Short Complete Proofs of the Serre Spectral Sequence Theorems

A new improved ”Simple complete proofs of the Serre spectral sequence theorems”. In ([B]) I set forth a proof of the Serre calculation of E and claimed among other things that unlike my previous attempts to prove this in graduate course lectures, this proof was routine. On presenting this material in class, I discovered it was not as routine as I had imagined. With the help of Pallavi Jayawart and Saso Strle I have drastically improved and simplified the presentation, reducing proofs of the Serre Spectral Sequence (SSS) theorems ([S]) to a collection of lemmas provable by straightforward mechanical checking which is left to the reader. In addition it offers some motivation for the definitions. A knowledge of the standard material on singular homology and cohomology, including the Eilenberg-Zilber theorem, is sufficient to prove the lemmas. We do homology first and then add variations, including cohomology, as exercises. 1. The Algebra of the SSS Suppose 0 = A−1 ⊂ A ⊂ · · · ⊂ A ⊂ · · · ⊂ A = UA is a sequence of subcomplexes of A, where A is a chain complex over a commutative ring Λ. For example, A = C∗(Xp) where A = C∗(X) is the singular chains on X with coefficients in Λ and φ = X−1 ⊂ · · · ⊂ X ⊂ · · · ⊂ X = UX are spaces. The long exact homology sequence of the pair (A, Ap−1) enables one to calculate, with some ambiguity H∗(A) from H∗(Ap−1) and H∗(A, Ap−1). Combining the calculations for each p > −1 gives a way of calculating H∗(A), with considerable ambiguity, from H∗(A , Ap−1) , p > −1. The algebra of the SSS is a way of organizing such a Typeset by AMS-TEX 1 2 EDGAR H. BROWN, JR. calculation. We give a diagram displaying all of these exact sequences: (1.1) Hp+q−1(A −1) y y .. ∂∗ −→ Hp+q(A) −→ Hp+q(A,A) −→ Hp+q−1(Ap−2)−→ yj∗ y Hp+q(A ) i∗ −→ Hp+q(A,A) ∂∗ −→> Hp+q−1(Ap−1)−→ y yj∗ .. Hp+q(A) An element u ∈ Hp+q(A, Ap−1) contributes to Hp+q(A) if it pulls back to Hp+q(A ) in which case the pull-back can be sent down to Hp+q(A). Thus we want to know when ∂∗u = 0. A first obstruction is the image of ∂∗u inHp+q−1(A p−1, Ap−2) (the idea is to emphasize the groups H∗(A, Ap−1) , p > −1. If ∂∗u goes to zero in Hp+q−1(Ap−1, Ap−2), it lifts to u′ ∈ Hp+q−1(Ap−2) and the second obstruction to ∂∗u = 0 is i∗u′ ∈ Hp+q−1(Ap−2, Ap−3). Continuing in this way, one has a sequence of obstructions to lifting ∂∗u to Hp+q−1(A−1) = 0 and hence to ∂∗u = 0. This motivates defining Z r = {u ∈ Hp+q(A, Ap−1) | ∂∗u lifts to Hp+q−1(A)}. Then Z p+1 = ker∂∗ and it gives some elements of Hp+q(A), namely D = image(Hp+q(A )→ Hp+q(A)). We make the process of lifting ∂∗u to Hp+a−1(A p−1) and then to Hp+q−1(Ap−r, Ap−r−1), giving an element dru, into a well defined map by dividing out by the indeterminacy, namely the image of ker(Hp+q−1(Ap−r)→ Hp+q−1(Ap−1)) in Hp+q−1(Ap−r, Ap−r−1). Let B r = {i∗v | v ∈ ker(Hp+q(A)→ Hp+q(A))} and let E r = Z p,q r /B p,q r . Lemma 1.2. u→ dru as above gives a well defined map dr : E r → Ep−r,q+r−1 r . Furthermore, dr = 0. Lemma 1.3. The inclusion Z r+1 ⊂ Z r induces an isomorphism E r+1 → ker(dr : E r → Ep−r,q+r−1 r )/ image(Ep+r,q−r+1 r → E r ). Lemma 1.4. Suppose An = An, n ≥ 0. Then for each p and q, E p,q r+1 = E p,q r for r large. Let E ∞ = E p,q r , r large. Then 0 = D−1,n+1 ⊂ D ⊂ · · · ⊂ Dp,n−p ⊂ · · · ⊂ D = Hn(A) and Hp+1(A p)→ Hp+q(A, Ap−1) induces an isomorphism Dp,q/Dp−1,q+1 → E ∞ . The sequence of bigraded chain complexes {(Er, dr)} is called the spectral sequence of the fibration {Ap} and is said to converge to H∗(A), written Er ⇒ H∗(A) when E ∞ ≈ Dp,q/Dp−1,q+1. In E, p is called the filtration degree and p+ q is the total degree. SHORT COMPLETE PROOFS OF THE SERRE SPECTRAL SEQUENCE THEOREMS 3 2. The SSS for a Continuous Map Suppose π : X → B is a continuous map, B is path connected, b0 ∈ B and F = π−1(b0). In this section we use π to define a fibration of C∗(X) and hence a spectral sequence as in §1. Let ∆q = {(t0, . . . , tq) ∈ R | ti ≥ 0, ∑ ti = 1} be the standard q-simplex with vertices, εi = (0, 0, . . . , 1, 0, . . . , 0); ∆q(X) = {T : ∆q → X}. Let ∆[q] = {(i0, . . . , ip) | 0 ≤ i0 ≤ i1 ≤ · · · ip ≤ q}. We identify (i0, . . . , ip) ∈ ∆[q]p with the linear map of ∆p to ∆q taking εj → εij . Note under composition T (i0, . . . , ip) ∈ ∆p(X). Let ∂iT = T (0, . . . , î, . . . , q), σiT = T (0, . . . , i, i, . . . , q). Let Γn = {T : ∆n → X | πT = S(i0, . . . , in), in 5 p, S ∈ ∆p(B, )}. Let Cq(X) be the normalized singular chains of X, that is, Cq(X) consists of all finte sums, ∑ λiTi, λi ∈ Λ, Ti ∈ ∆q(X), modulo the subcomplex generated by simplexes of the form σiS. Let A p,q ⊂ Cp+q(X) be the chains based on Γpp+q. Note, A is a subcomplex of C∗(X), A−1 = 0, A ⊂ A and A = Cp(X). Let φ : C∗(X)→ C∗(B)⊗ C∗(X) be given by