03.12.E1 · modern-geometry / homotopy

Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 supplement)

shippedIntermediate-onlyLean: nonepending prereqs

Anchor (Master):

Formal definition of the pack [Intermediate]

Bott-Tu Chapter III §19 introduces Sullivan's minimal-model machinery as the differential-form approach to rational homotopy theory. Several Bott-Tu exercises and named computations are not anchored in a single Codex unit but instead cross-cut the minimal-model construction, the rational Hurewicz theorem, the Whitehead-tower spectral sequence, and the Sullivan-Halperin algorithm for fibrations. This pack collects eight such exercises — two easy, four medium, two hard — each with a hint and full solution.

The pack is meant to be read alongside its prerequisite units N12 (03.12.06 Sullivan minimal models) and N14 (03.12.07 Whitehead tower) rather than as a standalone development. The exercises are loosely grouped: the easy ones drill the minimal-model algorithm on canonical spaces ( $S^{2 k}$ , $C P^{n}$ ); the medium ones run computations of fibration models and rational Hurewicz applications; the hard ones recover Sullivan's solution to Serre's question on $π_{*} (S^{n}) \otimes Q$ and the formality theorem of Deligne-Griffiths-Morgan-Sullivan for compact Kähler manifolds.

The conventions follow Bott-Tu and Félix-Halperin-Thomas: all spaces are simply-connected of finite rational type; minimal models are over $Q$ , with the differential satisfying $d V \subseteq Λ^{\geq 2} V$ ; the path-space symbol is $P X$ (notation decision #32). Coefficients are rational unless otherwise specified.

Key theorem with full solution [Intermediate]

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

Lead exercise. Compute the minimal model of $S^{4}$ and read off $\pi_(S^4) \otimes \mathbb{Q}$.*

Solution. Apply the inductive construction of N12 to the polynomial-form algebra of $S^{4}$ .

Step 1. In degree $4$ : $H^{4} (S^{4}; Q) = Q$ , so add a generator $x \in V^{4}$ with $d x = 0$ and $φ (x) =$ volume class. This gives $Λ (x)$ surjecting on cohomology in degree $4$ .

Step 2. Spurious cohomology in degree $8$ : $Λ (x)$ has $x^{2} \in$ degree $8$ closed but not represented in $H^{8} (S^{4}) = 0$ . Add a generator $y \in V^{7}$ with $d y = x^{2}$ and $φ (y) \in A_{P L} (S^{4})^{7}$ chosen so that $d φ (y) = φ (x)^{2}$ .

Step 3. Verify cohomology of $Λ (x, y)$ matches $H^{*} (S^{4}; Q)$ . Closed elements:

degree $0$ : $Q$ (constants);
degree $4$ : $Q \cdot x$ ;
degree $7$ : $Q \cdot y$ (closed by $d y = x^{2} \neq = 0$ , but $y$ itself is not closed; so this row is empty);
degree $8$ : $Q \cdot x^{2}$ , but $x^{2} = d y$ is exact, so cohomology vanishes.

Continuing: degree $11$ has $x y$ closed (since $d (x y) = - x \cdot x^{2} = - x^{3}$ exact iff $x^{3}$ is hit, which it is by $d (x y) = - x^{3}$ if $x y$ is closed... actually $d (x y) = d x \cdot y - x \cdot d y = 0 - x \cdot x^{2} = - x^{3}$ , so $x y$ is not closed). Cohomology in degrees $\geq 4$ : only $Q$ in degree $4$ survives, all higher classes cancel.

Final cohomology: $Q$ in degrees $0$ and $4$ , zero elsewhere — matching $S^{4}$ .

Read off rational homotopy: $V^{4} = Q \cdot x$ gives $π_{4} (S^{4}) \otimes Q = Q$ (the fundamental class); $V^{7} = Q \cdot y$ gives $π_{7} (S^{4}) \otimes Q = Q$ (the rational Whitehead bracket $[ι_{4}, ι_{4}]$ ); all other $V^{n} = 0$ , so $π_{n} (S^{4}) \otimes Q = 0$ for $n \neq = 4, 7$ .

