All Questions
Tagged with rational-homotopy-theory at.algebraic-topology
78 questions
3
votes
1
answer
246
views
Model structure on simply-connected topological spaces in which the weak equivalences are the rational homotopy equivalences
I recently started learning rational homotopy theory, and found the claim on page 7 of this survey that there is a suitable model category structure in which the weak equivalences are the rational ...
- 33
9
votes
0
answers
120
views
Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
- 371
3
votes
0
answers
110
views
String cobracket and co-Hochschild homology
Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of ...
- 985
3
votes
0
answers
79
views
Rational model for composition of linear isometries
There is a composition map on spaces of linear isometries (over $\mathbb{C}$ say)
$$
\mathcal{L}(\mathbb{C}^k, \mathbb{C}^\ell) \times \mathcal{L}(\mathbb{C}^\ell, \mathbb{C}^m) \longrightarrow \...
- 901
15
votes
0
answers
317
views
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}=...
- 151
5
votes
0
answers
192
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/...
- 1,184
5
votes
1
answer
346
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 ...
- 2,152
5
votes
0
answers
179
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 ...
- 349
5
votes
1
answer
223
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 ...
5
votes
1
answer
273
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 ...
- 1,374
6
votes
1
answer
374
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 ...
- 1,969
3
votes
0
answers
145
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 ...
- 285
7
votes
1
answer
662
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:...
- 5,859
16
votes
0
answers
325
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 ...
- 23.5k
4
votes
0
answers
113
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 ...
- 285
11
votes
1
answer
448
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.
- 965
14
votes
0
answers
318
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 ...
- 8,167
18
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(\...
user137162
6
votes
0
answers
235
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, ...
- 1,177
7
votes
1
answer
413
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 $\...
- 5,859
2
votes
0
answers
120
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}...
- 21
4
votes
1
answer
326
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 ...
- 5,901
22
votes
1
answer
679
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 ...
- 8,167
13
votes
6
answers
4k
views
What is the best way to study Rational Homotopy Theory
I studied basic algebraic topology elements:
fundamental group, higher homotopy groups, fibre bundles, homology groups, cohomology groups, obstruction theory, etc.
I want to study Rational Homotopy ...
9
votes
1
answer
556
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 $\...
- 305
30
votes
6
answers
3k
views
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 ...
- 6,229
4
votes
0
answers
170
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 ...
- 1,361
9
votes
1
answer
253
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 ...
- 1,452
28
votes
3
answers
2k
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 $f'$ ...
- 23.5k
3
votes
0
answers
123
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
$\...
- 1,361
9
votes
1
answer
386
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 ...
- 383
16
votes
1
answer
689
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 ...
- 161
38
votes
2
answers
2k
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 ...
- 12.5k
2
votes
1
answer
660
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 ...
- 239
4
votes
0
answers
197
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 ...
30
votes
1
answer
787
views
Is a filtered colimit of rational spaces again rational?
Let me first explain the statement of the question and then give some indication why the answer might be 'yes'.
By a space I mean, say, a simplicial set and by rational I mean rational in the sense of ...
- 383
5
votes
2
answers
266
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 ...
- 745
26
votes
2
answers
2k
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 its commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...
- 2,974
2
votes
1
answer
402
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 ...
- 809
2
votes
0
answers
320
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 ...
- 21
3
votes
1
answer
279
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?
- 4,600
12
votes
2
answers
799
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 $C_\infty$-...
- 35.9k
9
votes
2
answers
2k
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 ...
- 25.6k
2
votes
1
answer
196
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.
...
- 403
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 ...
- 18.9k
13
votes
2
answers
1k
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 (...
- 27.5k
3
votes
1
answer
239
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?
- 121
6
votes
2
answers
310
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 ...
- 121
4
votes
1
answer
243
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 ...
- 1,424
5
votes
0
answers
138
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 ...
- 523