03.13.E1 · modern-geometry / spectral-sequences

Spectral-sequence computation exercise pack (Bott-Tu Ch. III supplement)

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Formal definition of the pack [Intermediate]

Bott-Tu Chapter III covers spectral sequences (§14), the Leray-Serre spectral sequence and its applications (§15–§18), and characteristic classes derived through them (§19–§23). Several Bott-Tu exercises and named computations cross-cut multiple Codex units in 13-spectral-sequences/, and several others are computations the main units state without working out in full (the cohomology of $Ω S^{n}$ , Serre's $π_{4} (S^{3}) = Z /2$ truncation, Borel's $H^{*} (B U (n))$ , Eilenberg-Moore on the path-loop fibration).

This pack collects eighteen such exercises — five easy, nine medium, four hard — each with a hint and full solution. It is read alongside its three prerequisite units rather than as a standalone development. The exercises are loosely grouped: easy ones at the start drill the abstract machinery (bidegree, convergence symbol, exact-couple from a SES); medium exercises run canonical computations (Hopf fibration via Serre, Gysin on $C P^{k}$ , Borel computation of $H^{*} (B U (n))$ , Leray-Hirsch on $P (E)$ , Pontryagin via splitting); the hard ones at the end are the load-bearing computations a reader is most likely to need pen and paper for ( $π_{4} (S^{3}) = Z /2$ via Serre, Eilenberg-Moore on path-loop, transgression in a specific bundle, multiplicative structure on a Kähler-style spectral sequence).

The conventions follow Bott-Tu: cohomological grading throughout; $d_{r}$ of bidegree $(r, 1 - r)$ (notation decision #15); convergence symbol $\Rightarrow$ (decision #30); two-filtration spectral sequences $_{I} E$ and $_{I I} E$ (decision #29); Bott-Tu sign convention $d ψ = - π^{*} e (E)$ on the global angular form. Coefficients are integer unless otherwise specified.

Key theorem with full solution [Intermediate]

Before the pack proper, we work one canonical exercise in full as an exemplar.

Lead exercise. Verify that the Leray-Serre spectral sequence of the Hopf fibration $S^{1} \to S^{3} \to S^{2}$ collapses at $E_{3}$ , with $E_{3} = E_{\infty}$ matching $H^(S^3)$.*

Solution. With $Z$ coefficients, $$ E_2^{p, q} = H^p(S^2; H^q(S^1)) = \begin{cases} \mathbb{Z} & (p, q) \in {(0, 0), (0, 1), (2, 0), (2, 1)}, \ 0 & \text{else}. \end{cases} $$ The non-zero entries form the corners of a $2 \times 1$ rectangle. The differential $d_{2} : E_{2}^{p, q} \to E_{2}^{p + 2, q - 1}$ has bidegree $(2, - 1)$ . The only source-target pair where both are non-zero on $E_{2}$ is $E_{2}^{0, 1} \to E_{2}^{2, 0}$ .

By the universal Euler-class transgression formula, $d_{2}$ on this column is multiplication by $e (η) \in H^{2} (S^{2})$ , where $η$ is the Hopf bundle. The Euler class of $η$ equals the generator of $H^{2} (S^{2}) = Z$ (the Hopf bundle is the universal $U (1)$ -bundle over $C P^{1}$ , with $c_{1} = 1$ ). Hence $d_{2} : Z \to Z$ is multiplication by $\pm 1$ , an isomorphism. Both source and target are killed.

Surviving entries on $E_{3} = E_{\infty}$ : $E_{\infty}^{0, 0} = Z$ (total degree $0$ ), $E_{\infty}^{2, 1} = Z$ (total degree $3$ ). All other entries zero. Matching $H^{0} (S^{3}) = Z$ , $H^{3} (S^{3}) = Z$ , and $H^{k} (S^{3}) = 0$ otherwise. $□$

This is the canonical computation — the Hello, World of spectral sequences. The same template (read off the $E_{2}$ grid; identify which differentials can be non-zero by bidegree; identify the values of those differentials by transgression; compute kernel and cokernel) recovers all eighteen exercises below in some form.

