Complexifiable characteristic classes

Journal of Homotopy and R elate d Structur es, vol. 1(1), 2013, pp.1–11 COMPLEXIFIABLE CHARA CTERISTIC CLASSES ALEXA NDER D. RAHM ( c ommunic ate d by James D. Stasheff ) Abstr act W e examine the topo logical c haracter is tic cohomolo gy cla sses o f complexified vector bundles. In pa rticular, all the class es coming from the real v ector bundles underlying the complexification are deter- mined. This article is dedicated to Mark F. F esh bach (1950- 2010), for his v aluable work on cohomology rings of classifying spaces. 1. In t ro duction and stat ement of the r esults In the theory of character is tic classes (in the sense of Milno r and Stasheff [ 4 ], whom w e follow in terminology and notation in this article), it is well-known how the Che r n cla sses are mapped to even S tiefel-Whitney classes when conv er ting complex vector spa c e bundles to rea l vector space bundles by forgetting the complex str ucture. In the other dire c tion, w e hav e the fibre-wise complexifica tion: Given a real v ector bundle F → B with fibre R n , its complexification is the complex v ector bundle F C := F ⊗ R C → B obtained b y declaring complex multiplication on F ⊕ F in each fibre R n ⊕ R n by i ( x, y ) := ( − y , x ) for the imagina ry unit i . The Pontrjagin classe s o f a rea l v ector bundle a re (up to a sign) co ns tructed as Chern c lasses of its complexific a tion. Conv ersely , which cla sses of a re a l vector bundle can be attributed to its complex ification? These are the c omplexifiable characteristic classes which w e determine in this a rticle, under the request that they ar e characteristic classes in the sense of [ 4 ]. Consider a r eal vector bundle F → B and a complex vector bundle E → B ov er the same paracompa ct Hausdorff base spa ce B (we k eep the latter a ssumption on B throughout this article). Definition 1. A real vector bundle F is called a r e al gener ator bund le of E , if its com- plexification F C is isomorphic to E . In the case that such a bundle F exists, we call E r e al-gener ate d . Not every complex vector bundle is real-gener ated; a s the o dd degr ee Chern classes have the prop erty c 2 k +1 ( E ) = − c 2 k +1 ( E ) on the complex co njugate bundle E , it is an eas y exer- cise to show that no complex vector bundle with some nonzer o and non-2 -torsio n o dd Cher n class ca n admit a r eal gener ator bundle. This ma kes it see m pos sible that the sub catego ry F unded b y the Irish Research Council for Science, Engineering and T echn ology . Receiv ed July 14, 2012, revised Octob er 11, 2018; publi shed on Mon th Da y , Y ear. 2000 M athematics Sub ject Classification: 55R40, Homol ogy of classifying spaces, c haracteristic classes. Key words and phrases: Characte ristic classes, Classifying spaces of groups and H -s paces, Stable classes of v ector space bundles. c  2013, Alexander D. Rahm. Permission to cop y for priv ate use gran ted. Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 2 of real- generated vector bundles could admit informa tio n a dditional to its Chern clas s es, in terms of co mplexifiable classes o f the re a l gener a tor bundles. How ever, we will see that the Chern class es alre ady contain a ll of the r elev ant information. Definition 2. A characteristic class c o f real vector bundles is c omplexifiable if for all pairs ( F , G ) o f real v ector bundles with isomorphic complexification F C ∼ = G C , the identit y c ( F ) = c ( G ) holds. W e will no w g ive a complete class ification of the complexifiable characteristic class es. Denote by Z 2 := Z / 2 Z the gr oup with tw o elements. Theorem 1. L et c b e a p olynomial in the St iefel-Whitney classes w i , i ∈ N ∪ { 0 } . Then the fol lowing two c onditions ar e e quivalent: (i) The class c is an element of the sub-ring Z 2 [ w 2 i ] i ∈ N ∪{ 0 } of the p olynomials in the Stiefel-Whitney classes. (ii) The class c is c omplexifiable. The implicatio n (i) ⇒ (ii) fo llows eas ily from the fact that the squar e of the n -th Stiefel- Whitney clas s of a real vector bundle is the mo d-2 - reduction of the n -th Cher n class of the complexified vector bundle. The pro of of the implication (ii) ⇒ (i) is prepared with several int ermediary steps lea ding to it. One ingr edient , Lemma 1, follows essentially fro m work of Cartan on fibrations of H-spa ces (at Cartan’s time called Hopf spaces). But this only allows us to show that complexifia ble character istic class es in c o homology with Z 2 –co efficients are contained in the ide al genera ted by the s quares of the Stiefel-Whitney classes . T o show that they constitute exactly the subring g enerated b y the sq uares of the Stiefel-Whitney classes, which is muc h smaller, we need the technical decomp osition of Lemma 2 that we prove b y induction. By their natura lity , characteristic clas s es are uniquely determined on the universal bundle ov er the cla ssifying space ( B O for real vector bundles). As the co homology ring H ∗ ( B O , Z 2 ) is gener ated by the Stiefel-Whitney classes o f the universal bundle, all mo dulo 2 character- istic classes are polynomia ls in the Stiefel-Whitney cla sses, and Theorem 1 tells us which of them ar e complexifiable. W e build on this result to inv estigate which integral cohomo logy classes are complex- ifiable. T o expr ess our re s ult, we use F eshbac h’s descr iption [ 3 ] o f the cohomolog y ring of the clas sifying space B O with Z –co efficients. Generators for this ring are known since Thomas [ 5 ], [ 6 ], and all the relations be tw een its genera to rs are known since Br own [ 1 ] and F eshb ach [ 3 ]. Consider the Steenr o d sq uaring op era tion S q 1 and the mo d–2 –reduction homomorphism ρ : H ∗ ( B O , Z ) → H ∗ ( B O , Z 2 ) . As generator s for H ∗ ( B O , Z ), F eshbac h uses P ontrjagin classes and cla sses V I with index sets I that a re finite nonempty subsets of  1 2  ∪ N , admitting mo d–2 –reductions ρ ( V I ) = S q 1 [ i ∈ I ω 2 i ! , where ω i is the i -th Stiefel Whitney cla ss of the univ ersal bundle ov er B O . In particular , we hav e a generator V { 1 2 } . W e give the details of F eshbac h’s descriptio n in the app endix. Our final res ult now takes the following shape . Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 3 Theorem 2. L et C b e a p olynomial in V 2 I , with I arbitr ary, V { 1 2 } and the Pontrjagin classes. Then C is c omplexifiable. And conv er sely , we can say the following. Theorem 3. L et C b e a c omplexifiable inte gr al char acteristic class. Then for any r e al ve ctor bund le ξ , C ( ξ ) is c ompletely determine d by some Chern classes c k ( ξ C ) , k ∈ N . 