03.12.50 · modern-geometry / homotopy

The Cartan model for the minimal model of a homogeneous space

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Anchor (Master): Cartan 1950 *Notions d'algèbre différentielle* (Colloque de topologie, Bruxelles); Sullivan 1977 *Infinitesimal computations in topology* (Publ. IHÉS 47); Greub-Halperin-Vanstone vol. III Ch. X-XI; Félix-Halperin-Thomas §15; Onishchik *Topology of Transitive Transformation Groups* (1994)

Intuition Beginner

A homogeneous space is a quotient $G / H$ of a symmetry group $G$ by a subgroup $H$ — a sphere, a projective space, a flag manifold. These spaces are everywhere in geometry, and a recurring question is: what are their holes, and how are those holes organised? The Cartan model answers this with a recipe that turns the cohomology of two auxiliary spaces — the classifying spaces of $G$ and of $H$ — into the exact rational-homotopy blueprint of the quotient.

The idea, in one line: the cohomology of $B G$ is a polynomial algebra on a few generators, the cohomology of $B H$ is another, and there is a natural map from the first into the second. The Cartan model packages that single map into a small algebra whose cohomology is the rational cohomology of $G / H$ , and whose generators are the rational homotopy of $G / H$ .

What makes this remarkable is that the recipe always works and always gives a formal answer: for a homogeneous space, the holes alone determine the whole rational homotopy type. No hidden higher operations. This unit channels Greub-Halperin-Vanstone and Griffiths-Morgan Ch. X.

Visual Beginner

Picture three spaces stacked. At the bottom sits the quotient $G / H$ you care about. Above it sit the two classifying spaces $B H$ and $B G$ , with an arrow $B H \to B G$ between them. The Cartan recipe runs that top arrow backwards on cohomology and reads off the bottom space.

The side panel is the algebra. The polynomial generators of $B G$ map into those of $B H$ ; the elements that get killed and the room left over assemble into the model of $G / H$ . Reading the algebra recovers both the cohomology ring and the rational homotopy groups.

Worked example Beginner

Take the even sphere $S^{2}$ , which is the homogeneous space $S O (3) / S O (2)$ . The classifying space $B S O (2)$ has cohomology a polynomial ring on one generator $x$ in degree $2$ (the Euler class). The classifying space $B S O (3)$ has cohomology a polynomial ring on one generator $p$ in degree $4$ (the first Pontryagin class), and the natural map sends $p$ to $x^{2}$ .

The Cartan recipe now says: start from the cohomology of the smaller classifying space, here $Q [x]$ with $x$ in degree $2$ , and adjoin one new generator $y$ in degree $3$ whose job is to record where the degree- $4$ class of the bigger space went. So set the differential of $y$ equal to $x^{2}$ . The result is the algebra on $x$ (degree $2$ ) and $y$ (degree $3$ ) with $d y = x^{2}$ .

This is exactly the minimal model of $S^{2}$ from the Sullivan unit. Its cohomology is a copy of the rationals in degree $0$ and degree $2$ and nothing else. The two generators tell us the rational homotopy: one in degree $2$ , one in degree $3$ . The recipe rebuilt the sphere from the cohomology of two classifying spaces and a single map between them.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $G$ is a compact connected Lie group and $H \subseteq G$ a closed connected subgroup; write $ι : H ↪ G$ . All cohomology is taken with rational coefficients, and all CDGAs are graded-commutative over $Q$ in non-negative degrees, in the sense of the Sullivan-model unit 03.12.06.

Cohomology of classifying spaces. By Borel's theorem, $H^{*} (B G; Q) = Q [y_{1}, \dots, y_{r}]$ is a polynomial algebra, where $r$ is the rank of $G$ and $de g y_{i} = 2 d_{i}$ with $d_{1}, \dots, d_{r}$ the fundamental degrees of $G$ (the exponents shifted by one). Each $y_{i}$ is the transgression of an odd primitive generator $x_{i} \in H^{*} (G; Q)$ of degree $2 d_{i} - 1$ in the universal bundle $G \to E G \to B G$ , so that $H^{*} (G; Q) = Λ (x_{1}, \dots, x_{r})$ as recalled in 03.12.06. The classifying-space functor 03.08.04 sends $ι$ to a map $B ι : B H \to B G$ , inducing the restriction homomorphism $$ i^* := (B\iota)^* : H^(BG; \mathbb{Q}) \longrightarrow H^(BH; \mathbb{Q}). $$

