Rational Homotopy of Manifolds

• Rational homotopy theory is an algebraic framework that models manifolds via Sullivan and Quillen constructions, encoding key cohomological features.
• It relates formality and Koszul duality to manifold topology, using invariants like Massey and Pentagonal tensors to capture structure.
• The theory informs realization problems and classification by applying computational invariants and surgery techniques to assess smooth manifold types.

Rational homotopy theory of manifolds is the study of manifold homotopy types via algebraic models over $\mathbb{Q}$, emphasizing cohomological invariants and their explicit relationship to the topology and geometry of underlying smooth, topological, or algebraic manifolds. The development of minimal Sullivan and Quillen models provides canonical algebraic objects encoding rational information, revealing deep connections between formality, realization obstructions, Koszul structures, and the behavior of characteristic classes, especially in high dimensions and for manifolds with prescribed (truncated polynomial) cohomology rings (Fowler et al., 2014).

1. Algebraic Models: Sullivan and Quillen Framework

Let $M$ be a simply connected, smooth closed manifold. The Sullivan minimal model $(\Lambda V, d)$ is a free commutative differential graded algebra (CDGA) over $\mathbb{Q}$, generated by graded vector space $V$ (degree $>\!0$), with differential determined by the cohomology ring structure and higher-order Massey products. Rational homotopy groups are computed by the duals: $(sV)^* \cong \pi_*(M) \otimes \mathbb{Q}$ (Suciu, 2022). Formality occurs when $(\Lambda V, d) \simeq (H^*(M; \mathbb{Q}), 0)$, so the rational homotopy type is preserved by the cohomology algebra.

Quillen's Lie-algebraic models encode the rational homotopy groups, and via Koszul duality, formality properties can be seen as quadraticity conditions on the differentials or presentations of the model (Berglund et al., 2023). For certain classes, coformality arises—cohomology algebra is Koszul, which is equivalent to cohomology becoming free as a coalgebra under suitable grading.

2. Formality, Koszul Duality, and Characterization

Manifolds with "simple" rational cohomology, such as spheres ($H^*(S^n; \mathbb{Q}) = \mathbb{Q}[x]/(x^2)$, $|x| = n$) and projective spaces with truncated polynomial rings, serve as models for classical formality: higher Massey products vanish, and their Sullivan models have trivial differential (Berglund et al., 2023). The Koszul criterion tightly connects the quadratic presentation of cohomology and the structure of associated $L_\infty$ and $C_\infty$ models.

If $H^*(M)$ is a quotient of a free algebra by a regular sequence (as for homogeneous spaces, symmetric spaces, compact Kähler manifolds), then $M$ is formal (Suciu, 2022). For $M$ of the form $G/K$ (compact group quotients), Poincaré series encode the generator degrees in $H^*(M)$ and thereby control the rational homotopy groups (Xu-an et al., 2013).

3. Realization Problems and Arithmetic Obstructions

Given a rational Poincaré duality algebra $\mathcal{A}$, the central question is whether there exists a smooth manifold $M$ such that $H^*(M; \mathbb{Q}) = \mathcal{A}$ (Fowler et al., 2014). The Hirzebruch signature formula gives a fundamental obstruction—one must find Pontryagin classes $p_i \in H^{4i}(A)$ and a fundamental class $\mu$ so that

$$\langle L_k(p_1,...,p_k), \mu \rangle = \sigma(A) \in \mathbb{Z},$$

where $L_k$ is the $k$th term of the $L$-polynomial built from Bernoulli numbers.

Specific arithmetic phenomena arise for truncated polynomial algebras:

• For $\mathbb{Q}[x]/\langle x^3 \rangle$, with $|x|=2k$, existence constraints force the dimension $n=8(2^a + 2^b)$. Small cases ($n=32$) lack 2-connectedness or spin structures.
• Rational octonionic projective spaces $\mathbb{Q}[x]/\langle x^{m+1} \rangle$, $|x|=8$, yield existence theorems for odd $m>2$ via rational surgery, with vanishing signature and Pontryagin numbers.
• High-dimensional E$_8$ plumbing manifolds exhibit failure of rational realization when k exceeds binary weight thresholds, linking to signature denominators involving irregular primes.

These results demonstrate the intertwining of number theoretic properties, rational surgery, and geometric realization, uncovering precise dimensions and connectivity restrictions for the existence of smooth models with specified rational cohomology (Fowler et al., 2014).

4. Homotopy Invariants: Bianchi–Massey and Pentagonal Tensors

For highly connected manifolds ($n$-connected, dimension up to $5n-3$), Crowley–Nordström defined the Bianchi–Massey tensor $F$, a linear map from a subquotient of $H^*(M)^{\otimes 4}$, strictly refining Massey triple product information. The rational homotopy type is determined by $H^*(M)$ and $F_M$; vanishing of $F$ implies formality (Crowley et al., 2015).