2. Classes in cohomology with Z 2 –co efficien t s In this section, w e shall prov e Theorem 1, after developing all the to ols we need to do so. F or this ent ire se c tio n, we only c o nsider co homology with Z 2 –co efficients. W e write N for the natural num b ers without 0 . Let F → B b e a real vector bundle over a pa racompac t Hausdor ff base space . Let c b e a complexifiable p oly nomial in the Stiefel-Whitney classes w i . Let O b e the direct limit of the or thogonal groups , U the direct limit of the unitary g r oups and E U the universal total space to the clas sifying spa ce B U for stable complex v ec to r bundles. Let B O := E U / O , via the inclusion O ⊂ U induced b y the canonical inclusion R ⊂ C . Le t γ ( R ∞ ) be the univ ersal bundle ov e r B O , and denote its Stiefel-Whitney classes b y ω i := w i ( γ ( R ∞ )). Let ε be the trivial vector bundle. Lemma 1. Le t c b e a c omplexifiable class in c ohomolo gy with Z 2 –c o efficients. Then c ( γ ( R ∞ )) − c ( ε ) is c ontaine d in the ide al h ω 2 i i i ∈ N . Pr o of. W e use Cartan’s fibra tion o f H-spa ces [ 2 , p. 17-22] (fibration en espaces de Hopf ), U / O f / / B O p / / / / B U . The co homology ring H ∗ ( B O , Z 2 ) is the p olynomia l algebra Z 2 [ ω 1 , ω 2 , ... ] with genera tors the Stiefel-Whitney classes of the universal bundle. Car tan [ 2 , p. 17-22 ] has shown that f ∗ maps these genera tors ω i to the gener ators ν i := w i ( f ∗ γ ( R ∞ )) of the exterio r alg ebra H ∗ ( U / O , Z 2 ) = ^ ( Z 2 [ ν 1 , ν 2 , ... ]) , which is obtained b y dividing out the ideal h ν 2 i i i ∈ N of the poly nomial algebr a Z 2 [ ν 1 , ν 2 , ... ]. Hence, exactly the ideal h ω 2 i i i ∈ N is mapp ed to zero . So, h ω 2 i i i ∈ N = ker f ∗ . Comp osing f with the pr o jection p : B O → B U , w e o btain a co nstant map a nd there- fore a trivial bundle ( p ◦ f ) ∗ γ ( C ∞ ). This pullback of the complex universal bundle is the complexification of f ∗ γ ( R ∞ ): ( p ◦ f ) ∗ γ ( C ∞ ) = f ∗ p ∗ E U × U C ∞ = f ∗ E O × O C ∞ = f ∗ ( E O × O R ∞ ) C = f ∗ γ ( R ∞ ) C = ( f ∗ γ ( R ∞ )) C . So, f ∗ γ ( R ∞ ) admits a trivial co mplex ification, and all of the co mplexifiable classes c must treat it like the tr iv ial bundle ε : c ( f ∗ γ ( R ∞ )) = c ( ε ). A pullback of the trivial bundle is trivial to o, s o 0 = c ( f ∗ γ ( R ∞ )) − c ( f ∗ ε ) = f ∗ ( c ( γ ( R ∞ )) − c ( ε )) Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 4 by naturality . Whence, c ( γ ( R ∞ )) − c ( ε ) is a n element of the kernel of f ∗ , whic h we have ident ified with the ideal h ω 2 i i i ∈ N . The ab ov e lemma allows us to write the characteristic class c under investigation as a sum ov e r pro ducts with squa res o f Stiefel-Whitney clas ses, c ( γ ( R ∞ )) − c ( ε ) = m X j =1 ω 2 i j ∪ r j ( γ ( R ∞ )) , with r j some p olynomials in the Stiefel-Whitney classes. W e must inductively identif y squares o f Stiefel-Whitney classe s as factor s of the r e mainders r j , until we achieve the decomp osition claimed in the following lemma. Notation. F o r indices j 1 , ..., j