The Cartan-Koszul model. Form the free graded-commutative algebra $$ \Omega_{G/H} := \big( H^(BH; \mathbb{Q}) \otimes \Lambda(\bar{y}_1, \ldots, \bar{y}_r),; d \big), \qquad \deg \bar{y}_i = \deg y_i - 1 = 2 d_i - 1, $$ where $H^(BH; \mathbb{Q})$ carries zero differential and the exterior generators satisfy $$ d \bar{y}_i = i^(y_i) \in H^(BH; \mathbb{Q}). $$ This is the Cartan model (equivalently the relative Koszul complex of the homomorphism $i^{*}$ ). The differential is extended as a derivation; that $d^{2} = 0$ is immediate, since $d^{2} \overset{y}{ˉ}_{i} = d (i^{*} (y_{i})) = 0$ as $i^{*} (y_{i})$ lies in the differential-free factor.

Equivalently via primitives. Because each $y_{i}$ is the transgression of the primitive $x_{i}$ , the generator $\overset{y}{ˉ}_{i}$ may be identified with the image of $x_{i}$ ; the Cartan model is then $(H^{*} (B H) \otimes Λ P_{G}, d)$ , where $Λ P_{G} = Λ (x_{1}, \dots, x_{r})$ is the exterior algebra on the primitives $P_{G}$ of $H^{*} (G; Q)$ , and $d$ carries the primitive of degree $2 d_{i} - 1$ to the image in $H^{*} (B H)$ of the corresponding universal class — the Leray-Serre transgression 03.13.02 of the bundle $G \to G / H \to B H$ .

Pure Sullivan algebras and rank. A Sullivan algebra of the form $(Q [even] \otimes Λ (odd), d)$ with $d (even) = 0$ and $d (odd) \subseteq Q [even]$ is called pure. The Cartan model is pure. The homogeneous space $G / H$ is of equal rank if $rk H = rk G$ , equivalently $H$ contains a maximal torus of $G$ ; otherwise it has non-equal rank. This dichotomy governs the homology and is taken up in the next two sections.

Key theorem with proof Intermediate+

Theorem (Cartan 1950; Sullivan-Vigué 1976 — the Koszul model of $G / H$ ). Let $H \subseteq G$ be a closed connected subgroup of a compact connected Lie group. Then the Cartan model $\big( H^(BH) \otimes \Lambda(\bar{y}_1, \ldots, \bar{y}_r),, d\bar{y}i = i^*(y_i) \big) $i s a co mm u t a t i v ecoc hainm o d e l f or$ G/H $, t ha t i s, i t i s q u a s i - i so m or p hi c t o$ A{PL}(G/H) \otimes \mathbb{Q} $. C o n se q u e n tl y$ G/H$ is formal, and* $$ H^(G/H; \mathbb{Q}) \cong H\big( H^(BH) \otimes \Lambda(\bar{y}_1, \ldots, \bar{y}_r),, d \big). $$

Proof. Consider the fibration $G / H B H B ι B G$ , obtained because $G / H$ is the homotopy fibre of $B ι$ (the fibre of $B H \to B G$ is $G / H$ , since the fibre of $E G / H \to E G / G = B G$ is $G / H$ and $E G / H ≃ B H$ ). The pullback fibration $G \to E G / H \to B H$ identifies $E G / H ≃ B H$ with fibre $G$ .

Apply Sullivan's relative-minimal-model machinery (the Halperin twisted-tensor construction recalled in 03.12.06, Exercise 6) to the fibration $G \to B H^{'} \to G / H$ — more directly, build a model of $G / H$ as the fibre of $B ι$ . A model of $B G$ is the polynomial algebra $(Q [y_{1}, \dots, y_{r}], 0)$ and a model of $B H$ is $(H^{*} (B H), 0)$ , both having zero differential because classifying spaces of compact groups are formal with polynomial cohomology. The relative model of the map $B ι$ is obtained by adjoining to $H^{*} (B H)$ one generator $\overset{y}{ˉ}_{i}$ of degree $de g y_{i} - 1$ for each polynomial generator $y_{i}$ upstairs, with $d \overset{y}{ˉ}_{i}$ equal to the image $i^{*} (y_{i})$ of $y_{i}$ under the map of models. The fibre model is then the quotient setting the base generators to zero, which returns precisely $$ \big( H^(BH) \otimes \Lambda(\bar{y}_1, \ldots, \bar{y}_r),, d\bar{y}_i = i^(y_i) \big). $$

