ ( c ) , and thus c (Ω ) = c (Ω ) . In particularif there is a continuous (but not smooth) isomorphism of vector bundles between E and E , their classifying functions will be homotopic and their characteristicclasses will coincide.Now consider the closed sets C := (cid:8) ( x , . . . , x ) ∈ S | x ≥ (cid:9) ,C := (cid:8) ( x , . . . , x ) ∈ S | x ≤ , x ≥ (cid:9) and C := (cid:8) ( x , . . . , x ) ∈ S | x ≤ , x ≤ (cid:9) . It is clear that there is a retraction r i : U i → C i leaving C i fixed, for i = 1 , , .Notice that by applying on U ∩ U r first and then r , we obtain a retraction r : U ∩ U → C ∩ C leaving C ∩ C fixed. For U ∩ U we apply first r andthen r , we obtain a retraction r : U ∩ U → C ∩ C leaving C ∩ C fixed,and similarly we obtain r : U ∩ U → C ∩ C leaving C ∩ C fixed. Via thisrestrictions of ρ ij we obtain transition functions for the closed cover { C , C , C } : ˜ ρ ij : C i ∩ C j → SU (2) . This new transition functions are clearly homotopic to ρ ij via the retractions r ij .Thus, they characterized vector bundles over S whose classifying functions arehomotopic.Consider the identification S ∼ = (cid:8) ( x , . . . , x ) ∈ S | x = 0 (cid:9) . This setting allowus to apply Lemma 3.1 of [RV]. There they show that the bundle induced bythese three cocycles is isomorphic to the vector bundle with clutching function ϕ : S → SU (2) defined for x = ( x , . . . , x ) by ϕ ( x ) := (cid:40) ρ ( r ( x )) ρ ( r ( x )) x ≥ .ρ ( r ( x )) ρ ( r ( x )) x ≤ . The function ϕ can clearly be extended continuously to the whole disk D − , sincewe defined r on D − . This implies that ϕ is null homotopic, and thus, the vectorbundle given by these cocycles is trivial.Now lets consider the same construction but using the cocycles given by σ ij = ρ − ij . They give rise to the ( − -th associated bundle by definition. Once again HARACTERISTIC CLASSES FOR TC STRUCTURES 36 allow us to use Lemma 3.1 of [RV]. We conclude that this bundle can be obtain,up to isomorphims, by the clutching function given by φ ( y ) := (cid:40) ρ − ( r ( x )) ρ − ( r ( x )) x ≥ .ρ − ( r ( x )) ρ − ( r ( x )) x ≤ . In this case this function cannot be extended continuously to D − if ρ and ρ donot commute everywhere in D . So φ is not necessarily null homotopic.6.5. Existence of a non trivial TC structure:
From the previous part, weneed to show that it is possible to obtain a non null homotopic clutching function φ . For this it is enought to display two functions ρ , ρ : D → SU (2) such thatthey commute in ∂D ∼ = S , giving us a non zero Chern class for the bundle withclutching function φ : S → SU (2) .We can describe φ in terms of the northern and southern hemispheres of S , D + and D − , respectively. Each of them can be identify with the 3-dimensional disc D . Then we get that φ ( y ) := (cid:40) ρ − ρ − in D + ,ρ − ρ − in D − . For brevity allow us to write the matrices of SU (2) as ( a, b ) := (cid:20) a − ¯ bb ¯ a (cid:21) . Proposition 37.