Exercises [Intermediate]

Exercise 3 (easy, proof). Exact couple from a SES of complexes.

Let $0 \to A^{*} f B^{*} g C^{*} \to 0$ be a short exact sequence of cochain complexes. Construct an exact couple whose associated spectral sequence has $E_{1} = H^{*} (C)$ and $d_{1} =$ a connecting morphism.

Hint

The long exact sequence in cohomology supplies the exactness data; take $A^{p, q} = H^{p + q} (B)$ and $E^{p, q} = H^{p + q} (C)$ .

Answer

The long exact sequence $$ \cdots \to H^n(A) \to H^n(B) \to H^n(C) \xrightarrow{\delta} H^{n+1}(A) \to \cdots $$ unrolls into an exact triangle. Take $A = ⨁_{n} H^{n} (B)$ as the upper vertex (with bigrading absorbed into the filtration), $E = ⨁_{n} H^{n} (C)$ as the lower vertex, with maps:

$i : A \to A$ the identity (no filtration shift) — for the simplest filtration the map is identity-shift
$j : A \to E$ induced by $g_{*}$
$k : E \to A$ the connecting morphism $δ$ .

The differential $d_{1} := j \circ k$ on $E$ is the composition $g_{*} \circ δ : H^{*} (C) \to H^{*} (A) \to H^{*} (C)$ (where the last map is $f_{*}$ pulled back), and squares to zero. The spectral sequence iterates from $E_{1} = H^{*} (C)$ . The Bockstein construction is the avatar.

Exercise 5 (easy, proof). Collapse from a row hypothesis.

Let ${E_{r}^{p, q}}$ be a first-quadrant cohomological spectral sequence. Show that if $E_{2}^{p, q} = 0$ for $q \geq 1$ , then $E_{2} = E_{\infty}$ and $H^{n} ≅ E_{2}^{n, 0}$ via the bottom-edge homomorphism.

Hint

$d_{r}$ from row $q = 0$ goes to negative $q$ (outside the first quadrant); $d_{r}$ landing in row $q = 0$ from the rest of the support is precluded if all higher rows vanish on $E_{2}$ .

Answer

Suppose $E_{2}^{p, q} = 0$ for all $q \geq 1$ . The differential $d_{r} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ has source in row $q = 0$ landing in row $q = 1 - r < 0$ for $r \geq 2$ , outside the first quadrant. Hence $d_{r}$ is zero on row $q = 0$ at every page.

A differential landing in row $q = 0$ on page $r$ has source $(p - r, r - 1)$ , in row $r - 1 \geq 1$ . Since $E_{2}$ on this row vanishes, so do all later $E_{r}$ on this row, so the source is already zero. No differential lands in row $q = 0$ either.

Thus $E_{2}^{p, 0} = E_{3}^{p, 0} = \dots = E_{\infty}^{p, 0}$ , the only non-zero entries. The filtration on $H^{n}$ has only one non-zero graded piece (at $p = n$ ), so $H^{n} ≅ E_{\infty}^{n, 0} = E_{2}^{n, 0}$ , with the isomorphism the bottom-edge homomorphism. $□$

Exercise 6 (medium, symbolic). Hopf fibration via Serre.

Run the Leray-Serre spectral sequence on the Hopf fibration $S^{3} \to S^{7} \to S^{4}$ to compute $H^{*} (S^{7})$ .

Hint

The fibre $S^{3}$ has cohomology in degrees $0$ and $3$ . The base $S^{4}$ has cohomology in degrees $0$ and $4$ . The bidegree of $d_{r}$ matters.

Answer

$E_{2}^{p, q} = H^{p} (S^{4}; H^{q} (S^{3}))$ has non-zero entries at $(0, 0), (0, 3), (4, 0), (4, 3)$ , all $Z$ . The differential $d_{r}$ has bidegree $(r, 1 - r)$ . The only source-target pair on the first quadrant where both endpoints are non-zero is $E_{2}^{0, 3} \to E_{2}^{4, 0}$ via $d_{4}$ . The differential $d_{4}$ in bidegree $(4, - 3)$ acts as multiplication by the Euler class of the Hopf bundle $S^{3} \to S^{7} \to S^{4}$ , which is the generator of $H^{4} (S^{4})$ (this is the quaternionic Hopf bundle, with Euler class $\pm 1$ ).