The transgression identification with the Leray-Serre spectral sequence 03.13.02 of $G \to G / H \to B H$ confirms that $d \overset{y}{ˉ}_{i}$ is the transgression $τ (x_{i})$ of the primitive $x_{i}$ , equal to $i^{*} (y_{i})$ in $H^{2 d_{i}} (B H)$ . Since the model is built from cohomology rings with zero differentials joined only through the transgression, it has no further perturbations, hence is a model of $G / H$ . Formality follows because this model maps quasi-isomorphically onto its own cohomology: the pure differential admits the standard bigrading by even-generator word length, and the associated spectral sequence of the pure model degenerates to exhibit the quasi-isomorphism to $(H^{*} (G / H), 0)$ . ^{[Greub-Halperin-Vanstone vol. III; Félix-Halperin-Thomas §15]} $□$

Bridge. This construction builds toward the equal-rank computation and Poincaré-polynomial formulae of the Advanced section, and the same transgression mechanism appears again in 03.06.20, where the Borel-Hirzebruch presentation of $H^{*} (G / T)$ is exactly the equal-rank shadow of the Koszul homology computed here. The foundational reason the model is so rigid is exactly the formality of classifying spaces: $B G$ and $B H$ have polynomial cohomology with zero differential, so the only data is the single map $i^{*}$ , and this is dual to the picture in which the regular sequence $i^{*} (y_{1}), \dots, i^{*} (y_{r})$ inside $H^{*} (B H)$ carries the whole computation. The central insight is that a homogeneous space is the Koszul complex of one ring map; putting these together, the rational homotopy theory of $G / H$ generalises the Sullivan minimal model 03.12.06 of $G$ itself to the relative setting of the pair $(G, H)$ .

Exercises Intermediate+

Exercise 1 (easy, symbolic). The model of $S^{2 n}$ .

Realise the even sphere as $S^{2 n} = S O (2 n + 1) / S O (2 n)$ and write its Cartan model. Confirm it recovers $H^{*} (S^{2 n}; Q)$ .

Hint

$H^{*} (B S O (2 n); Q)$ contains the Euler class $e$ in degree $2 n$ ; the Pontryagin generator of $B S O (2 n + 1)$ restricting to $e^{2}$ is the relevant transgression.

Answer

The cohomology $H^{*} (B S O (2 n); Q)$ has the Euler class $e$ of degree $2 n$ among its generators, and the top Pontryagin class $p_{n}$ of $B S O (2 n + 1)$ restricts to $e^{2}$ . After the cancellations among the lower Pontryagin classes (which restrict to themselves), the surviving Koszul data is the model $Λ (x_{2 n}, y_{4 n - 1})$ with $de g x = 2 n$ , $de g y = 4 n - 1$ , $d x = 0$ , $d y = x^{2}$ . Its cohomology is $Q$ in degrees $0$ and $2 n$ — matching $S^{2 n}$ . The generators give $π_{2 n} \otimes Q = Q$ and $π_{4 n - 1} \otimes Q = Q$ , agreeing with the sphere model of 03.12.06.

Exercise 2 (medium, symbolic). Complex projective space.

Write $C P^{n} = U (n + 1) / (U (n) \times U (1))$ and compute its Cartan model. Recover $H^{*} (C P^{n}; Q) = Q [x] / (x^{n + 1})$ .

Hint

$H^{*} (B U (k); Q) = Q [c_{1}, \dots, c_{k}]$ ; the map sends the Chern classes of $U (n + 1)$ to the products of Chern classes of the two factors.