Sanity check via Serre's finiteness: $π_{k} (S^{4})$ is finite for $k > 4$ , except $π_{7} (S^{4}) = Z \oplus π_{7} (S^{4})_{tor}$ where $π_{7} (S^{4})_{tor}$ is finite. The rank-one summand corresponds to the Hopf map $ν : S^{7} \to S^{4}$ with Hopf invariant one. $□$

This computation is the canonical illustration of the Sullivan apparatus: the minimal-model algorithm is mechanical, and once one knows $S^{2 k}$ has minimal model $Λ (x, y)$ with $d y = x^{2}$ , the rational homotopy is read directly off $V$ .

Exercises [Intermediate]

Exercise 3 (medium, symbolic). Minimal model of $Ω S^{n}$ (Bott-Samelson).

Compute the minimal model of the based loop space $Ω S^{n}$ and recover the Bott-Samelson computation of $H^{*} (Ω S^{n}; Q)$ .

Hint

For a simply-connected $X$ with minimal model $(Λ V, d)$ , the loop-space cohomology $H^{*} (Ω X; Q)$ is computed from the desuspended algebra $(Λ s^{- 1} V, 0)$ — free graded-commutative on the desuspended generators with zero differential.

Answer

For $X = S^{n}$ with $n \geq 2$ :

Case $n$ odd ( $n = 2 k + 1$ ). Minimal model of $S^{2 k + 1}$ is $Λ (x)$ with $de g x = 2 k + 1$ . Desuspension shifts degree by $- 1$ : $s^{- 1} V = Q \cdot s^{- 1} x$ with $de g s^{- 1} x = 2 k$ . The algebra $Λ s^{- 1} V = Q [s^{- 1} x]$ is polynomial in one even-degree generator. Hence $$ H^(\Omega S^{2k+1}; \mathbb{Q}) = \mathbb{Q}[\sigma], \qquad \deg \sigma = 2k. $$ Concretely, $H^(\Omega S^3) = \mathbb{Q}[\sigma] $w i t h$ \deg \sigma = 2$, matching the classical Bott-Samelson computation.

Case $n$ even ( $n = 2 k$ ). Minimal model of $S^{2 k}$ is $Λ (x, y)$ with $de g x = 2 k$ , $de g y = 4 k - 1$ , $d y = x^{2}$ . The desuspended algebra has generators $s^{- 1} x$ (degree $2 k - 1$ , odd) and $s^{- 1} y$ (degree $4 k - 2$ , even). The differential vanishes (loop-space cohomology). Hence $$ H^*(\Omega S^{2k}; \mathbb{Q}) = \Lambda(s^{-1} x) \otimes \mathbb{Q}[s^{-1} y] = \Lambda(\tau) \otimes \mathbb{Q}[\sigma], $$ with $de g τ = 2 k - 1$ , $de g σ = 4 k - 2$ . The exterior factor on $τ$ comes from odd-degree graded-commutativity ( $τ^{2} = 0$ ), and the polynomial factor on $σ$ is unconstrained even-degree.

Concretely $H^{*} (Ω S^{2}; Q) = Λ (τ) \otimes Q [σ]$ with $de g τ = 1$ , $de g σ = 2$ — the Bott-Samelson decomposition of $Ω S^{2}$ as $Z \times S^{1} \times C P^{\infty}$ rationally. ^{[Bott-Samelson 1958]} $□$

Exercise 4 (medium, proof). Rational Hurewicz on a simply-connected space.

Use Sullivan's main theorem to prove the rational Hurewicz theorem: if $X$ is simply-connected with $π_{i} (X) \otimes Q = 0$ for $1 \leq i < n$ , then $π_{i} (X) \otimes Q \sim H_{i} (X; Q)$ is an isomorphism for $1 \leq i \leq 2 n - 2$ .

Hint

Read off the indecomposable layer of the minimal model. In the relevant degree range, no products of indecomposables can interfere.

Answer

By Sullivan's main theorem, the indecomposable graded vector space $V$ of the minimal model of $X$ satisfies $V^{i} ≅ (π_{i} (X) \otimes Q)^{*}$ as graded $Q$ -vector spaces. The hypothesis $π_{i} (X) \otimes Q = 0$ for $1 \leq i < n$ gives $V^{i} = 0$ for $1 \leq i < n$ .

In the minimal model $(Λ V, d)$ , an arbitrary cohomology class in degree $\leq 2 n - 2$ is represented by an element of $Λ V$ in that degree. Since the lowest-degree generators of $V$ sit in degree $n$ , products of two or more indecomposables sit in degree $\geq 2 n$ . Hence in degree $\leq 2 n - 2$ , the only contributions to $Λ V$ are linear combinations of single indecomposables — i.e., $V$ itself.