Hence $d_{4}$ is an isomorphism and kills both endpoints. Surviving entries on $E_{5} = E_{\infty}$ : $E_{\infty}^{0, 0} = Z$ (degree $0$ ) and $E_{\infty}^{4, 3} = Z$ (degree $7$ ). Matching $H^{*} (S^{7}) = Z$ in degrees $0$ and $7$ and zero elsewhere.

Exercise 8 (medium, symbolic). Borel computation of $H^(BU(n))$.*

Use the Serre spectral sequence of the universal fibration $U (n) \to E U (n) \to B U (n)$ (with $E U (n)$ contractible) to derive $H^{*} (B U (n); Q) = Q [c_{1}, \dots, c_{n}]$ , $de g c_{i} = 2 i$ .

Hint

$H^{*} (U (n); Q) = Λ [x_{1}, x_{3}, \dots, x_{2 n - 1}]$ exterior on odd-degree generators. Each $x_{2 i - 1}$ transgresses to $c_{i}$ .

Answer

$H^{*} (U (n); Q)$ is the exterior algebra on generators $x_{1}, x_{3}, \dots, x_{2 n - 1}$ in odd degrees (Hopf's theorem on cohomology of compact Lie groups). The Serre SS has $E_{2} = H^{*} (B U (n); Q) \otimes Λ [x_{1}, \dots, x_{2 n - 1}]$ converging to $H^{*} (E U (n); Q) = Q$ in degree $0$ .

Each generator $x_{2 i - 1}$ must die. The only way for $x_{2 i - 1}$ to be killed is by being transgressed to a class in the bottom row. Hence $d_{2 i} (x_{2 i - 1}) = c_{i}$ for some class $c_{i} \in H^{2 i} (B U (n); Q)$ . By multiplicativity, $d_{2 i}$ on $x_{2 i - 1} \cdot α$ is $c_{i} α \pm x_{2 i - 1} d_{2 i} (α)$ , allowing one to inductively see that the bottom-row classes $c_{i}$ generate $H^{*} (B U (n); Q)$ and the kernel is exactly the polynomial algebra on ${c_{i}}$ .

After all transgressions the entire $E_{\infty} = Q$ in degree $0$ . Hence $H^{*} (B U (n); Q) = Q [c_{1}, \dots, c_{n}]$ , with $c_{i}$ the transgressed images of the universal odd-degree generators of $H^{*} (U (n); Q)$ — these are the Chern classes of the universal bundle $E U (n) \to B U (n)$ .

Exercise 9 (medium, symbolic). Cohomology of $Ω S^{n}$ rationally.

Use the Serre spectral sequence of the path-loop fibration $Ω S^{n} \to P S^{n} \to S^{n}$ (with $P S^{n}$ contractible) to compute $H^{*} (Ω S^{n}; Q)$ for $n \geq 2$ .

Hint

Reverse engineer: the abutment is concentrated in degree $0$ , so all classes outside $(0, 0)$ must be killed by some differential. The only differential with non-vanishing range is $d_{n}$ .

Answer

$E_{2}^{p, q} = H^{p} (S^{n}) \otimes H^{q} (Ω S^{n}; Q)$ . Non-zero columns at $p = 0$ and $p = n$ only. The single non-vanishing differential is $d_{n} : E_{n}^{0, q} \to E_{n}^{n, q - n + 1}$ .