Answer

Write $H^{*} (B (U (n) \times U (1))) = Q [a_{1}, \dots, a_{n}, t]$ with $de g a_{i} = 2 i$ , $de g t = 2$ . The restriction $i^{*}$ sends the total Chern class of $U (n + 1)$ to $(1 + a_{1} + \dots + a_{n}) (1 + t)$ , so $i^{*} (c_{k})$ is the degree- $2 k$ part. The full Koszul model adjoins exterior generators dual to $c_{1}, \dots, c_{n + 1}$ . After eliminating the $a_{i}$ against $i^{*} (c_{1}), \dots, i^{*} (c_{n})$ (a regular sequence expressing each $a_{i}$ in terms of $t$ ), only the relation from $i^{*} (c_{n + 1}) = t \cdot a_{n} \mapsto t^{n + 1}$ survives, giving $Λ (x, z)$ with $de g x = 2$ , $de g z = 2 n + 1$ , $d z = x^{n + 1}$ . Cohomology: $Q [x] / (x^{n + 1})$ , matching $C P^{n}$ and the model of 03.12.06.

Exercise 3 (medium, symbolic). The full flag manifold.

For the flag manifold $U (n) / T^{n}$ , identify the Cartan model and use the equal-rank case to compute the Poincaré polynomial.

Hint

Here $H = T^{n}$ is a maximal torus, so $rk H = rk G$ and all exterior generators cancel. Use the Borel presentation 03.06.20.

Answer

Since $T^{n}$ is a maximal torus, the model is equal rank: $H^{*} (B T^{n}) = Q [t_{1}, \dots, t_{n}]$ and $i^{*}$ sends the generators $c_{k}$ of $H^{*} (B U (n))$ to the elementary symmetric polynomials $e_{k} (t)$ . These form a regular sequence, every exterior generator $\overset{c}{ˉ}_{k}$ is used to kill $e_{k} (t)$ , and the Koszul homology collapses to $$ H^*(U(n)/T^n; \mathbb{Q}) = \mathbb{Q}[t_1, \ldots, t_n] / (e_1, \ldots, e_n), $$ the Borel presentation of 03.06.20, the coinvariant algebra of the symmetric group $S_{n}$ . Its Poincaré polynomial is the Gaussian factorial $$ P_t = \prod_{k=1}^{n} \frac{1 - t^{2k}}{1 - t^2} = [n]{t^2}!, $$ and evaluating at $t = 1$ gives $\chi = n! = |W{U(n)}| / |W_{T^n}|$, concentrated in even degrees.

Exercise 4 (medium, proof). Equal rank means even cohomology.

Show that if $rk H = rk G$ then $H^{*} (G / H; Q)$ is concentrated in even degrees and equals $H^{*} (B H) / (i^{*} H^{+} (B G))$ .

Hint

Equal rank makes $i^{*} (y_{1}), \dots, i^{*} (y_{r})$ a regular sequence in $H^{*} (B H)$ , so the Koszul complex has homology only in degree zero of the exterior grading.

Answer

When $rk H = rk G$ , both $H^{*} (B G)$ and $H^{*} (B H)$ are polynomial on $r$ generators, and $i^{*}$ makes $H^{*} (B H)$ a finitely generated free module over $i^{*} H^{*} (B G)$ (Borel). The elements $i^{*} (y_{1}), \dots, i^{*} (y_{r})$ then form a regular sequence in the Cohen-Macaulay ring $H^{*} (B H)$ . The Cartan model is the Koszul complex on this regular sequence tensored over $H^{*} (B H)$ ; a regular sequence has Koszul homology concentrated in exterior degree zero, equal to the quotient $H^{*} (B H) / (i^{*} y_{1}, \dots, i^{*} y_{r}) = H^{*} (B H) / (i^{*} H^{+} (B G))$ . Since $H^{*} (B H)$ lives in even degrees and the relations are even, the quotient is concentrated in even degrees. $□$

Exercise 5 (hard, proof). Euler characteristic in the equal-rank case.

Prove that for an equal-rank pair, $χ (G / H) = ∣ W_{G} ∣/∣ W_{H} ∣ > 0$ , where $W_{G}, W_{H}$ are the Weyl groups.

Hint

Use the Hopf-Leray theorem: the Poincaré series of the coinvariant algebra is the quotient of the two graded-regular-sequence Poincaré series. Evaluate the limit at $t \to 1$ by L'Hôpital, or use the degree product.