For 8-manifolds, Nagy–Nordström introduced a pentagonal Massey tensor $P$ (quintic tensor), with vanishing characterizing formality together with $F$ in the range $m \leq 5n-2$ (Nagy et al., 2021). These serve as a complete set of invariants beyond the cohomology ring, especially controlling the classification and non-formal cases of simply-connected 7- and 8-manifolds.

The table below summarizes tensor invariants and their dimensional ranges:

Tensor Invariant	Dimensional Range	Vanishing Characterizes Formality
Bianchi–Massey ($F$)	$m \leq 5n-3$	Yes
Pentagonal Massey ($P$)	$m \leq 5n-2$	$F{=}0$, $P{=}0$ sufficient

Iterated generalizations are anticipated for higher-order tensors controlling formality near $m \sim (r+1)n-2$ (Nagy et al., 2021).

5. Manifold Constructions: Blow‐Ups, Configuration Spaces, and Homogeneous Examples

Blow-up constructions in symplectic and algebraic geometry undergo rational homotopy decompositions after looping, yielding rational equivalences: $$\Omega \widetilde{N} \simeq_\mathbb{Q} S^1 \times \Omega \mathrm{Ne} \times \Omega \Sigma^2 F,$$ where $F$ is the homotopy fibre of the sphere-bundle boundary inclusion. Rational dichotomy (elliptic/hyperbolic) is refined by the splitting of the homotopy types of the components; thus, the rational type of a blow-up reflects its base and fibre structure (Huang et al., 2023).

The rational model of configuration space $F(M,2)$ for a simply connected closed manifold $M$ depends only on the rational homotopy type of $M$ in even dimensions. In odd dimensions, a family of CDGA models indexed by $x \in H^{n-2}(M; \mathbb{Q})$ completely parametrizes the rational homotopy types, with the untwisted case ($x=0$) holding for even-dimensional and 2-connected $M$ (Bulens, 2015).

The rational cohomology and formality of indefinite Kac–Moody groups $G$ and flag manifolds $F = G/B$ are controlled by their Poincaré series expansions and the calculation of generators and relations in $H^*(G; \mathbb{Q})$; rational formality is assured by the vanishing of Massey products and the absence of higher differentials (Xu-an et al., 2013).

6. Characteristic Classes, Operadic Decompositions, and Rational Homeomorphism Groups

Pontryagin–Weiss classes are detected using rational homotopy decompositions of classifying spaces for homeomorphism groups. After looping, the stabilisation map ${\rm Top}(d) \rightarrow {\rm Top}$ splits rationally for $d \geq 6$, giving infinite-dimensional rational homotopy groups and nontrivial evaluations of stable Pontryagin classes on bundles over spheres (Krannich et al., 28 Apr 2025). Operadic models (pro-operad completions of $E_d$) and tensor product structures clarify the splitting and additivity properties in high-dimensional embedding calculus contexts.

For automorphism groups of highly connected manifolds, graph-complex structures (Kontsevich’s Lie graph complex) deliver direct algebraic models of their stable rational cohomology, incorporating the interplay of symmetric operadic brackets and Miller–Morita–Mumford classes (Berglund et al., 2023).

7. Classification, Applications, and Open Directions

Classification of simply connected manifolds up to rational homotopy equivalence, especially in dimensions 7 and 8, is now algorithmically governed by cohomological data and explicit tensor invariants ($F$, $P$). Rational maps between formal homogeneous spaces (e.g., Grassmannians, flag varieties) are classified by CDGA homomorphisms, which are often forced to vanish in positive degrees, quantifying the scarcity of nontrivial continuous maps in these regimes (Chakraborty et al., 2015).

Rational homotopy theory accurately predicts finiteness and formality properties, elliptic/hyperbolic dichotomies, and the geometric content of cohomology jump loci. Future extensions involve higher-order tensor obstructions, the geometric realization of algebraic invariants (especially in the context of truncated polynomial algebras and non-integral models), and the possible classification of rational manifold types by graph-complex and Koszul duality parameters.

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References:

• (Fowler et al., 2014) Smooth manifolds with prescribed rational cohomology ring
• (Nagy et al., 2021) Rational homotopy and simply-connected 8-manifolds
• (Crowley et al., 2015) The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3
• (Huang et al., 2023) Homotopy of blow ups after looping
• (Berglund et al., 2023) Higher structures in rational homotopy theory
• (Krannich et al., 28 Apr 2025) Pontryagin-Weiss classes and a rational decomposition of spaces of homeomorphisms
• (Xu-an et al., 2013) Poincaré series and rational cohomology rings of Kac-Moody groups and their flag manifolds
• (Bulens, 2015) Rational model of the configuration space of two points in a simply connected closed manifold
• (Suciu, 2022) Formality and finiteness in rational homotopy theory
• (Chakraborty et al., 2015) Rational homotopy of maps between certain complex Grassmann manifolds
• (Huang, 2021) Loop homotopy of $6$-manifolds over $4$-manifolds