Let $σ \in H^{n - 1} (Ω S^{n}; Q)$ be the transgression source: $d_{n} (σ) = ι_{n} \in H^{n} (S^{n})$ the fundamental class. By multiplicativity, $$ d_n(\sigma^k) = k \sigma^{k-1} \cdot d_n(\sigma) = k \sigma^{k-1} \iota_n. $$ For this to kill all higher classes, $σ$ must generate $H^{*} (Ω S^{n}; Q)$ as either a polynomial algebra (when $σ^{k} \neq = 0$ for all $k$ , i.e., $n$ odd, where $σ$ has even degree $n - 1$ ) or as polynomial in $τ$ with exterior on $σ$ (when $n$ even, where $σ$ has odd degree). Computing: for $n = 2 m + 1$ odd, $σ$ has degree $2 m$ (even) and $H^{*} (Ω S^{2 m + 1}; Q) = Q [σ]$ , polynomial. For $n = 2 m$ even, $σ$ has degree $2 m - 1$ (odd) and $σ^{2} = 0$ rationally, so we need an additional even-degree generator $τ = σ \cdot d_{n} (σ)$ of degree $4 m - 2$ , giving $H^{*} (Ω S^{2 m}; Q) = Λ [σ_{2 m - 1}] \otimes Q [τ_{4 m - 2}]$ .

Exercise 11 (medium, symbolic). Splitting principle for Pontryagin classes.

For a real vector bundle $E_{R}$ of rank $2 k$ with Chern roots $\pm ξ_{1}, \dots, \pm ξ_{k}$ on the complexification, write the total Pontryagin class $p (E_{R})$ in terms of $ξ_{i}$ .

Hint

$p_{i} (E_{R}) = (- 1)^{i} c_{2 i} (E_{R} \otimes C)$ and the complexification has paired roots $\pm ξ_{i}$ .

Answer

The complexification $E_{R} \otimes C$ has total Chern class $$ c(E_{\mathbb{R}} \otimes \mathbb{C}) = \prod_{i=1}^k (1 + \xi_i)(1 - \xi_i) = \prod_{i=1}^k (1 - \xi_i^2). $$ Expanding: $c_{2 i} (E_{R} \otimes C) = (- 1)^{i} e_{i} (ξ_{1}^{2}, \dots, ξ_{k}^{2})$ , hence $$ p_i(E_{\mathbb{R}}) = e_i(\xi_1^2, \ldots, \xi_k^2), \qquad p(E_{\mathbb{R}}) = \prod_{i=1}^k (1 + \xi_i^2). $$ This is the splitting-principle formula for Pontryagin classes (decision: pair-symmetric in the squared roots).

Exercise 14 (medium, proof). Whitney via splitting on flag bundle.

Use the splitting principle to prove the Whitney sum formula for Pontryagin classes: $p (E \oplus F) = p (E) p (F)$ modulo $2$ -torsion.

Hint

Pull both bundles back to a flag bundle where each splits into lines, then use the Chern-root formula.

Answer

Pull back to $F ℓ (E_{R} \oplus F_{R})$ , where the bundle splits as a sum of real line bundles, and the complexification splits with paired roots $\pm ξ_{i}$ (from $E$ ) and $\pm η_{j}$ (from $F$ ). The total Pontryagin class on the flag bundle: $$ \sigma^* p(E_{\mathbb{R}} \oplus F_{\mathbb{R}}) = \prod_i (1 + \xi_i^2) \cdot \prod_j (1 + \eta_j^2) = \sigma^* p(E_{\mathbb{R}}) \cdot \sigma^* p(F_{\mathbb{R}}). $$ Since $σ^{*} : H^{*} (B) \to H^{*} (F ℓ)$ is injective in characteristic $\neq = 2$ (where the rank- $2$ flag bundle for an oriented bundle pair has Euler class $2 ξ$ — non-zero divisor away from $2$ -torsion), the formula descends to $B$ . The mod- $2$ caveat is a standard subtlety handled by passing to mod- $2$ characteristic classes (Stiefel-Whitney) on the real side. $□$

Exercise 15 (hard, symbolic). $π_{4} (S^{3}) = Z /2$ via Postnikov truncation.

Compute $π_{4} (S^{3})$ using the Postnikov truncation $K (Z, 3) \to S^{3} ⟨ 3 ⟩ \to S^{3}$ and the Serre spectral sequence on the relevant fibration.