Answer

By Exercise 4, $H^{*} (G / H; Q) = H^{*} (B H) / (i^{*} H^{+} (B G))$ , a complete intersection. Its Poincaré series is $$ P_{G/H}(t) = \frac{\prod_{i=1}^{r} (1 - t^{2 d_i^G})}{\prod_{j=1}^{r} (1 - t^{2 d_j^H})}, $$ where $d_{i}^{G}$ are the fundamental degrees of $G$ and $d_{j}^{H}$ those of $H$ (here $r = rk G = rk H$ ). Since cohomology is even, the Euler characteristic is $P_{G / H} (1)$ . Writing each factor $1 - t^{2 d} = (1 - t^{2}) (1 + t^{2} + \dots + t^{2 d - 2})$ and cancelling the $r$ common factors of $(1 - t^{2})$ between numerator and denominator, the limit $t \to 1$ gives $$ \chi(G/H) = \frac{\prod_i d_i^G}{\prod_j d_j^H} = \frac{|W_G|}{|W_H|}, $$ using the identity $∣ W ∣ = \prod_{k} d_{k}$ (product of the degrees equals the Weyl-group order). This is a positive integer, the number of $W_{H}$ -cosets in $W_{G}$ . $□$

Exercise 6 (hard, proof). Non-equal rank forces zero Euler characteristic.

Show that if $rk H < rk G$ then $χ (G / H) = 0$ , and exhibit a surviving odd-degree cohomology class.

Hint

Now $i^{*} (y_{1}), \dots, i^{*} (y_{r})$ cannot be a regular sequence: there are $r = rk G$ of them but only $rk H < r$ algebraically independent elements available in $H^{*} (B H)$ .

Answer

Let $s = rk H < r = rk G$ . The ring $H^{*} (B H)$ has Krull dimension $s$ , so among the $r$ elements $i^{*} (y_{1}), \dots, i^{*} (y_{r})$ at most $s$ can be part of a regular sequence; at least $r - s \geq 1$ of them are redundant. Each redundancy contributes a cocycle of the form $\overset{y}{ˉ}_{k}$ (or a correction $\overset{y}{ˉ}_{k} - θ$ ) whose differential is already in the ideal generated by the others, so it survives to an odd-degree cohomology class — concretely, the highest such $\overset{y}{ˉ}_{k}$ pairs an odd primitive of $G$ that has no transgressive target in $H^{*} (B H)$ . The existence of a nonzero odd class forces $χ (G / H) = \sum_{k} (- 1)^{k} dim H^{k} = 0$ , since the pure model is balanced: the Koszul homology of a non-regular sequence carries equal even and odd contributions in the resolution, cancelling the alternating sum. The sphere $S^{2 n} = S O (2 n + 1) / S O (2 n)$ is the model case, with surviving class $y$ in degree $4 n - 1$ . $□$

Advanced results Master

The pure Sullivan algebra and its bigrading. The Cartan model is a pure Sullivan algebra: $(Q [V_{ev}] \otimes Λ V_{odd}, d)$ with $d V_{ev} = 0$ and $d V_{odd} \subseteq Q [V_{ev}]$ , where $V_{ev}$ is spanned by the even generators of $H^{*} (B H)$ and $V_{odd}$ by the $\overset{y}{ˉ}_{i}$ . The lower grading by exterior word length in $V_{odd}$ makes $d$ homogeneous of degree $- 1$ , so the model is a chain complex of free $Q [V_{ev}]$ -modules — exactly the Koszul complex of the sequence $(i^{*} y_{1}, \dots, i^{*} y_{r})$ . Félix-Halperin-Thomas §15 develop the homotopy theory of pure models and prove that $G / H$ , being pure, is formal and elliptic: both $H^{*} (G / H; Q)$ and $π_{*} (G / H) \otimes Q$ are finite-dimensional.

The rational dichotomy. Equal rank ( $rk H = rk G$ , e.g. $G / T$ , the flag manifolds, and more generally $G / H$ with $H$ of maximal rank) makes $(i^{*} y_{i})$ a regular sequence: the Koszul homology concentrates in exterior degree zero, $H^{*} (G / H)$ is even, and $χ (G / H) = ∣ W_{G} ∣/∣ W_{H} ∣ > 0$ . Non-equal rank ( $rk H < rk G$ ) leaves $r - rk H$ odd generators surviving and forces $χ (G / H) = 0$ . The even spheres are equal rank ( $S O (2 n + 1) / S O (2 n)$ has $rk S O (2 n + 1) = n = rk S O (2 n)$ , $χ = 2$ ), while the odd spheres $S^{2 n + 1} = S O (2 n + 2) / S O (2 n + 1)$ are non-equal rank ( $rk S O (2 n + 2) = n + 1 > n$ , $χ = 0$ ), exactly tracking even versus odd Euler characteristic.

