All Questions
Tagged with rational-homotopy-theory at.algebraic-topology
78 questions
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
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
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
29
votes
1
answer
1k
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 ...
- 466
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
27
votes
1
answer
3k
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 ...
- 23.5k
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
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
19
votes
0
answers
2k
views
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 ...
- 6,229
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
18
votes
1
answer
991
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 ...
- 6,229
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
16
votes
2
answers
1k
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 $H$-...
- 7,637
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
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
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
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 ...
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
13
votes
3
answers
966
views
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) ...
- 23.5k
13
votes
1
answer
1k
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 ...
- 3,423
13
votes
1
answer
469
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 ...
- 40.3k
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
11
votes
1
answer
804
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?
- 3,880
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
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
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
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
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
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
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
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
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
7
votes
1
answer
839
views
Schematization of a topological space
I wanted to understand or at least to know if what follows make sense.
Given a connected toplogical space $X$, I want to associate a scheme. In the following way.
For a space $X$ and $A(X)$ the ...
- 235
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
7
votes
1
answer
597
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 ...
- 35.9k
6
votes
1
answer
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 ...
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
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
6
votes
1
answer
637
views
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 $f(D)...
- 23.5k
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
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
6
votes
0
answers
284
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.
...
- 12.5k
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
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
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
2
answers
879
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 ...
- 12.5k
5
votes
1
answer
634
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 ...
- 54.6k
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
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