Hint

Replace $S^{3}$ with its Postnikov tower; $π_{4} (S^{3}) = π_{4} (S^{3} ⟨ 3 ⟩)$ since $π_{3}$ is killed in the truncation. Compute $H^{*} (S^{3} ⟨ 3 ⟩)$ using Serre on the path-loop fibration above $K (Z, 3)$ .

Answer

The Postnikov $3$ -truncation $S^{3} \to K (Z, 3)$ has homotopy fibre $S^{3} ⟨ 3 ⟩$ , the $3$ -connected cover of $S^{3}$ . The fibration $$ S^3 \langle 3 \rangle \to S^3 \to K(\mathbb{Z}, 3) $$ has $π_{4} (S^{3} ⟨ 3 ⟩) = π_{4} (S^{3})$ (since $K (Z, 3)$ has homotopy concentrated in degree $3$ ).

By Hurewicz on $S^{3} ⟨ 3 ⟩$ (which is $3$ -connected), $π_{4} (S^{3} ⟨ 3 ⟩) = H_{4} (S^{3} ⟨ 3 ⟩)$ . Compute the latter using the Serre SS of the path-loop $Ω K (Z, 3) \to P K (Z, 3) \to K (Z, 3)$ in the relevant Whitehead-tower extension.

The cohomology of $K (Z, 3)$ in low degrees: $H^{3} = Z$ (fundamental class $ι_{3}$ ), $H^{6} = Z$ generated by $ι_{3}^{2}$ (cup square; $K (Z, 3)$ has divided power algebra structure rationally and a $Sq^{2}$ -connection mod $2$ ). The Serre SS of the path-loop fibration with fibre $Ω K (Z, 3) = K (Z, 2)$ shows that the transgression $τ (x) = ι_{3}$ for $x$ the degree- $2$ generator of $K (Z, 2)$ , and $τ (x^{2}) = 2 x ι_{3}$ — so divided powers control the integer cohomology.

The mod- $2$ analysis: $Sq^{2} ι_{3} =$ a non-zero class in $H^{5} (K (Z, 3); Z /2)$ , which detects the obstruction in the Whitehead extension. Working through Serre's argument (Bott-Tu §18 outline; full in Serre 1951 thesis), the result is $π_{4} (S^{3}) = Z /2$ , generated by the Hopf map $η : S^{4} \to S^{3}$ (suspension of $S^{3} \to S^{2}$ ).

This is one of the most consequential single computations in Serre's PhD thesis. The full argument requires the Steenrod algebra structure on $H^{*} (K (Z, 3); Z /2)$ via $Sq^{2} ι_{3}$ . $□$

Exercise 16 (hard, symbolic). Eilenberg-Moore on the path-loop fibration.

Use the Eilenberg-Moore spectral sequence to compute $H^{*} (Ω S^{n}; Z)$ in a single page.

Hint

The Eilenberg-Moore SS of the pullback $Ω Y \to P Y \to Y$ has $E_{2} = Tor_{H^{*} (Y)}^{*, *} (Z, Z)$ . For $Y = S^{n}$ , $H^{*} (S^{n}) = Z [ι_{n}] / ι_{n}^{2}$ , and $Tor_{Z [ι] / ι^{2}} (Z, Z)$ is computable.

Answer

For $Y = S^{n}$ , $H^{*} (Y) = Z [ι_{n}] / (ι_{n}^{2})$ — exterior on a single class of degree $n$ . The Eilenberg-Moore SS of $Ω S^{n} \to P S^{n} \to S^{n}$ has $$ E_2 = \operatorname{Tor}{H^(S^n)}^{, *}(\mathbb{Z}, \mathbb{Z}) = \operatorname{Tor}{\mathbb{Z}[\iota_n] / (\iota_n^2)}(\mathbb{Z}, \mathbb{Z}). $$ For $n$ odd: $Z [ι_{n}] / (ι_{n}^{2})$ is exterior on an odd-degree generator. The Koszul resolution of $Z$ over this exterior algebra is the polynomial algebra $Z [σ]$ on a class $σ$ of degree $n - 1$ , so $Tor = Z [σ]$ . The spectral sequence collapses and $H^{*} (Ω S^{n}) = Z [σ]$ , polynomial on $σ$ of degree $n - 1$ .

