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

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### 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 ...

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211 views

### Convergence of a sum with the ranks of homotopy groups

Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. ...

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160 views

### The image of the Hurewicz map for rational loop spaces

Let $K$ be the rationalization of a simply-connected finite CW complex. Then the Samelson product gives $\pi_*(\Omega K)$ the structure of a graded Lie algebra, and the Hurewicz map
$h: \pi_*(\Omega ...

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41 views

### Relation between three simplicial Lie algebras associated to a simply connected CW comlpex?

Given a CW complex with 1 vertex, and no 1-cells, one can associate to it three simplicial Lie algebras, as follows:
First, fix some notation. If $\mathcal C$ is the category of groups, or Lie ...

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62 views

### Lusternik-Schnirelmann category of rational spaces of infinite type

Suppose I have an $n$-dimensional (simply-connected) rational space $X$ with L-S category $k$. Must there be a finite rational subcomplex $K \subseteq X$ with $\mathrm{cat}(K) = k$?
Even better, I ...

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66 views

### natural co-product on minimal Sullivan model

Let M be a compact manifold. The diagonal $M \rightarrow M \times M$ induces
co-product on singular cohomolgy $H^*(M) \rightarrow H^*(M) \times H^*(M)$ via Poincare duality.
I would like to know if ...

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**1**answer

101 views

### Hochschild chain model for the evaluation map at half

Let M be a manifold and $\Lambda(M)$ its free loop space, $A= C^*(M)$ denotes the cochain algebra of $M$. We know that Hochschild chain model for the evaluation $ ev_0: \Lambda(M) \rightarrow M$ is ...

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### 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 ...

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**1**answer

244 views

### A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no ...

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168 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, ...

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465 views

### rationalization of classifying spaces

This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an ...

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**3**answers

611 views

### The cohomology plus what characterizes the rational homotopy type?

For simplicity let me work only with connected and simply connected spaces. "Space" will mean a space of this type.
A space is rational if its homotopy groups are rational vector spaces ...

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**1**answer

554 views

### Are there geometrically formal manifolds, which are not rationally elliptic?

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if it's commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...

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307 views

### What's a good reference for the following obstruction theory yoga?

Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...

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**1**answer

134 views

### Zero-divisors in a graded Lie algebra

Let $\mathfrak{g}$ be positively graded Lie algebra over $\mathbb{Q}$, concentrated in even degrees.
Question: If $\mathfrak{g}$ is not free, must there exist linearly independent elements ...

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225 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 ...

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**1**answer

188 views

### sufficient conditions for rational homotopy equivalence

Is it true that if a finite CW complex $X$ is simply connected, and $\tilde{H}_i(X, \mathbb{Q}) =0$ for $i \neq D$, then $X$ is rationally homotopy equivalent to a bouquet of $D$-dimensional spheres?
...

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337 views

### Equivariant cohomology of finite group actions and invariant cohomology classes

Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certainly not an ...

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552 views

### Equivariant Cohomology for actions with finite stabilizers

Let $X$ be a reasonable topological space (let's say it has the homotopy type of a CW complex) and let $G$ be a topological group acting on that space. Let $E_G \rightarrow B_G$ be the universal ...

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**1**answer

431 views

### rational homotopy of a manifold

Given a finite dim rational homotopy type satisfying Poincaré duality,
what is the best reference to when it is the rational homotopy type of a fin dim manifold?

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### 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
$$ ...

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180 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.
...

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297 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, ...

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1k views

### Mysterious property of $\mathbb{Q}$

Hi,
I am currently working through the paper by Bousfield and Gugenheim on rational homotopy theory, and have come to a point where they show why it is important to work over $\mathbb{Q}$, and not ...

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**1**answer

620 views

### Homotopy type of the self-homotopy equivalences of a bouquet of spheres

Before I state the questions I have in mind, let me give some background. If one considers $S^2$ then it is known due to Kneser that $\textrm{Homeo}^{+}(S^2)$ has the homotopy type of $SO(3)$. By ...

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733 views

### Is the polynomial de Rham functor a Quillen equivalence?

It is known that the rational homotopy theory of spaces (e.g. simplicial sets) is equivalent in some sense to the homotopy theory of cdgas over $\mathbb{Q}$. This has been expressed in various forms ...

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369 views

### Minimal models with local coefficients

Let $X$ be a path-connected nilpotent space (meaning $\pi_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi_1(X)\to \mathrm{Gl}(V)$ be a ...

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559 views

### Reference for functors in Kadeishvili's C_\infty paper

In his paper Cohomology $C_\infty$-algebra and rational homotopy type, Tornike Kadeishvili describes how the rational cohomology of a simply-connected space carries the structure of a ...

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550 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 ...

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666 views

### Higher homotopy algebraic structure on the homology of an operad

Given a DGA $A$, then by standard techniques such as homological perturbation theory, the ring structure on the homology $H(A)$ extends to a minimal $A_\infty$-algebra structure such that $H(A)$ is ...

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1k views

### A non-formal space with vanishing Massey products?

Let $X$ be a polyhedron. For each $n$-dimensional face $f$ of $X$ fix a homeomorphism $\sigma_f:\triangle^n\to f$ where $\triangle^n$ is the standard $n-$simplex so that whenever $f$ is a face of $f'$ ...

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### Poincare duality and the $A_\infty$ structure on cohomology

If $X$ is a topological space then the rational cohomology of $X$ carries a canonical $A_\infty$ structure (in fact $C_\infty$) with differential $m_1: H^\ast(X) \to H^{\ast+1}(X)$ vanishing and ...

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### 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 ...

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528 views

### Characterizing the rationalization of spaces.

In the category of rational spaces, loop spaces split as products of Eilenberg-Mac Lane
spaces and SUSPENSIONS split as wedges of (rational) spheres. I wonder if anything of the following form is ...

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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 ...

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### Rational homotopy type of a complement

Let $X$ and $X'$ be smooth closed manifolds. Take closed subpolyhedra $D\subset X$ and $D'\subset X'$ (with respect to some triangulations) and let $f:X\to X'$ be a homotopy equivalence such that ...

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### Rational homotopy theory of a punctured manifold

Let $M$ be a smooth simply connected manifold and let $N$ be $M$ minus a point. Is it possible to construct an explicit Sullivan model for $N$ (i.e. a commutative differential graded algebra (cdga) ...