The rational-homotopy-theory tag has no usage guidance.

**33**

votes

**0**answers

1k views

### Finite complexes whose homotopy groups are not “finitely generated”

I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that ...

**23**

votes

**0**answers

588 views

### Software for rational homotopy theory

Does anybody know a software manipulating commutative differential graded algebras, and providing a computation of the minimal model? I tried to use the package DGAlgebras of Macaulay2, but I got ...

**16**

votes

**0**answers

1k views

### Mixed Hodge structure on the rational homotopy type

A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential ...

**14**

votes

**0**answers

885 views

### Hodge star and harmonic simplicial differential forms

Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set?
Let me recall some background.
Hodge Theory on a Riemannian manifold
A ...

**6**

votes

**0**answers

187 views

### Rational Hodge Theory

I am currently learning about the formality result for Kaehler manifolds proved in DGMS. Their result uses some basic Hodge theory, and so only gives formality over $\mathbf{R}$. Later, however, ...

**6**

votes

**0**answers

308 views

### Formality of algebraic varieties via l-adic cohomology?

The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by
Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, ...

**5**

votes

**0**answers

84 views

### Example request: equivariant formality versus formality for homogeneous spaces

Recall that a continuous action of a compact Lie group $T$ on a space $X$ is said to be equivariantly formal if the Borel equivariant cohomology $H^*(X \times_T ET;\mathbb Q)$ surjects through ...

**5**

votes

**0**answers

187 views

### Reference request: splittings in rational homotopy theory

It is well known that for simply-connected rational spaces,
every suspension splits as a wedge of rational spheres and
every loop space splits as a product of rational Eilenberg-Mac Lane spaces.
...

**4**

votes

**0**answers

249 views

### Is there a picture I should have in my head of rational homotopy equivalence?

My understanding is that one thinks to rational homotopy theory for computational advantage. However, thinking about things in terms of localizations still lacks some amount of intuition for me.
In ...

**4**

votes

**0**answers

203 views

### Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds

Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ ...

**3**

votes

**0**answers

103 views

### Reconstructing a nilpotent Lie algebra from its cohomology with $A_{\infty}$-structure

Let $L$ be a nilpotent Lie algebra (over a field of char 0) and $CE^{\bullet}(L)$ be it's Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...

**3**

votes

**0**answers

118 views

### Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...

**3**

votes

**0**answers

61 views

### Is the functor of PA forms known to be equivalent to the functor of PL forms for noncompact spaces?

In the following paper:
Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić, Real homotopy theory of semi-algebraic sets, Algebr. Geom. Topol. 11 (2011), no. 5, 2477–2545.
the authors ...

**3**

votes

**0**answers

92 views

### Representing rational homotopy class by geometric objects

Given a smooth manifold $M$, we can study its rational homotopy type by looking at differential forms. I am wondering if there is a way to represent each $rational$ homotopy class by geometric ...

**2**

votes

**0**answers

103 views

### A question about the definition of complete dg Lie algebras in a paper of Lazarev and Markl

In their paper Disconnected Rational Homotopy Theory, Lazarev and Markl give the following definition (page 23):
Definition: A complete differential graded Lie algebra is an inverse limit of ...