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)
$$
\mathcal{L}(\mathbb{C}^k, \mathbb{C}^\ell) \times \mathcal{L}(\mathbb{C}^\ell, \mathbb{C}^m) \longrightarrow \...
- 901
15
votes
0
answers
317
views
Does there exist a closed 4-manifold whose $\pi_2$ contains torsion elements
Does there exist a closed 4-manifold $X$ such that $\pi_2(X)$ contains torsion elements?
And, if so, does there exist a closed 4-manifold $X$ such that $\pi_2(X)\neq 0$ but $\pi_2(X)\otimes \mathbb{Q}=...
- 151
5
votes
0
answers
192
views
When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?
The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces.
I am looking for classes of examples of connected topological groups/...
- 1,184
5
votes
0
answers
179
views
Zigzag vs direct map in rational homotopy theory
I was reading these notions from "Rational Homotopy Theory" by Felix, Halperin, and Thomas.
The notion of weak homotopy type is as follows: two spaces $X$ and $Y$ are said to be weak ...
- 349
5
votes
1
answer
346
views
Analogues of Sullivan Theory at a prime for coformality
In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.
If I am ...
- 2,152
3
votes
0
answers
145
views
Formality of Sullivan Representatives
Suppose we have a map $f : \mathcal{A} \to \mathcal{B}$ between two formal, simply connected CDGAs, with induced map on cohomology $H(f) : H(\mathcal{A}) \to H(\mathcal{B})$. Further, suppose we have ...
- 285
4
votes
0
answers
113
views
Does integration induce a Kan fibration between the mapping spaces of CDGAs and cochain complexes over $\mathbb{Q}$?
For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices
$$
\Phi : A \to B \otimes \Omega^*(\Delta^n)
$$
and simplices maps ...
- 285
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
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
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
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
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
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
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
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
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
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
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
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
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 ...
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
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
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
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
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
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
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
5
votes
0
answers
191
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 ...
- 403
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
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
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
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
9
votes
0
answers
186
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 ...
- 5,040
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
108
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$ ...
- 404
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
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
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
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
8
votes
0
answers
134
views
Rational homotopy type of Hilbert scheme components
What is known about rational homotopy type of irreducible component of $Hilb^n(\mathbb C^k)$ containing configuration space? I've searched arXiv for a while and found nothing but surmise that Betti ...
- 4,600
1
vote
0
answers
409
views
Rational homotopy and l-adic cohomology
In rational homotopy theory there is a basic construction which, given a prime number $l$ and a $CW$-complex $X$, produces a localized space $X_l$ equipped with a map $X\rightarrow X_l$ that induces ...
user95283
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
4
votes
2
answers
377
views
cohomology of configuration space of punctured variety
Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points
$$
F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j ...
- 1,424
4
votes
0
answers
333
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 ...
- 5,040
3
votes
0
answers
82
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 ...
- 5,040