Formal dimension and Poincaré duality. The pure model gives the formal dimension (top nonzero cohomological degree) of $G / H$ as $dim (G / H) = \sum_{j} (2 d_{j}^{H} - 1) - \sum$ , more transparently $dim G - dim H$ . For equal-rank pairs the formal dimension equals the manifold dimension and Poincaré duality holds visibly in the complete-intersection quotient. The Poincaré polynomial is the ratio of degree-product factors derived in Exercises 3 and 5.

Relation to the Weil model and equivariant cohomology. The Cartan model is the $H = e$ degenerate case of the broader Cartan-Weil model of equivariant cohomology: $H_{G}^{*} (pt) = H^{*} (B G)$ , and the model of $G / H$ arises as $H_{H}^{*} (G) = H_{G}^{*} (G / H \times E G)$ collapsing. Greub-Halperin-Vanstone vol. III develop this from the algebra of a reductive pair $(g, h)$ , with the Cartan model the cochain incarnation of relative Lie-algebra cohomology with coefficients in the symmetric algebra of the base.

Synthesis. Putting these together, the Cartan model is the foundational reason homogeneous spaces are simultaneously formal, elliptic, and completely computable: the entire rational invariant is the Koszul homology of one ring map $i^{*}$ , and this is exactly the relative refinement of the Sullivan picture 03.12.06 for the pair $(G, H)$ . The central insight is that the equal-rank/non-equal-rank dichotomy generalises the even/odd-sphere split and is dual to the Weyl-group index $∣ W_{G} ∣/∣ W_{H} ∣$ ; the regular-sequence criterion appears again in the Borel-Hirzebruch presentation 03.06.20, which is precisely the equal-rank shadow, and builds toward the symmetric-space theory 07.04.07 where the geometry of $G / H$ refines its rational homotopy. The bridge is the transgression of 03.13.02: the differential $d \overset{y}{ˉ}_{i} = i^{*} (y_{i})$ is nothing but the Leray-Serre transgression read algebraically.

Full proof set Master

Proposition (formality and ellipticity of homogeneous spaces). Let $H \subseteq G$ be a closed connected subgroup of a compact connected Lie group, with $r = rk G$ and $s = rk H$ . Then $G / H$ is formal and elliptic, with $\dim_{\mathbb{Q}} \pi_(G/H) \otimes \mathbb{Q} = \dim_{\mathbb{Q}}(V_{\mathrm{ev}} \oplus V_{\mathrm{odd}}) < \infty $, an d t h e h o m o t o p y E u l er c ha r a c t er i s t i c$ \chi_\pi(G/H) = \dim V_{\mathrm{ev}} - \dim V_{\mathrm{odd}} = s - r \leq 0$.*

Proof. The Cartan model $(Q [V_{ev}] \otimes Λ V_{odd}, d)$ is a finite-dimensional-in-each-degree Sullivan model of $G / H$ by the Key Theorem; minimalisation does not change the indecomposable spaces' total dimension because the model is already generated by finitely many elements ( $dim V_{ev} = s$ generators of $H^{*} (B H)$ , $dim V_{odd} = r$ generators $\overset{y}{ˉ}_{i}$ ). Hence $π_{*} (G / H) \otimes Q$ is finite-dimensional and concentrated in finitely many degrees, so $G / H$ is rationally elliptic.

For formality: the pure model carries the bigrading by exterior word length described above, and the differential lowers this grading by one. The induced spectral sequence of the bigrading converges, and on $E_{1}$ the differential is the Koszul differential of the sequence $(i^{*} y_{i})$ . Choosing a maximal regular subsequence and splitting off the redundant generators, the model decomposes up to quasi-isomorphism as a tensor product of a complete-intersection even algebra (formal, zero differential on cohomology) and an exterior algebra on the surviving odd generators (zero differential). A tensor product of formal CDGAs is formal, so $G / H$ is formal. ^{[Sullivan 1977; Félix-Halperin-Thomas §15]}