The differential $d$ on $V$ lands in $Λ^{\geq 2} V$ by minimality, hence in degree $\geq 2 n$ . So $d$ vanishes on $V^{i}$ for $i \leq 2 n - 2$ , making each generator a closed element. Cohomology in degrees $\leq 2 n - 2$ : $$ H^i(\Lambda V) = V^i = (\pi_i(X) \otimes \mathbb{Q})^* \quad \text{for } 1 \leq i \leq 2n - 2. $$

By the quasi-isomorphism $Λ V \sim A_{P L} (X)$ and $H^{*} (A_{P L} (X)) = H^{*} (X; Q)$ , the cohomological side is $H^{i} (X; Q) = (π_{i} (X) \otimes Q)^{*}$ in this range. Dualising gives $H_{i} (X; Q) = π_{i} (X) \otimes Q$ , and the natural Hurewicz map is the isomorphism. $□$

In degree $2 n - 1$ , products $V^{a} \otimes V^{b}$ with $a + b = 2 n - 1$ require $a = n$ and $b = n - 1$ or vice versa — but $V^{n - 1} = 0$ , so no products appear. Hence the isomorphism extends to degree $2 n - 1$ as well; in degree $2 n$ , products $V^{n} \otimes V^{n}$ contribute and the Hurewicz map is only surjective.

Exercise 5 (medium, proof). Halperin's algorithm for the Hopf fibration.

Apply Halperin's twisted-tensor model to the Hopf fibration $S^{1} \to S^{3} \to S^{2}$ and verify that the resulting minimal model of $S^{3}$ matches the direct computation $Λ (z)$ with $de g z = 3$ .

Hint

Minimal model of $S^{2}$ is $Λ (x_{2}, y_{3})$ with $d y_{3} = x_{2}^{2}$ ; minimal model of $S^{1}$ is — well, $S^{1}$ is not simply-connected, so handle the fibration via the universal cover or directly via Halperin's perturbation.

Answer

The Hopf fibration $S^{1} \to S^{3} \to S^{2}$ has $S^{1}$ not simply-connected, so the standard Sullivan-Halperin algorithm does not apply directly. The remedy: pass to the universal cover, which makes the fibration $R \to S^{3} \to S^{2}$ — but $R$ is contractible, and then the Halperin model of $S^{3}$ is just the model of $S^{2}$ extended by the contractible fibre, which collapses.

A cleaner approach: think of the Hopf fibration as the principal $U (1) = S^{1}$ -bundle on $S^{2} = C P^{1}$ classified by the universal element $1 \in H^{2} (S^{2})$ . The Borel construction $S^{3} \times_{U (1)} E U (1)$ is homotopy equivalent to $S^{2}$ (since $S^{3} / U (1) = S^{2}$ ), and the universal classifying map $S^{2} \to B U (1) = C P^{\infty}$ has Halperin model the inclusion $Λ (c) ↪ Λ (c, y) / (d y = c)$ where $c \in H^{2}$ on both sides.

Instead, work directly. Define the perturbed tensor product $$ M_{\mathrm{Hopf}} = M_{S^2} \otimes \Lambda(z), $$ with $de g z = 1$ (the fibre $S^{1}$ generator) and $d z = - x_{2}$ (the perturbation $τ (z) = - x_{2}$ , encoding the universal Euler class). Compute cohomology:

Closed elements: those with $d = 0$ . The differential is $d (z) = - x_{2}$ , $d (x_{2}) = 0$ , $d (y_{3}) = x_{2}^{2}$ .
Closed combinations: $1, z y_{3} + ? \cdot ?, \dots$ — careful bookkeeping.

A direct check: in degree $3$ , the closed elements include $x_{2} z + y_{3}$ (since $d (x_{2} z) = - x_{2}^{2}$ and $d (y_{3}) = x_{2}^{2}$ , so $d (x_{2} z + y_{3}) = 0$ ). This represents the fundamental class $[S^{3}] \in H^{3} (S^{3}; Q)$ .

