Questions tagged [rational-homotopy-theory]

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5 votes
1 answer
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Does the Lie algebra structure on rational homotopy groups reflect similar information to the formal group structure in characteristic p?

It's well known (c.f. Quillen and Sullivan) that the rational homotopy theory of spaces is equivalent to the homotopy theory of rational DG-algebras; in particular, rational spaces and rational ...
16 votes
0 answers
288 views

Rational equivalence of smooth closed manifolds

All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they ...
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7 votes
1 answer
406 views

Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory

Here is a collection of facts that all seem true, but together seem to give a nonsensical solution: After $T(n)$-localization, all natural transformations $F \sim G$ between homogenous functors $F,G:...
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2 votes
0 answers
104 views

Homotopy groups of homotopy fixed points of a $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local orthogonal spectrum

Let $G$ be a finite group and $X$ an orthogonal $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local spectrum with an $G$-action that is trivial on $\pi_*X$. I want to show that then the map $X^{hG}...
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14 votes
0 answers
280 views

Do Sullivan's and Wilkerson's algebraic group structures on rational homotopy automorphisms coincide?

For a 1-connected space $X$ with the homotopy type of a finite CW-complex, the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X)$ are known to have the structure of an arithmetic group up to finite ...
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6 votes
0 answers
213 views

Different rational homotopy type with generators of different degree but cohomology algebras same

There are manifolds (rationally elliptic) $M_1$ and $M_2$ of different rational homotopy types but their rational cohomology algebras coincide. Such examples were discussed in Nishimoto, Shiga, ...
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2 votes
0 answers
94 views

Geometric fixed points of induction spectrum

I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....
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1 vote
0 answers
42 views

A question related to injective envelope for a system of DGA's

I was trying to read Fine and Triantafillou's paper "On the equivariant formality of Kahler manifolds with finite group action". They have defined the enlargement at $H$ of a system of DGA's ...
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8 votes
0 answers
225 views

Closed geodesics on closed manifolds

There is (or was) a notorious open problem (see e.g. this survey by Keith Burns and Vladimir Matveyev from 2013) in differential geometry: Conjecture. Let $M$ be a closed (compact, with empty boundary)...
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6 votes
0 answers
221 views

Model structure on dg-algebras over an "equivariant fundamental category"?

For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)...
2 votes
0 answers
42 views

Projective resolution of a dual coefficient system

I was trying to read the paper "Equivariant minimal models" by G. Triantafillou(1982) and was trying to compute cohomology of a system of DGA with rational coefficient system. Given a finite ...
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4 votes
1 answer
253 views

Rational homotopy type of rational mapping spaces

I was interested in the question of figuring out the rational homotopy type of mapping spaces (regular or rational) between two algebraic varieties over $\mathbb{C}$. I encountered the following paper ...
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4 votes
0 answers
160 views

Relation between $Tor_{C^*(BG;K)}(K,K) $ and $K^G$?

Let $K$ be an algebraically closed field and $G$ a group. Given a dg-algebra $A$, a left $A$-module $M$ and a right $A$-module $N$ let $Tor_A(M,N)$ denote the homology of the derived tensor product $M ...
3 votes
0 answers
109 views

Homology of a fiber as a cotorsion product

Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let $\...
7 votes
1 answer
353 views

Is there a topological interpretation of a module over $\Omega_{PL}(X)$?