The homotopy Euler characteristic is $χ_{π} = dim V_{ev} - dim V_{odd} = s - r$ , which is $\leq 0$ with equality precisely in the equal-rank case; this matches the rational-ellipticity dichotomy $χ_{π} \leq 0$ always, with $χ_{π} = 0 ⟺ χ (G / H) > 0$ (Friedlander-Halperin). $□$

Proposition (Koszul homology computes $H^{*} (G / H)$ ). With notation as above, $H^(G/H; \mathbb{Q}) $i s i so m or p hi c a s a g r a d e d r in g t o t h e h o m o l o g y o f t h eK osz u l co m pl e x o f t h ese q u e n ce$ (i^* y_1, \ldots, i^* y_r) $o v er$ H^(BH; \mathbb{Q}) $. I n t h ee q u a l - r ank c a se t hi s i s t h eco m pl e t e in t er sec t i o n$ H^(BH)/(i^* y_1, \ldots, i^* y_r)$.*

Proof. The Cartan model is by definition the Koszul complex $K_{∙} (i^{*} y_{1}, \dots, i^{*} y_{r}; H^{*} (B H))$ , with the exterior generators $\overset{y}{ˉ}_{i}$ providing the Koszul exterior algebra and $d \overset{y}{ˉ}_{i} = i^{*} y_{i}$ the Koszul differential. Its cohomology is the model's cohomology, equal to $H^{*} (G / H; Q)$ by the Key Theorem. When $s = r$ the sequence is regular (Exercise 4), so the higher Koszul homology vanishes and only $H_{0} = H^{*} (B H) / (i^{*} y_{i})$ survives, a complete intersection of the predicted Poincaré series. When $s < r$ , the Koszul homology in higher exterior degree is nonzero and contributes the surviving odd classes of Exercise 6. $□$

Connections Master

Sullivan minimal models 03.12.06 — the Cartan model is the relative version of the minimal-model construction. Where 03.12.06 gives the exterior-algebra model $Λ (x_{1}, \dots, x_{r})$ of the compact Lie group $G$ itself, the present unit lifts this to the pair $(G, H)$ : the same primitives $x_{i}$ reappear as the exterior generators $\overset{y}{ˉ}_{i}$ , now with nonzero differential $i^{*} (y_{i})$ instead of zero. Connection type: foundation-of.
Borel-Hirzebruch cohomology of $G / T$ 03.06.20 — the equal-rank case with $H = T$ a maximal torus recovers exactly the Borel presentation $H^{*} (G / T; Q) = Q [t_{1}, \dots, t_{r}] / (positive-degree W -invariants)$ . That unit gives the cohomology ring; this unit explains it as the Koszul homology of the regular sequence $i^{*} (y_{i}) = e_{i} (t)$ and supplies the minimal model that 03.06.20 never constructs. Connection type: equivalence (the ring presentation is the exterior-degree-zero Koszul homology).
Classifying space 03.08.04 — the entire construction is driven by the map $B ι : B H \to B G$ and its induced restriction $i^{*}$ on Borel cohomology. The polynomial structure of $H^{*} (B G; Q)$ and the formality of classifying spaces are what make the Cartan model so rigid. Connection type: foundation-of.
Leray-Serre spectral sequence 03.13.02 — the differential $d \overset{y}{ˉ}_{i} = i^{*} (y_{i})$ is the algebraic incarnation of the transgression in the Leray-Serre spectral sequence of the bundle $G \to G / H \to B H$ . The Koszul model is the minimal-model repackaging of that spectral sequence, which here degenerates because the only differentials are the transgressions. Connection type: foundation-of.
Riemannian symmetric spaces 07.04.07 — symmetric spaces $G / H$ are the geometrically distinguished homogeneous spaces, and the Cartan model computes their rational cohomology directly from the symmetric pair. Compact symmetric spaces of equal rank (such as the complex and quaternionic Grassmannians) have even cohomology and positive Euler characteristic by the dichotomy proved here. Connection type: bridging-theorem.

Historical & philosophical context Master

The construction originates with Élie Cartan's son Henri Cartan, whose 1950 Brussels colloquium lectures Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie introduced the differential-algebraic apparatus — the Weil algebra, the Cartan model, the transgression — for computing the cohomology of homogeneous spaces and of spaces acted on by a Lie group ^{[Cartan 1950]}. Cartan's model predated the language of minimal models by a quarter century, yet it is in hindsight the first explicit Sullivan-type construction: a small commutative differential graded algebra, built from the cohomology of classifying spaces, whose cohomology is that of the quotient. The colloquium, organised in the aftermath of the war, also carried Leray's foundational spectral-sequence papers; the transgression that Cartan used and the transgression of the Leray spectral sequence 03.13.02 are the same map, a coincidence the two authors recognised at once.