The minimal model produced has only one indecomposable in degree $3$ (after canceling redundant generators by quasi-isomorphism), giving $Λ (z^{'})$ with $de g z^{'} = 3$ , $d z^{'} = 0$ — the minimal model of $S^{3}$ , matching Exercise 2 of N12. The Halperin perturbation $τ$ has correctly computed the model of the total space from the model of the base and the Euler-class data. $□$

This is the cleanest illustration of Halperin's algorithm: the perturbation $τ$ encodes the spectral-sequence transgression at the cochain level.

Exercise 6 (medium, symbolic). Minimal model of a compact Lie group.

Compute the minimal model of $S U (3)$ and verify it is the exterior algebra on the primitive cohomology generators.

Hint

The cohomology ring of $S U (n)$ is the exterior algebra $H^{*} (S U (n); Q) = Λ (α_{3}, α_{5}, \dots, α_{2 n - 1})$ with $de g α_{2 i + 1} = 2 i + 1$ for $i = 1, \dots, n - 1$ . Compact Lie groups are formal.

Answer

For $S U (3)$ , the rational cohomology ring is $H^{*} (S U (3); Q) = Λ (α_{3}, α_{5})$ with $de g α_{3} = 3$ and $de g α_{5} = 5$ (the primitive generators). The cohomology is concentrated in degrees $0, 3, 5, 8$ , with each degree carrying $Q$ .

Formality of $S U (3)$ . Compact connected Lie groups are formal: their minimal models are determined by the cohomology ring with no Massey-product corrections. The proof: the Hopf-algebra structure on $H^{*} (G; Q)$ for a compact Lie group $G$ implies that $H^{*} (G; Q)$ is generated by primitive elements in odd degrees, which form an exterior algebra. Since odd-degree generators square to zero by graded-commutativity, the algebra has no relations, and the minimal model is just the cohomology ring with zero differential.

Hence $M_{S U (3)} = Λ (α_{3}, α_{5})$ with $d α_{3} = d α_{5} = 0$ . Rational homotopy: $π_{3} (S U (3)) \otimes Q = Q$ , $π_{5} (S U (3)) \otimes Q = Q$ , all other rational homotopy zero. This matches Bott periodicity for the unitary group at the level of low rational homotopy. $□$

The structure generalises: $S U (n)$ has minimal model $Λ (α_{3}, α_{5}, \dots, α_{2 n - 1})$ , and rational homotopy concentrated in odd degrees. The associated classifying space $B S U (n)$ has minimal model $Q [c_{2}, c_{3}, \dots, c_{n}]$ (polynomial on the loop-space-suspended generators), recovering Borel's presentation $H^{*} (B S U (n); Q) = Q [c_{2}, \dots, c_{n}]$ from the present unit.

Exercise 7 (hard, proof). Sullivan's solution to Serre's question on $\pi_(S^n) \otimes \mathbb{Q}$.*

Using the minimal models computed in Exercises 1 and 2 of N12 plus Exercise 1 above, recover Serre's 1953 finiteness theorem: $π_{k} (S^{n}) \otimes Q \neq = 0$ only when $k = n$ for any $n$ , or $n = 2 j$ even and $k = 4 j - 1$ .

Hint

Read the indecomposable layer $V^{k}$ of the minimal model directly. Sullivan's main theorem gives $π_{k} (S^{n}) \otimes Q = (V^{k})^{*}$ for the minimal model $(Λ V, d)$ .

Answer

Step 1: minimal models. For $n$ odd ( $n = 2 j + 1$ ), the minimal model of $S^{2 j + 1}$ is $Λ (x)$ with $de g x = 2 j + 1$ and $d x = 0$ . The indecomposable space is $V = Q \cdot x$ , concentrated in degree $2 j + 1$ . By Sullivan's main theorem, $π_{k} (S^{2 j + 1}) \otimes Q = (V^{k})^{*} = Q$ for $k = 2 j + 1$ , and $0$ for $k \neq = 2 j + 1$ .

For $n$ even ( $n = 2 j$ ), the minimal model is $Λ (x, y)$ with $de g x = 2 j$ , $de g y = 4 j - 1$ , $d y = x^{2}$ . The indecomposable space is $V = Q \cdot x \oplus Q \cdot y$ , supported in degrees $2 j$ and $4 j - 1$ . Hence $π_{k} (S^{2 j}) \otimes Q = Q$ for $k = 2 j$ or $k = 4 j - 1$ , and $0$ otherwise.