s ∈ N and i j 1 , ..., i ( j 1 ,...,j s ) ∈ N , w e shall write ~ j s := ( j 1 , ..., j s ) and I ( ~ j s ) := { i ~ j 1 , ..., i ~ j s } . W e set ~ j 0 := 0. Note that the c la sses c ( ε ) , r ~ j ( ε ) of the tr ivial bundle ε that we are going to us e now, a re just co efficients in H 0 ( B O , Z 2 ) ∼ = { 0 , 1 } . Lemma 2. Any c omplexifiable char acteristic class c admits a de c omp osition c ( γ ( R ∞ )) − c ( ε ) = m ~ j k − 1 P j k =1 ω 2 i ~ j k r ~ j k ( γ ( R ∞ )) ! ∪ k − 1 S n =1 m ~ j n − 1 P j n =1 ω 2 i ~ j n ! + k − 1 P s =1 s S n =1 m ~ j n − 1 P j n =1 ω 2 i ~ j n r ~ j n ( ε ) for some k , m ~ j 0 , ..., m ~ j k − 1 ∈ N ∪ { 0 } , some i ~ j 1 , ..., i ~ j k ∈ N , some r ~ j k ( γ ( R ∞ )) ∈ H ∗ ( B O , Z 2 ) , and s ome c o efficients r ~ j 1 ( ε ) , ..., r ~ j k − 1 ( ε ) ∈ { 0 , 1 } , such that the fol lowing ine quality holds: 2 P p ∈ I ( ~ j k ) p > deg c. Remark A. Once that this lemma is establis hed, we use that the degr ee must b e the same on b oth sides in order to deduce that the sum over all terms containing a factor S p ∈ I ( ~ j k ) ω 2 p exceeding the degree o f c via the requested inequality must already b e ze r o. So in fact, the decomp osition is of the form c ( γ ( R ∞ )) − c ( ε ) = k − 1 X s =1 s [ n =1 m ~ j n − 1 X j n =1 ω 2 i ~ j n ∪ r ~ j n ( ε ) , meaning that c ( γ ( R ∞ )) is a polyno mial in so me squares ω 2 p , p ∈ N ∪ { 0 } , which implies Theorem 1, (ii) ⇒ (i). Before g iving the pro of of Lemma 2, we shall introduce tw o notations just to make that pro of more reada ble. Definition 3. An index vector ~ j app e ars in a given decomp ositio n of c ( γ ( R ∞ )) − c ( ε ) if b oth 2 P p ∈ I ( ~ j ) p ! 6 deg c Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 5 and this decomp os ition a dmits a summand of the form r ~ j ( γ ( R ∞ )) ∪ S p ∈ I ( ~ j ) ω 2 p . Note that the terms r ~ j ( γ ( R ∞ )) ∪ S p ∈ I ( ~ j ) ω 2 p ! with 2 P p ∈ I ( ~ j ) p > deg c ! m ust v anish in any decompos itio n of c ( γ ( R ∞ )) − c ( ε ). That is why we do not let them contribute in the last definition. Definition 4. Set ℓ := min ~ j app e ars max I ( ~ j ) . Consider an index vector ~ j appea ring in a given decomp osition of c ( γ ( R ∞ )) − c ( ε ). If max I ( ~ j ) = ℓ , then we call r ~ j ( γ ( R ∞ )) − r ~ j ( ε ) a lower de gr e e r emainder . As seen in Lemma 1, c ( γ ( R ∞ )) − c ( ε ) lies in k er f ∗ = h ω 2 i i i ∈ N , so there is a deco mpo s ition c ( γ ( R ∞ )) − c ( ε ) = m X j 1 =1 ω 2 i ~ j 1 ∪ r ~ j 1 ( γ ( R ∞ )) , for some m ∈ N ∪ { 0 } , some i ~ j 1 ∈ N , and some r ~ j 1 ( γ ( R ∞ )) ∈ H ∗ ( B O , Z 2 ). W e will sho w that ther e is a lo wer degr e e remainder r ~ j 1 ( γ ( R ∞ )) − r ~ j 1 ( ε ) in this decompos ition that lies in ker f ∗ . Then, that lower degree rema inder admits a decomp osition as a linear co m bination of squares ω 2 i ~ j 2 with coe fficient s r ~ j 2 ( γ ( R ∞ )) in H ∗ ( B O , Z 2 ), leading to a new decomp osition of c ( γ ( R ∞ )) − c ( ε ). So, inductively , we will r e place a lo wer degree remainder in any given decomp osition of c ( γ ( R ∞ )) − c ( ε ) by a linear combination the co efficients of which are remainders with longer index vectors. That is wh y after a finite num ber of these steps, the