Associated to a DGCA (differential graded commutative algebra) $A$, we can associate to it the category of modules over $A$. Hence, for a space $X$ we can consider the category of modules over $\...
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11 votes
1 answer
343 views

Rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$

What is the rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$? Here $\Omega$ denotes based loop space.
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9 votes
1 answer
183 views

Almost free circle actions on spheres

$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$I am looking for any reference regarding the following problem: Problem: Consider a smooth almost-free action of $S^1$ on a smooth sphere $S^n$. Then ...
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7 votes
0 answers
108 views

Existence of relative equivariant minimal models

In equivariant rational homotopy theory the existence of minimal models (i.e. the equivariant generalization of minimal Sullivan models) has been established by Triantafillou (jstor:1999119) and Scull ...
2 votes
1 answer
100 views

Kan replacement of finite $\mathbb{Q}$-type simplicial set

Sorry if this question is maybe a bit basic, but it is on a rather specialized topic so I think it is more appropriate for MO than SE. Suppose that $X$ is a simplicial set that has finitely many non-...
4 votes
0 answers
187 views

Modelling rational spaces of finite $\mathbb{Q}$-type with spaces with finite simplices in every simplicial dimension

EDIT 2 Original question below. I will award the outstanding bounty for an answer to the following question (question (2) in the OP). Let $X$ be a Kan complex which is connected, nilpotent, and of ...
2 votes
0 answers
260 views

Rational homotopy theory [closed]

I am trying to read the paper "Rational homotopy theory " by Quillen and am stuck with the notion of complete augmented algebra. He had defined the complete augmented algebra and I don't understand ...
7 votes
1 answer
435 views

Are exterior algebras intrinsically formal as associative dg algebras?

(Cross-posted from mathematics stackexchange.) Fix a finite dimensional vector space $V$ over a field of characteristic zero, and let $R=Sym(V[1])$ be the free graded commutative algebra generated by ...
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9 votes
1 answer
291 views

Different definitions of formality for groups

Let $X$ be a space with fundamental group $G$. Recall that the de Rham fundamental group of $X$ is the inverse limit of the Malcev completions of the nilpotent truncations of $G$. This has a Lie ...
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6 votes
0 answers
123 views

On the weak homotopy type of a differentiable (Chen) space

Suppose that $M$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ). Assume that $M$ also has the structure of a topological space and that the two ...
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2 votes
0 answers
182 views

Homology of homotopy fiber of inclusion

We consider an inclusion $j: A \hookrightarrow X$. Let $A_j = A \times_X PX$ be the homotopy fiber (which is the fiber of the fibration associated to $j$). The space $PX$ there is the Moore path space ...
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22 votes
1 answer
550 views

When does rationalization commute with homotopy fixed points?

Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a ...
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5 votes
0 answers
122 views

Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
1 vote
0 answers
117 views

PD model for CDGA over the integers

Lambrechts and Stanley constructed Poincaré duality model for cdga over a field with simply connected cohomology. Are there construction of PD model for CDGA over integer coefficients?
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16 votes
3 answers
2k views

Homology of spectra vs homology of infinite loop spaces

Let $X$ be a CW complex and let $\Sigma^\infty X$ denote its suspension spectrum. By definition, the $n$th singular homology group of $\Sigma^\infty X$ with coefficients in $\mathbb{Z}$ is $\pi_n(\...
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2 votes
1 answer
187 views

Rationalization of topological groups and degree maps

Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_\mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space. ...
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5 votes
0 answers
169 views

DG-Modules over CDG-algebras in the sense of rational homotopy theory

I don't know if this question is elementary or not. Suppose we have a rationalization $X_\mathbb{Q}$ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a minimal ...
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9 votes
1 answer
527 views

Formality over $\mathbb{R}$ vs formality over $\mathbb{Q}$

On ncatlab page on formality, it is stated that Deligne--Griffiths--Morgan--Sullivan proved that the real homotopy type of a closed Kaehler manifold is formal. Later, Sullivan "improved" this to $\...
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3 votes
0 answers
96 views

It there a nice way to describe the structure of Malcev-complete groups?

Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
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1 vote
1 answer
325 views

Formality of surfaces

The de Rham dg algebra $\Omega(F)$ of a closed orientable surface $F$ is formal (that is, weakly equivalent to its cohomology algebra). This is a special case of the fact of formality of Kähler ...
5 votes
2 answers
248 views

Naturality of PD model of a CDGA

In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are ...
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-5 votes
1 answer
376 views

Computing $H^*(BDiff(W_{\infty},D^{\infty});\mathbb{Q})$ via Mumford-Morita-Miller classes

Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, "Stable Moduli spaces of High Dimensional Manifolds". For each characteristic class of oriented $...
3 votes
1 answer
245 views

Formal complex manifold without dd^c

Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma?
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0 votes
1 answer
150 views

Compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of a product of Lie groups (and their quotients)

As the following product is a bit unfamiliar to me: How do we compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of the product of Lie groups: $M=SO(n_1)\times U(n_2)\times SU(n_3)\times (...
2 votes
1 answer
346 views

Relationship between the Betti numbers $b_i(M;\mathbb{Q})$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{Q}$

What is the relationship between the Betti numbers $b_i(M;\mathbb{Q})=rkH_i(M;\mathbb{Q})$ of a simply connected closed Riemannian manifold $M$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}...
8 votes
0 answers
158 views

Does real formality descend to rational formality for operads?

A classical theorem in rational homotopy theory says that a space is formal over $\mathbb{Q}$ iff it is formal over any field of characteristic zero. In other words, the algebra $A^*_{PL}(X)$ is ...
2 votes
1 answer
634 views

Show that the rational cohomology ring $H^*(M;\mathbb{Q})$ needs at least two generators

Let $M$ be a simply connected closed Riemannian manifold. How does one find a necessary condition going both ways that may be imposed on $M$ (perhaps on the curvature of $M$ and on torsion) which ...
4 votes
0 answers
86 views

Finite spatial realization of relative minimal sullivan models

In their book "Algebraic Models in Geometry" (Felix, Oprea, Tanre) the authors claim that: "Each finite type relative minimal cdga $(∧V ⊗∧W,D)$ is the relative minimal model of a fibration $p: E → B$ ...
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15 votes
1 answer
613 views

Rational homotopy groups of a projective hypersurface

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^n$. Has anyone computed the rational homotopy groups $\pi_i(X)\otimes \mathbb{Q}$ of $X$? I tried Google, but did not find anything. One ...
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7 votes
2 answers
348 views

Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?

By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}...
3 votes
0 answers
156 views

Completion of coalgebras

Is it possible to complete commutative dg associative (conilpotent) coalgebras over $\mathbb{Q}$ in a way so that when we complete the symmetric coalgebra $Sym(V)$ it becomes completed with respect to ...
3 votes
1 answer
210 views

Jacobian and configuration space and massey products

Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and ...
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6 votes
2 answers
284 views

a comparison between LS and cohomological dimension

Let $X$ a simply connected elliptic space. Assume $\pi_\star(X)\otimes\Bbb{Q}$ is concentrated in odd degrees. So, we have $dim~\pi_\star(X)\otimes\Bbb{Q}=TC(X_\Bbb{Q})=catX_\Bbb{Q}$ (ie) the ...
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3 votes
0 answers
91 views

odd spectral sequence

let $(\Lambda V,d)=(\Lambda (a_1,...a_n),d)$ be a graded commutative differential algebra, and $(\tau;a_1,...a_n)$ be a connected, finite c-finite tower with odd spectral sequence $(E_i,d_i)$. Assume $...
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3 votes
1 answer
213 views

Is a simply connected elliptic space rationally homotopy equivalent to a loop space or a suspension?

Let $X$ be an elliptic simply connected space. Is it rationally homotopy equivalent to the suspension of some connected space $Y$? If not, is it rationally homotopy equivalent to a loop space?
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6 votes
1 answer
366 views

Homotopy equivalence vs gauge equivalence

Let $(\mathfrak{g},[-,-])$ be a pronilpotent Lie algebra (considered of degree zero). We can consider $(\mathfrak{g},[-,-])$ as a differential graded Lie algebra endowed with the $0$-differential. Let ...
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