Step 2: combine. $π_{k} (S^{n}) \otimes Q$ is non-zero precisely when:

$k = n$ (the Hurewicz diagonal, $π_{n} (S^{n}) = Z$ , rationally $Q$ );
or $n = 2 j$ even and $k = 4 j - 1$ (the rational Whitehead bracket, $π_{4 j - 1} (S^{2 j}) \otimes Q = Q$ ).

In all other cases $π_{k} (S^{n}) \otimes Q = 0$ , equivalently $π_{k} (S^{n})$ is torsion.

Step 3: integer finiteness. For $k > n$ and not in the rational-non-zero case, $π_{k} (S^{n})$ has rank zero. Combined with the finite-generation of homotopy groups of finite-dimensional simply-connected CW complexes (Serre 1951), $π_{k} (S^{n})$ is finite. For $n$ even and $k = 4 j - 1$ , $π_{k} (S^{n}) = Z \oplus π_{k} (S^{n})_{tor}$ where the torsion part is finite. $□$

This is the cleanest statement of Sullivan's solution: the rational answer is read directly off the minimal model, and the integer answer follows from finite-generation. The proof reduces Serre's 1953 spectral-sequence argument (which is intricate) to the elementary observation that the minimal model has only finitely many indecomposable generators in each degree. ^{[Sullivan 1977; Serre 1953]}

The exception $n = 2 j, k = 4 j - 1$ corresponds to the Hopf-invariant generators: $η : S^{3} \to S^{2}$ ( $j = 1$ ), $ν : S^{7} \to S^{4}$ ( $j = 2$ ), $σ : S^{15} \to S^{8}$ ( $j = 4$ ), and (for $j \neq = 1, 2, 4$ ) Whitehead-bracket generators $[ι_{2 j}, ι_{2 j}]$ . By Adams 1960, only $j = 1, 2, 4$ have Hopf-invariant-one generators; the higher cases are realised by Whitehead brackets with non-zero rational image.

Exercise 8 (hard, proof). Formality of compact Kähler manifolds (Deligne-Griffiths-Morgan-Sullivan 1975).

Sketch the proof that every simply-connected compact Kähler manifold is formal: its minimal model is determined by its cohomology ring with zero differential, no Massey-product corrections.

Hint

Use the $\partial \overset{ˉ}{\partial}$ -lemma on a compact Kähler manifold: if $α$ is $d$ -closed and $d^{c}$ -closed, and is $d$ -exact or $d^{c}$ -exact, then $α = d d^{c} β$ for some $β$ . This makes the de Rham complex rigid as a DGA.

Answer

Let $X$ be a simply-connected compact Kähler manifold. Equip the complex de Rham complex $Ω_{C}^{*} (X) = Ω^{*} (X) \otimes C$ with the Kähler structure: holomorphic / antiholomorphic decomposition $d = \partial + \overset{ˉ}{\partial}$ , the Hodge $⋆$ , and $d^{c} = i (\overset{ˉ}{\partial} - \partial) /2$ .

Step 1: $\partial \overset{ˉ}{\partial}$ -lemma. The Hodge theory of compact Kähler manifolds (Andreotti-Vesentini, Weil) gives the $\partial \overset{ˉ}{\partial}$ -lemma: if $α$ is both $d$ -closed and $d^{c}$ -closed, and $α$ is either $d$ -exact or $d^{c}$ -exact, then $α = d d^{c} β = - 2 i \partial \overset{ˉ}{\partial} β$ for some $β$ . This is the key Kähler-geometric input.

Step 2: zigzag of quasi-isomorphisms over $C$ . Define two intermediate DGAs:

A_{∙} := ker d^{c} \cap Ω_{C}^{*} (X), B_{∙} := H_{d^{c}}^{*} (Ω_{C}^{*} (X), d^{c}) .

Two natural maps:

Inclusion $i : A_{∙} ↪ Ω_{C}^{*} (X)$ — closed-under- $d^{c}$ forms include into all forms.
Projection $π : A_{∙} \to B_{∙}$ — $d$ -cocycles in $ker d^{c}$ project to $d^{c}$ -cohomology classes; $π$ kills $d$ -coboundaries that are $d^{c}$ -exact, which by $\partial \overset{ˉ}{\partial}$ -lemma is equivalent to $\partial \overset{ˉ}{\partial}$ -exact, hence to $d d^{c}$ -exact, hence in the image of $A_{∙}$ by the chain-homotopy from the $\partial \overset{ˉ}{\partial}$ -equation.