index vectors ~ j will no longer app ear, b eca us e the sums 2 P p ∈ I ( ~ j ) p ! will exceed the degree o f c . This is the moment when all lower degr ee remainders ar e eliminated a nd the decomp osition describ ed in Lemma 2 is achiev ed. T o carr y out this strateg y , we first need to intro duce the following truncation pro cedure. T runca ted stable in v ariance With Lemma 3, we shall giv e a sense to “the truncatio n o f the equatio n c ( F ⊕ G ) = c ( G ) at the dimension ℓ ”. Define the bundles F := pr ∗ 1 f ∗ γ ( R ∞ ) − → U / O × B O and G := pr ∗ 2 γ ( R ∞ ) − → U / O × B O , where pr i is the pro jection on the i -th factor of the bas e s pace U / O × B O . Let ℓ ∈ N . Consider the map ( id, emb l ) : ( U / O × B O ℓ ) ֒ → ( U / O × B O ) where emb l : B O ℓ ֒ → B O is the natural embedding into the direct limit. Then the bundle G l := ( id, emb l ) ∗ G admits Stiefel-Whitney class es that are in bijectiv e co rresp ondence with those of the ℓ -dimensio nal universal bundle γ l ( R ∞ ) → B O ℓ . Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 6 T o b e precise, G l ∼ = pr B O ℓ ∗ γ l ( R ∞ ) and the situation is γ l ( R ∞ )   G l ∼ = pr B O ℓ ∗ γ l ( R ∞ )   G := pr ∗ 2 γ ( R ∞ )   γ ( R ∞ )   B O ℓ ( U / O × B O ℓ ) pr B O ℓ o o   ( id,emb l ) / / ( U / O × B O ) pr 2 / / B O . Esp ecially , w p ( G l ) v anishes for p > ℓ . Compare the latter s ta temen ts with [ 4 ]. Lemma 3. U nder the ab ove assumptions, the fol lowing e quation holds: max I ( ~ j ) 6 ℓ X ~ j appears r ~ j ( F ⊕ G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) = max I ( ~ j ) 6 ℓ X ~ j appears r ~ j ( G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) . We wil l c al l it the equation c ( F ⊕ G ) = c ( G ) truncated at dimens io n ℓ . Pr o of. The bundle F inherits from f ∗ γ ( R ∞ ) the pro per ty of admitting a triv ial complexifi- cation. As c is co mplexifiable, we have c ( F ⊕ G ) = c ( G ) . Applying the induced cohomolo gy map ( id, emb l ) ∗ to this equation, we obtain c ( id ∗ F ⊕ emb ∗ l G ) = c ( emb ∗ l G ) and hence c ( F ⊕ G l ) = c ( G l ) . By the universality o f γ ( R ∞ ), and the naturality o f all c haracter is tic classes with resp ect to the class ifying maps o f G l and F ⊕ G l , any given decomp os ition c ( γ ( R ∞ )) − c ( ε ) = X ~ j r ~ j ( γ ( R ∞ )) [ p ∈ I ( ~ j ) ω 2 p gives analo gous decomp ositions c ( G l ) − c ( ε ) = X ~ j r ~ j ( G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) and c ( F ⊕ G l ) − c ( ε ) = X ~ j r ~ j ( F ⊕ G l ) [ p ∈ I ( ~ j ) w 2 p ( F ⊕ G l ) . By Theorem 1, (i) ⇒ (ii) the square w 2 p is complexifiable and hence inv ariant under adding the bundle F of tr ivial complexification : w 2 p ( F ⊕ G l ) = w 2 p ( G l ) . Thu s, the equation c ( F ⊕ G l ) = c ( G l ) can be rew r itten using that all s ummands containing a factor w p ( G l ) with p > ℓ v a nish: max I ( ~ j ) 6 ℓ X ~ j r ~ j ( F ⊕ G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) = max I ( ~ j ) 6 ℓ X ~ j r ~ j ( G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) . Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 7 In order not to exceed the deg ree o f c , also a ll ter ms with 2 P p ∈ I ( ~ j ) p > deg c must v anish: max I ( ~ j ) 6 ℓ X ~ j appears r ~ j ( F ⊕ G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) = max I ( ~ j ) 6 ℓ X ~ j appears r ~ j ( G l ) [ p ∈ I ( ~ j ) w 2 p ( G l ) . So, this las t equatio n is the eq ua tion c ( F ⊕ G ) = c ( G ) tr uncated at the dimension ℓ . Pr o of of L emma 2. W e car r y out the pr o of b y induction over the index ve ctor identifying a lower de gr e e r emainder . Base case . Lemma 1 implies c ( γ ( R ∞ )) − c ( ε ) = m P j 1 =1 ω 2 i ~ j 1 ∪ r ~ j 1 ( γ ( R ∞ )), with r ~ j 1 some p o lynomials in the Stiefel-Whitney c la sses. Rename i 1 , ..., i m such that i 1 < i 2 < ... < i m . W e truncate the equation c ( F ⊕ G ) = c ( G ) a t the dimensio n i 1 , and obtain i j 1 6 i 1 X ~ j 1 app ears r ~ j 1 ( F ⊕ G i 1 ) ∪ w 2 i ~ j 1 ( G i 1 ) = i j 1 6 i 1 X ~ j 1 app ears r ~ j 1 ( G i 1 ) ∪ w 2 i ~ j 1 ( G i 1 ) . As i 1 < i 2 < ... < i m , this is just r 1 ( F ⊕ G i 1 ) ∪ w 2 i 1 ( G i 1 ) = r 1 ( G i 1 ) ∪ w 2 i 1 ( G i 1 ). Injectivit y of the multiplication map ∪ w 2 i 1 ( G i 1 ) in H ∗ ( U / O × B O i 1 , Z 2 ) then holds r 1 ( F ⊕ G i 1 ) = r 1 ( G i 1 ). Then we pull this back with ( id × const ) : U / O → ( U / O × B O i 1 ) , (where the map const ta kes just one, a rbitrary , v alue), to obtain r 1 ( f ∗ γ ( R ∞ ) ⊕ ε ) = r 1 ( ε ) . Due to the Whitney sum formula, the Stiefel-Whitney cla sses in whic h r 1 is a p o lynomial are stable under adding a trivial bundle; and the ab ov e left ha nd term eq uals r 1 ( f ∗ γ ( R ∞ )) . Using naturality of characteristic classes with resp ect to pullbacks, this shows that r 1 ( γ ( R ∞ )) − r 1 ( ε ) lies in ker f ∗ . So w e can replace it with a linear (o ver the field with 2 elements) combination of strictly quadratic ter ms , providing a new decomp os itio n, c ( γ ( R ∞ )) − c ( ε ) = ω 2 i 1 m 1 X j 2 =1 ω 2 i (1 ,j 2 ) r (1 ,j 1 ) ( γ ( R ∞ )) + ω 2 i 1 r 1 ( ε ) + m X j 1 =2 ω 2 i j 1 r j 1 ( γ ( R ∞ )) . Induction hypothes is . Consider a given deco mpo sition c ( γ ( R ∞ )) − c ( ε ) =   X ~ j k r ~ j k ( γ ( R ∞ )) [ p ∈ I ( ~ j k ) ω 2 p   + k − 1 X s =1 s [ n =1 m ~ j n − 1 X j n =1 ω 2 i ~ j n ∪ r ~ j n ( ε ) . Inductiv e claim . The deco mpo sition of the induction h y p o thesis admits a low er degree remainder that lies in ker f ∗ . W e show this in the inductive s tep. Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 8 Inductiv e step . W e tr uncate the equatio n c ( F ⊕ G ) = c ( G ) at the dimension ℓ := min ~ j app e ars max I ( ~ j ) . Then the rema ining terms of c ( G l ) − c ( ε ) do all have the commo n factor w 2 l ( G l ). This is not a zer o divisor in H ∗ ( U / O × B O ℓ , Z 2 ) and furthermor e its m ultiplication ma p ∪ w 2 l ( G l ) is injective. Now, in c ( F ⊕ G l ) = c ( G l ), this injectivity implies max I ( ~ j ) 6 ℓ X ~ j appears r ~ j ( F ⊕ G l ) [ p ∈ I ( ~ j ) \{ ℓ } w 2 p ( G l ) = max I ( ~ j ) 6 ℓ X ~ j appears r ~ j ( G l ) [ p ∈ I ( ~ j ) \{ ℓ } w 2 p ( G l ) . ♦ If there is just o ne low er degree rema inder r ~ j ( γ ( R ∞ )) − r ~ j ( ε ), then we use the injectivity of the multiplication ma p ∪ S p ∈ I ( ~ j ) \{ ℓ } w 2 p ( G l ) ! on H ∗ ( U / O × B O ℓ , Z 2 ) to obtain r ~ j ( F ⊕ G l ) = r ~ j ( G l ). Then we pull this back with ( id × const ) : U / O → ( U / O × B O ℓ ) to obtain