MAT 566: Differential TopologyFall 2006Notes on the Splitting PrincipleThe purpose of these notes is to formally state the Splitting Principle and comment on its justifi-cation. There are two possible proofs, but one of them is based on a fact that we have not proved.Therefore, one should consider the other approach as the proof.Throughout these notes, all vector bundles should be assumed to be complex, all cohomology ringsare with arbitrary coefficients, P∞denotes the infinite complex projective space CP∞, and Gnisthe infinite complex Grassmannian GrnC∞. Alternatively, all vector bundles should be assumed tobe real, all cohomology rings are with Z2-coefficients, P∞denotes the infinite real projective spaceRP∞, and Gnis the infinite real Grassmannian GrnR∞. A base space B will be assumed to beparacompact. Let HQ(B) be the product (rather than just sum) of all cohomology groups of B.So, an element of HQ(B) is a possibly infinite seriesa0+ a1+ . . . , where ai∈ Hi(B).Basic Splitting Principle: Suppose for every vector bundle E −→ B of rank k we have assignedclasses p(E), q(E) ∈ HQ(B) that are natural with respect to continuous maps. In other words,p(f∗E) = f∗p(E) ∈ HQ(B0) and q(f∗E) = f∗q(E) ∈ HQ(B0)for every continuous map f : B0−→ B and vector bundle E −→ B. If p(E) = q(E) for every splitvector bundle E (over every base B), then p(E)=q(E) for every vector bundle E.General Splitting Principle: Suppose for every r-tuple of vector bundles (E1, . . . , Er) of ranks(k1, . . . , kr) over every base B we have assigned classesp(E1, . . . , Er), q(E1, . . . , Er) ∈ HQ(B)that are natural with respect to continuous maps. In other words,pf∗E1, . . . , f∗Er= f∗p(E1, . . . , Er) ∈ HQ(B0) andqf∗E1, . . . , f∗Er) = f∗q(E1, . . . , Er) ∈ HQ(B0)for every continuous map f : B0−→ B and r-tuple of vector bundles E1, . . . , Er−→ B (of ranksk1, . . . , kr). Ifp(E1, . . . , Er) = q(E1, . . . , Er)for every r-tuple of split vector bundles E1, . . . , Er(over every base B), thenp(E1, . . . , Er) = q(E1, . . . , Er)for every r-tuple of vector bundles E1, . . . , Er.Approach I: The first proof of the splitting principle is based on the following claim, which wehave not proved.Claim (basic version): For every vector bundle E −→ B, there exists a topological space˜B and acontinuous map π :˜B −→ B such that the homomorphismπ∗: H∗(B) −→ H∗(˜B)is injective and the vector bundle π∗E −→˜B splits.Claim (general version): For every r-tuple of vector bundles E1, . . . , Er−→ B, there exists atopological space˜B and a continuous map π :˜B −→ B such that the homomorphismπ∗: H∗(B) −→ H∗(˜B)is injective and the vector bundle π∗Ei−→˜B splits for every i =1, . . . , r.Assuming the basic version of the claim, the basic Splitting Principle is proved as follows. Given avector bundle E −→ B, let π :˜B −→ B be as in the claim. Since π∗E −→˜B splits, p(π∗E) = q(π∗E).Thus, by the naturality of p and q,π∗p(E) = p(π∗E) = q(π∗E) = π∗q(E) ∈ HQ(˜B).Since the homomorphism π∗is injective, it follows thatp(E) = q(E) ∈ HQ(B).The general splitting principle is proved in exactly the same way using the general version of theclaim.Why is the claim true? For every vector bundle E −→ B, there exists a fibration PE −→ B calledthe projectivization of E. It is obtained by replacing each fiber Ebof E by its projectivization(taken over C or R as appropriate), i.e. PEx. Since a (linear) isomorphism between vector spacesV and W induces a diffeomorphism between PV and PW , the linear trivialization (transition) mapsfor E induce trivialization (transition) maps for PE. If k is the rank E, the fibers of the fibrationp: PE −→ B are the (k−1)-dimensional projective spaces. Under our assumptions on the coefficientring, the fibration p : PE −→ B admits a cohomology extension of the fiberθ : H∗(Pk−1) −→ H∗(PE).This means that θ is a homomorphism such thatι∗b◦ θ : H∗(Pk−1) −→ H∗(PEb)is an isomorphism for every b ∈ B, where ιb: Eb−→ E is the inclusion map. Thus, by the ThomIsomorphism Theorem, the homomorphismΦ: H∗(B)⊗H∗(Pk−1) −→ H∗(PE), α⊗β −→ p∗(α) ∪ θ(β),2is an isomorphism. In particular, the homomorphismp∗: H∗(B) −→ H∗(PE)is injective. So, the key missing argument is a construction of θ. This is not very simple; in fact,θ need not be unique.Let’s assume the conclusion of the previous paragraph. Suppose E −→ B is a vector bundle ofrank k. Letπ1: PE −→ Bbe its projectivization. The vector bundle π∗1E −→ PE contains the tautological line bundle:γ1≡ γ =(`, v)∈ π∗1E ⊂ PE×E : v ∈ `⊂ Eπ1(`).Since B is paracompact, we obtain a splittingπ∗1E = E1⊕ γ1,where E1is a vector bundle of rank k−1. If k = 2, we are done, as π∗1E is a split vector bundle andthe homomorphismπ∗1: H∗(B) −→ H∗(PE)is injective. If k >2, let π2: PE1−→ PE be the projectivization of E1. Then,π∗2E1= E2⊕ γ2=⇒ π∗2π∗1E = E2⊕ γ2⊕ π∗2γ1for some vector bundle E2−→ PE2of rank k−2. After taking k−1 projectivizations, we obtain afibrationπ ≡ π1◦. . .◦πk−1:˜B ≡ PEk−2−→ Bsuch thatπ∗E = Ek−1⊕ γk−1⊕ π∗k−1γk−2⊕ π∗k−1π∗k−2γk−3⊕ . . . ⊕ π∗k−1. . .π∗2γ1is a sum of line bundles and the homomorphismπ∗= π∗k−1◦. . .◦π∗1: H∗(B) −→ H∗(˜B)is injective.The last paragraph implies the basic case of the claim (assuming θ exists). The general case isobtained by repeating the same construction for vector bundles E2, . . . , Erpull-backed to˜B. So,we have to do the construction of the previous paragraph r times.Approach II: Let γk−→ Gkbe the tautological k-plane bundle. The second proof of the splittingprinciple is based on the following claim.Claim (basic version): There exists a continuous map f : (P∞)k−→ Gksuch that the homomor-phismf∗: H∗(Gk) −→ H∗(P∞)k3is injective andf∗γk= (γ1)k=j=kMj=1π∗jγ1−→ (P∞)k.Claim (general version): Let k1, . . . , krbe positive integers. There exists a continuous mapf : (P∞)k1+...+kr−→ Gk1×. . .×Gkrsuch that the homomorphismf∗: H∗Gk1×. . .×Gkr−→ H∗(P∞)k1+...+kris injective and the vector bundle f∗(π∗iγki)−→ (P∞)k1+...+krsplits for every i =1, . . . ,