For $n$ even: the Koszul resolution of $Z$ over $Z [ι_{n}] / (ι_{n}^{2})$ with $ι_{n}$ in even degree gives $Tor = Λ [σ_{n - 1}] \otimes Z [τ_{2 n - 2}]$ , exterior on an odd-degree class of degree $n - 1$ tensored with polynomial on an even-degree class of degree $2 n - 2$ . Hence $H^{*} (Ω S^{2 k}) = Λ [σ_{2 k - 1}] \otimes Z [τ_{4 k - 2}]$ .

The Eilenberg-Moore SS collapses at $E_{2}$ for $Y = S^{n}$ because the $Tor$ is concentrated in a single bigrading per total degree. The Serre SS computation of Exercise 9 gives the same answer with more pages of work.

Exercise 17 (hard, proof). Multiplicative structure on a Kähler-style spectral sequence.

Let $K^{*}$ be a differential graded algebra with a multiplicative bounded filtration $F^{*}$ (i.e., $F^{p} \cdot F^{q} \subset F^{p + q}$ ). Show that each page $E_{r}$ of the associated spectral sequence is a bigraded ring, with $d_{r}$ a derivation, and $E_{\infty}$ matches the associated graded of the cup product on $H^{*} (K^{*})$ .

Hint

Define the product on $E_{r}^{p, q} \otimes E_{r}^{p^{'}, q^{'}} \to E_{r}^{p + p^{'}, q + q^{'}}$ via cocycle representatives in $Z_{r}^{p, q} \cdot Z_{r}^{p^{'}, q^{'}} \subset Z_{r}^{p + p^{'}, q + q^{'}}$ , then show $d_{r}$ is a derivation by Leibniz on $K^{*}$ .

Answer

Take cocycle representatives $x \in Z_{r}^{p, q}$ (i.e., $x \in F^{p} K^{p + q}$ with $d x \in F^{p + r} K^{p + q + 1}$ ) and $y \in Z_{r}^{p^{'}, q^{'}}$ . By multiplicativity of the filtration, $x y \in F^{p + p^{'}} K^{p + p^{'} + q + q^{'}}$ , and by Leibniz on $K$ : $$ d(xy) = (dx) y + (-1)^{|x|} x (dy) \in F^{p+p'+r} K + F^{p+p'+r} K = F^{p+p'+r} K. $$ Hence $x y \in Z_{r}^{p + p^{'}, q + q^{'}}$ . The class $[x y] \in E_{r}^{p + p^{'}, q + q^{'}}$ is well-defined modulo $Z_{r - 1}^{p + p^{'} + 1, q + q^{'} - 1} + B_{r - 1}^{p + p^{'}, q + q^{'}}$ (changing $x$ within its class by an element of $Z_{r - 1}^{p + 1, q - 1} + B_{r - 1}^{p, q}$ changes $x y$ by an element of the indicated submodule).

The differential $d_{r}$ on $E_{r}$ acts by $d_{r} [x] = [d x] \in E_{r}^{p + r, q - r + 1}$ , and the Leibniz rule on $K^{*}$ gives $$ d_r ([x][y]) = [d(xy)] = [(dx) y + (-1)^{|x|} x (dy)] = (d_r [x])[y] + (-1)^{|x|} [x] (d_r [y]). $$ So $d_{r}$ is a graded derivation. The page-advance to $E_{r + 1}$ inherits this product. In the limit, $E_{\infty} = gr H^{*} (K^{*})$ as a bigraded ring, with the product matching the associated graded of the cup product on the abutment.

This is the structural reason for multiplicative collapse arguments: if the bottom-row classes (where $d_{r}$ acts by derivation) generate the entire $E_{2}$ , no $d_{r}$ can be non-zero by Leibniz, hence collapse. $□$

Exercise 18 (hard, symbolic). Transgression in a Stiefel manifold fibration.