r ~ j ( f ∗ γ ( R ∞ ) ⊕ ε ) = r ~ j ( ε ). Using natura lit y , we see no w that the low er degree remainder r ~ j ( γ ( R ∞ )) − r ~ j ( ε ) lies in ker f ∗ . ♦ Otherwise, we truncate the remaining equa tion a gain at the dimension ℓ ′ := max I ( ~ j )= ℓ min ~ j app e ars max( I ( ~ j ) \ { ℓ } ) , so as to obtain max( I ( ~ j ) \{ ℓ } ) 6 ℓ ′ X ~ j appears r ~ j ( F ⊕ G ℓ ′ ) [ p ∈ ( I ( ~ j ) \{ ℓ } ) w 2 p ( G ℓ ′ ) = max( I ( ~ j ) \{ ℓ } ) 6 ℓ ′ X ~ j app e ars r ~ j ( G ℓ ′ ) [ p ∈ ( I ( ~ j ) \{ ℓ } ) w 2 p ( G ℓ ′ ) . Now we pro ceed ana lo gously with the choice marked with the “ ♦ ” s igns and, after finitely many steps, find a low er degree r emainder in k er f ∗ . This low er degree remainder ca n be replaced by a linear combination of squa r es, holding a new decomp osition of c ( γ ( R ∞ )) − c ( ε ). This completes the induction. ✷ Pr o of of The or em 1, (ii) ⇒ (i). Let c b e a co mplexifiable characteristic class. By Remar k A and the universality of γ ( R ∞ ), the decomp osition of Lemma 2 yields the de c o mpo sition c = c ( ε ) + k − 1 X s =1 s [ n =1 m ~ j n − 1 X j n =1 w 2 i ~ j n ∪ r ~ j n ( ε ) . Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 9 As c ( ε ) , r ~ j 1 ( ε ) , ..., r ~ j k − 1 ( ε ) a r e elements o f { 0 , 1 = w 0 = w 2 0 } , the cla ss c is in the sub-ring Z 2 [ w 2 i ] i ∈ N ∪{ 0 } of the p olynomial ring o f Stiefel-Whitney classes. This completes the pro o f of Theor em 1 . 3. Classes in cohomology with integral co efficien ts W e will build on our results obtained fo r Z 2 –co efficients and use the mod– 2–reduction homomorphism ρ : H ∗ ( − , Z ) → H ∗ ( − , Z 2 ) to pr ov e the theorems with Z –c o efficients stated in the in tro duction. Define the element V I ∈ H ∗ ( B O , Z ) as in the appendix, and let v I be the characteristic class tha t is V I on the universal bundle. Lemma 4. F or any r e al bund le ξ , the m o d– 2 –r e duc e d class ρ ( v 2 I ( ξ )) e qu als   X i ∈ I ∩{ 1 2 } w 2 1 ∪ [ j ∈ I \{ i } w 4 j + X i ∈ I \{ 1 2 } ( w 4 i +2 + w 2 ∪ w 4 i ) ∪ [ j ∈ I \{ i } w 4 j   ( ξ ⊕ ξ ) . Pr o of. By F eshbac h’s descr iption (in the a ppendix ), the mo d–2–reductio n is ρ  v 2 I ( ξ )  = S q 1 [ i ∈ I w 2 i ( ξ ) !! 2 . W e expa nd this e xpression unt il it is a po lynomial in the Stiefel-Whitney classes. Then we rearr ange the expression using the Whitney sum fo rmula and the symmetr y of the terms. Pr o of of The or em 2. F o r v { 1 2 } and the Pontrjagin cla sses, the result is obvious. Now let F → B , G → B be r eal bundles with F C ∼ = G C . F orgetting the co mplex structure, this is F ⊕ F ∼ = G ⊕ G . By naturality of the Stiefel-Whitney classes, for any finite nonempt y index set I ⊂ ( { 1 2 } ∪ N ), the p oly no mial given in Lemma 4 is the same for the a rguments ( F ⊕ F ) and ( G ⊕ G ). Applying Lemma 4, this means that ρ ( v 2 I ( F )) = ρ ( v 2 I ( G )). As V 2 I is in the torsion o f H ∗ ( B O , Z ), restr icted on whic h ρ is injective [ 3 ]p. 513, this proves the theorem: v 2 I ( F ) = v 2 I ( G ). Pr o of of The or em 3. F es hb ach [ 3 ]p. 51 3 shows that H ∗ ( B O , Z ) = Z [ π i ] i ∈ N ⊕ { 2– torsion } , where π i is the i -th P ontrjagin clas s of the universal bundle. Then C = P ( p i ) + T with P a p olynomial in the Pon trja g in classes p i and T some 2 -torsio n class. So for e very real bundle ξ , ρ ( C )( ξ ) = P ( ρ ( p i ( ξ ))) + ρ ( T )( ξ ) . Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 10 By definition o f the Pon trjagin classes, p i ( ξ ) = ( − 1) i c 2 i ( ξ C ) ; and using the r eduction ρ  c 2 i ( ξ C )  = w 4 i ( ξ ⊕ ξ ) from Chern cla sses to Stiefel-Whitney class es, further the Whitney sum formula and the symmetry o f the s ummands, we deduce ρ ( C )( ξ ) = P ( w 2 2 i ( ξ )) + ρ ( T )( ξ ) . It follows fro m Theo r em 1 that the mo d-2-re ductio n ρ ( C )( ξ ) is a p olynomia l in the squar es of Stiefel-Whitney cla sses; and hence also ρ ( T )( ξ ) is a poly no mial Q ( w 2 j ( ξ )) in the squa r es of Stiefel-Whitney classe s. As accor ding to [ 3 ]p. 513, ρ is injective on the tor sion elements, there is an inv erse fo r the restricted map ρ | { 2 − torsion } , lifting ρ ( T ) back to T . So, from T ( ξ ) = ρ | { 2 − torsion } − 1  Q ( w 2 j ( ξ ))  , we obtain C ( ξ ) = P  ( − 1) i c 2 i ( ξ C )  + ρ | { 2 − torsion } − 1  Q  ρ ( c j ( ξ C ))  . The a uthor w ould like to thank Gr aham E llis a nd Tho mas Schic k for supp ort and en- courage men t, the latter a lso for p osing the ques tions tre ated in this article and g iving a dvice on them. App endix. The cohomology r ing of B O with Z –co efficien ts The cohomo lo gy r ing of B O with Z –co efficients is known since Thomas [ 5 ], [ 6 ] and with all r elations b etw een its gener ators since Brown [ 1 ] a nd F eshbach [ 3 ]. It can b e derived as follows. Define the set o f gene r ators of H ∗ ( B O n , Z ) as in [ 3 , definitio n 1]: It co ns ists of the Pon tr jagin classes p i of the univ e r sal bundle o ver B O n , a nd classes V I with I ra nging over all finite nonempt y subsets of  1 2  ∪  k ∈ N     0 < k < n + 1 2  with the proviso that I doe s no t contain both 1 2 and n 2 , for n > 1. According to [ 3 , theorem 2], H ∗ ( B O n , Z ) is for all n 6 ∞ isomorphic to the polyno mial ring ov er Z generated b y the above specified elements mo dulo the ideal genera ted by the following six types o f rela tions. In all re lations except the first, the ca rdinality of I is less than or equal to that o f J and g reater than one. On the index sets I and J , we per form s et-theoretic unions ( ∪ ), int ersections ( ∩ ) and differences ( \ ). By co n ven tion, p 1 2 where it o ccurs means V { 1 2 } . Also , if  n 2 , 1 2  ⊂ I ∪ J , then V I ∪ J shall mean V { n 2 } V ( I ∪ J ) \ { n 2 , 1 2 } . As F eshbac h remark s , most of the restrictions o n I and J are to avoid repea ting r elations. Journal of Homotopy and Re late d Structure s, vol. 1(1), 2013 11 1) 2 V I = 0. 2) V I V J + V I ∪ J V I ∩ J + V I \ J V J \ I Q i ∈ I ∩ J p i = 0 (for I ∩ J 6 = ∅ , I * J ). 3) V I V J + P i ∈ I V { i } V ( J \ I ) ∪{ i } Q j ∈ I \{ i } p j = 0 (for I ⊂ J ). 4) V I V J + P i ∈ I V { i } V ( I ∪ J ) \{ i } = 0 (for I ∩ J = ∅ ; if I and J hav e the same cardinality , then the smallest e le men t o f I is to b e less than that of J ). 5) P i ∈ I V { i } V I \{ i } = 0. 6) V { 1 2 } p n 2 + V 2 { n 2 } = 0, if n is even. Then ρ ( V I ) = S q 1 ( S i ∈ I w 2 i ). References [1] Edgar H. 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