The synthesis with rational homotopy theory came through Sullivan's 1977 Infinitesimal computations in topology and the parallel work of Greub, Halperin, and Vanstone, whose three-volume Connections, Curvature, and Cohomology (1972-1976) gave the definitive algebraic treatment of the cohomology of homogeneous spaces via the reductive pair $(g, h)$ ^{[Greub-Halperin-Vanstone vol. III]}. Halperin's later work with Vigué-Poirrier established that homogeneous spaces are formal, and the structural theory of pure and elliptic Sullivan algebras in Félix-Halperin-Thomas turned the Cartan model into a chapter of rational homotopy theory proper. The philosophical lesson is one of rigidity: among all spaces, homogeneous spaces are those whose rational homotopy type is purely a matter of linear algebra over the cohomology of two classifying spaces, with no room for the higher-order phenomena (Massey products, non-formality) that complicate the general theory.

Bibliography Master

Cartan, H., "Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie", in Colloque de topologie (espaces fibrés), Bruxelles 1950, Masson, Paris, 1951, 15–27.
Sullivan, D., "Infinitesimal computations in topology", Publications mathématiques de l'I.H.É.S. 47 (1977), 269–331.
Greub, W., Halperin, S., & Vanstone, R., Connections, Curvature, and Cohomology, Volume III: Cohomology of Principal Bundles and Homogeneous Spaces, Academic Press, 1976.
Griffiths, P. & Morgan, J., Rational Homotopy Theory and Differential Forms, Progress in Mathematics 16, Birkhäuser, 1981. Ch. X §10.2.
Félix, Y., Halperin, S., & Thomas, J.-C., Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer, 2001. §15 (homogeneous spaces and pure Sullivan algebras).
Onishchik, A. L., Topology of Transitive Transformation Groups, Johann Ambrosius Barth, Leipzig, 1994.
Borel, A., "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts", Annals of Mathematics 57 (1953), 115–207.

Prerequisites

03.12.06
03.08.04
03.06.20
03.13.02

Tier anchors

beginner: Greub-Halperin-Vanstone *Connections, Curvature, and Cohomology* vol. III (Academic Press 1976) Ch. XI — the Weil/Cartan model of a homogeneous space, read informally
intermediate: Griffiths-Morgan *Rational Homotopy Theory and Differential Forms* (Birkhäuser 1981) Ch. X §10.2; Félix-Halperin-Thomas *Rational Homotopy Theory* (Springer GTM 205, 2001) §15 (homogeneous spaces, the Koszul model and pure Sullivan algebras)
master: Cartan 1950 *Notions d'algèbre différentielle* (Colloque de topologie, Bruxelles); Sullivan 1977 *Infinitesimal computations in topology* (Publ. IHÉS 47); Greub-Halperin-Vanstone vol. III Ch. X-XI; Félix-Halperin-Thomas §15; Onishchik *Topology of Transitive Transformation Groups* (1994)

References

Cartan — Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie · Colloque de topologie (espaces fibrés), Bruxelles 1950, 15–27 — the originating Weil/Cartan-model construction
Sullivan — Infinitesimal computations in topology · Publ. Math. IHÉS 47 (1977), 269–331 — minimal models; formality of homogeneous spaces
Greub-Halperin-Vanstone — Connections, Curvature, and Cohomology, Volume III · Academic Press 1976; Ch. X–XI — cohomology of homogeneous spaces, the Cartan/Koszul model and the reductive pair (G,H)
Griffiths-Morgan — Rational Homotopy Theory and Differential Forms · Progress in Mathematics 16, Birkhäuser 1981; Ch. X §10.2 — minimal models of Lie groups and homogeneous spaces
Félix-Halperin-Thomas — Rational Homotopy Theory · Graduate Texts in Mathematics 205, Springer 2001; §15 — homogeneous spaces, pure Sullivan algebras, formal dimension

Estimated time

beginner: 18m
intermediate: 48m
master: 85m