Compute the transgression for the fibration $V_{n, k} (C) \to B U (n - k) \to B U (n)$ where $V_{n, k} (C) = U (n) / U (n - k)$ is the complex Stiefel manifold of $k$ -frames in $C^{n}$ . Express the result in terms of Chern classes.

Hint

The fibration is the universal $V_{n, k}$ -bundle. The cohomology of $V_{n, k} (C)$ is exterior on classes $x_{2 (n - k) + 1}, \dots, x_{2 n - 1}$ (Hopf's theorem applied to Lie groups). Transgression sends $x_{2 j - 1}$ to $c_{j}$ of the universal rank- $n$ bundle.

Answer

$V_{n, k} (C) = U (n) / U (n - k)$ has cohomology $$ H^(V_{n, k}(\mathbb{C}); \mathbb{Q}) = \Lambda[x_{2(n - k) + 1}, x_{2(n - k) + 3}, \ldots, x_{2n - 1}], $$ exterior algebra on $k$ generators in odd degrees. The Serre SS of $V_{n, k} \to B U (n - k) \to B U (n)$ has $E_2 = H^(BU(n)) \otimes H^(V_{n, k}) $, co n v er g in g t o$ H^(BU(n - k)) = \mathbb{Q}[c_1, \ldots, c_{n-k}]$.

The transgression of $x_{2 j - 1}$ (for $n - k < j \leq n$ ) lands in degree $2 j$ on the base, where $H^{2 j} (B U (n)) =$ degree- $2 j$ part of $Q [c_{1}, \dots, c_{n}]$ . By naturality with the universal classifying-space fibration $U (n) \to E U (n) \to B U (n)$ where each odd-degree generator transgresses to the corresponding Chern class: $$ \tau(x_{2j - 1}) = c_j(\text{universal bundle}) $$ restricted to the appropriate quotient. Specifically the kernel of $H^{*} (B U (n)) \to H^{*} (B U (n - k))$ is generated by $c_{n - k + 1}, \dots, c_{n}$ , exactly the Chern classes hit by the transgression of the new generators of $V_{n, k}$ .

This computation is the structural reason classifying-space cohomology behaves well under quotients: the Serre SS of $G / H \to B H \to B G$ shows $H^{*} (G / H)$ as the cokernel of the natural map $H^{*} (B G) \to H^{*} (B H)$ in nice cases, with transgression supplying the explicit generators. It underlies the construction of the characteristic classes of vector bundles with structure-group reduction.

Bott-Tu Pass 4 — Agent C — EP2. Eighteen exercises (5 easy / 9 medium / 4 hard) on spectral-sequence machinery, Leray-Serre, Gysin, Leray-Hirsch, splitting principle, and Eilenberg-Moore. Cross-cuts the three Batch units 03.13.01–03 and supplies the worked-example backfill for Bott-Tu §14–§18 (gap block 18 of §2.2). Notation conventions decisions #15, #29, #30 invoked throughout.

Prerequisites

03.13.01
03.13.02
03.13.03

Tier anchors

beginner
intermediate: Bott-Tu §14–§18 exercise problems; McCleary Ch. 1–6 worked examples; Hatcher §4.2–§4.3 supplement
master

References

TODO_REF
Bott-Tu — Differential Forms in Algebraic Topology · §14–§18 exercise problems and worked examples
TODO_REF
McCleary — A User's Guide to Spectral Sequences · Ch. 1–6 worked examples on Serre, Gysin, Leray-Hirsch, Eilenberg-Moore
TODO_REF
Hatcher — Algebraic Topology · §4.2 (Postnikov towers), §4.3 (Whitehead tower) — homotopy applications
TODO_REF
Borel — Sur la cohomologie des espaces fibrés principaux · Ann. of Math. 57 (1953), 115–207 — Borel's PhD thesis
TODO_REF
Serre — Homologie singulière des espaces fibrés · Ann. of Math. 54 (1951), 425–505 — original calculations of $\pi_4(S^3)$

Reviewer

TBD — cross-cuts §14–§18 of Bott-Tu and §5–§6 of McCleary

Estimated time

intermediate: 7h