Both $i$ and $π$ are quasi-isomorphisms with respect to $d$ :

$i$ quasi-iso: by $\partial \overset{ˉ}{\partial}$ -lemma, every $d$ -closed form is $d$ -cohomologous to one in $ker d^{c}$ .
$π$ quasi-iso: $H^{*} (A, d) ≅ H^{*} (B, d^{c})$ via the Hodge structure, with $H^{*} (B, d^{c})$ tautologically equal to $B$ as a graded vector space (since $d^{c}$ vanishes on $B$ ).

Hence $Ω_{C}^{*} (X) i A_{∙} π B_{∙}$ is a zigzag of DGA quasi-isomorphisms, and the right end $B_{∙}$ is a graded $C$ -algebra with zero differential. Comparing cohomology, $B_{∙} ≅ H^{*} (X; C)$ as graded algebras.

Step 3: descend to $Q$ . The zigzag is over $C$ , but the de Rham complex is the complexification of a $Q$ -form of itself ( $A_{P L}$ at the rational level). By uniqueness of the minimal model: $$ M_X \otimes \mathbb{C} \cong M_{H^(X; \mathbb{C})}. $$ The right-hand side is the minimal model of the cohomology ring as a DGA with zero differential. Since both sides are intrinsically defined as $Q$ -algebras and the isomorphism is compatible with the cohomology structures, descent gives $$ M_X \cong M_{H^(X; \mathbb{Q})}. $$

Conclusion. The minimal model of $X$ is determined by its cohomology ring; $X$ is formal. $□$

Corollary. Two simply-connected compact Kähler manifolds with isomorphic rational cohomology rings are rationally homotopy equivalent. The Massey-product / formality dichotomy is therefore invisible on the Kähler side — a striking rigidity theorem.

The result is one of the principal applications of the Sullivan apparatus and is a foundational theorem of Hodge-theoretic geometry. ^{[Deligne-Griffiths-Morgan-Sullivan 1975]}

Bott-Tu Pass 4 — Agent E — EP3. Eight exercises on Sullivan minimal models and rational homotopy: minimal models of $S^{2 k}$ and $C P^{n}$ (drill); $Ω S^{n}$ via desuspension and Bott-Samelson; rational Hurewicz from indecomposable layer; Halperin's algorithm on the Hopf fibration; minimal model of a compact Lie group as exterior algebra on primitives; Sullivan's solution to Serre's question on $\pi_(S^n) \otimes \mathbb{Q} $; D e l i g n e - G r i f f i t h s - M or g an - S u l l i v an f or ma l i t y t h eor e m f or co m p a c t K \overset{a}{¨} h l er mani f o l d s v ia t h e$ \partial \bar{\partial}$-lemma.*

Prerequisites

03.12.06 pending
03.12.07 pending
03.04.06 pending

Tier anchors

beginner
intermediate: Bott-Tu §19 minimal-model exercises; Félix-Halperin-Thomas §10–§14 worked examples; Sullivan 1977 *Publ. IHÉS* 47; Hatcher §4.2 + §4.3
master

References

TODO_REF
Bott-Tu — Differential Forms in Algebraic Topology · §19 — minimal-model computations on $S^n$, $\mathbb{C}P^n$, Lie groups
TODO_REF
Sullivan — Infinitesimal computations in topology · Publ. Math. IHÉS 47 (1977), 269–331 — minimal-model machinery and the rational-homotopy main theorem
TODO_REF
Félix-Halperin-Thomas — Rational Homotopy Theory · Graduate Texts in Mathematics 205, Springer 2001; §10 ($A_{PL}$ functor), §12 (minimal models), §15 (Whitehead products)
TODO_REF
Deligne-Griffiths-Morgan-Sullivan — Real homotopy theory of Kähler manifolds · Inventiones 29 (1975), 245–274 — formality of compact Kähler manifolds
TODO_REF
Bott-Samelson — Application of the theory of Morse to symmetric spaces · American Journal of Mathematics 80 (1958), 964–1029 — the Bott-Samelson decomposition of $\Omega G$

Reviewer

TBD — cross-cuts §19 of Bott-Tu and §10–§15 of Félix-Halperin-Thomas

Estimated time

intermediate: 3h