Questions tagged [rational-homotopy-theory]
The rational-homotopy-theory tag has no usage guidance.
44
questions with no upvoted or accepted answers
19
votes
0
answers
1k
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 ...
16
votes
0
answers
323
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 ...
15
votes
0
answers
280
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}=...
14
votes
0
answers
310
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 ...
10
votes
0
answers
318
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)...
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 ...
8
votes
0
answers
132
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 ...
7
votes
0
answers
129
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 ...
7
votes
0
answers
420
views
Reconstructing a nilpotent Lie algebra from its cohomology with $A_{\infty}$-structure
Let $L$ be a nilpotent Lie algebra (over a field of char 0) and $CE^{\bullet}(L)$ be its Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...
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, ...
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-)...
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 ...
6
votes
0
answers
298
views
Rational Hodge Theory
I am currently learning about the formality result for Kaehler manifolds proved in DGMS. Their result uses some basic Hodge theory, and so only gives formality over $\mathbf{R}$. Later, however, ...
6
votes
0
answers
282
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.
...
6
votes
0
answers
405
views
Formality of algebraic varieties via l-adic cohomology?
The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by
Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, 245-...
5
votes
0
answers
163
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/...
5
votes
0
answers
166
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 ...
5
votes
0
answers
132
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 ...
5
votes
0
answers
186
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 ...
5
votes
0
answers
190
views
Example request: equivariant formality versus formality for homogeneous spaces
Recall that a continuous action of a compact Lie group $T$ on a space $X$ is said to be equivariantly formal if the Borel equivariant cohomology $H^*(X \times_T ET;\mathbb Q)$ surjects through ...
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 ...
4
votes
0
answers
167
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 ...
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 ...
4
votes
0
answers
101
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 $p: E → B$ ...
4
votes
0
answers
319
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 ...
4
votes
0
answers
345
views
Is there a picture I should have in my head of rational homotopy equivalence?
My understanding is that one thinks to rational homotopy theory for computational advantage. However, thinking about things in terms of localizations still lacks some amount of intuition for me.
In ...
4
votes
0
answers
234
views
Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds
Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ \...
3
votes
0
answers
127
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 ...
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
$\...
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 ...
3
votes
0
answers
167
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
0
answers
98
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 $...
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 ...
3
votes
0
answers
124
views
Representing rational homotopy class by geometric objects
Given a smooth manifold $M$, we can study its rational homotopy type by looking at differential forms. I am wondering if there is a way to represent each $rational$ homotopy class by geometric ...
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}...
2
votes
0
answers
139
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....
2
votes
0
answers
53
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 ...
2
votes
0
answers
263
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 ...
2
votes
0
answers
209
views
A question about the definition of complete dg Lie algebras in a paper of Lazarev and Markl
In their paper Disconnected Rational Homotopy Theory, Lazarev and Markl give the following definition (page 23):
Definition: A complete differential graded Lie algebra is an inverse limit of finite-...
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 ...
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 ...
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?
1
vote
0
answers
401
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 ...
-1
votes
0
answers
177
views
Different notions of formality
Quoting from https://en.wikipedia.org/wiki/Rational_homotopy_theory a simply connected complex is called formal if its cohomology algebra is a model (in the sense of Sullivan) for the cochain complex. ...