Questions tagged [rational-homotopy-theory]

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Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements

Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements? And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...
Boyu Zhang's user avatar
5 votes
0 answers
152 views

When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces. I am looking for classes of examples of connected topological groups/...
ThorbenK's user avatar
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1 vote
0 answers
150 views

Can formality be read from the cohomology algebra

A cgda $(A,d)$ is formal if it is weakly equivalent to $(H(A),0)$. There are several equivalent conditions for this. Similarly, a space $X$ is formal if the cgda $(A_{PL}(X),d)$ of polynomial ...
CuriousUser's user avatar
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5 votes
0 answers
163 views

Zigzag vs direct map in rational homotopy theory

I was reading these notions from "Rational Homotopy Theory" by Felix, Halperin, and Thomas. The notion of weak homotopy type is as follows: two spaces $X$ and $Y$ are said to be weak ...
bishop1989's user avatar
5 votes
1 answer
338 views

Analogues of Sullivan Theory at a prime for coformality

In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model. If I am ...
Andrea Marino's user avatar
3 votes
0 answers
121 views

Formality of Sullivan Representatives

Suppose we have a map $f : \mathcal{A} \to \mathcal{B}$ between two formal, simply connected CDGAs, with induced map on cohomology $H(f) : H(\mathcal{A}) \to H(\mathcal{B})$. Further, suppose we have ...
kelly maggs's user avatar
4 votes
0 answers
105 views

Does integration induce a Kan fibration between the mapping spaces of CDGAs and cochain complexes over $\mathbb{Q}$?

For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices $$ \Phi : A \to B \otimes \Omega^*(\Delta^n) $$ and simplices maps ...
kelly maggs's user avatar
5 votes
1 answer
189 views

Rational G-spectrum and geometric fixed points

For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
Steven Patrak's user avatar
4 votes
1 answer
227 views

Monoidal Dold–Kan correspondence for non-connected CDGA

Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0? I understand that there is a technical problem with the original proof due to ...
Grisha Taroyan's user avatar
6 votes
1 answer
303 views

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 ...
Doron Grossman-Naples's user avatar
16 votes
0 answers
322 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 ...
algori's user avatar
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7 votes
1 answer
640 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:...
Connor Malin's user avatar
  • 5,191
2 votes
0 answers
114 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}...
Urs's user avatar
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14 votes
0 answers
305 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 ...
skupers's user avatar
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6 votes
0 answers
227 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, ...
piper1967's user avatar
  • 1,059
2 votes
0 answers
134 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....
N.B.'s user avatar
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1 vote
0 answers
45 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 ...
piper1967's user avatar
  • 1,059
10 votes
0 answers
286 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)...
Moishe Kohan's user avatar
  • 9,624
6 votes
0 answers
246 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-)...
Urs Schreiber's user avatar
2 votes
0 answers
52 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 ...
piper1967's user avatar
  • 1,059
4 votes
1 answer
299 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 ...
user127776's user avatar
  • 5,821
4 votes
0 answers
166 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 ...
Hadrian Heine's user avatar
3 votes
0 answers
122 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 $\...
Hadrian Heine's user avatar
7 votes
1 answer
410 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 $\...
Connor Malin's user avatar
  • 5,191
11 votes
1 answer
420 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.
qqqqqqw's user avatar
  • 915
9 votes
1 answer
212 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 ...
CuriousUser's user avatar
  • 1,420
7 votes
0 answers
127 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 ...
Urs Schreiber's user avatar
2 votes
1 answer
114 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-...
Daniel Robert-Nicoud's user avatar
4 votes
0 answers
196 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 ...
Daniel Robert-Nicoud's user avatar
2 votes
0 answers
290 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 ...
GURI920826's user avatar
7 votes
1 answer
558 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 ...
TBRS's user avatar
  • 73
9 votes
1 answer
367 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 ...
Tina's user avatar
  • 373
6 votes
0 answers
132 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 ...
John Klein's user avatar
  • 18.6k
2 votes
0 answers
256 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 ...
Timo E.'s user avatar
  • 21
22 votes
1 answer
639 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 ...
skupers's user avatar
  • 7,923
5 votes
0 answers
131 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 ...
Bashar Saleh's user avatar
1 vote
0 answers
125 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?
Arun 's user avatar
  • 725
17 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(\...
user avatar
2 votes
1 answer
191 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. ...
Fat ninja's user avatar
  • 393
5 votes
0 answers
180 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 ...
Fat ninja's user avatar
  • 393
9 votes
1 answer
547 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 $\...
man's user avatar
  • 305
3 votes
0 answers
114 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 ...
J. Darné's user avatar
  • 263
1 vote
1 answer
383 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 ...
Semen Podkorytov's user avatar
5 votes
2 answers
261 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 ...
Arun 's user avatar
  • 725
-5 votes
1 answer
397 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 $...
Sergio Charles's user avatar
3 votes
1 answer
274 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?
Denis T's user avatar
  • 4,371
0 votes
1 answer
165 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 (...
Sergio Charles's user avatar
2 votes
1 answer
413 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}}...
Sergio Charles's user avatar
9 votes
0 answers
177 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 ...
Najib Idrissi's user avatar
2 votes
1 answer
654 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 ...
Sergio Charles's user avatar