Questions tagged [rational-homotopy-theory]
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111 questions
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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 ...
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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, ...
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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
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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-...
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Equivariant Cohomology for actions with finite stabilizers
Let $X$ be a reasonable topological space (let's say it has the homotopy type of a CW complex) and let $G$ be a topological group acting on that space. Let $E_G \rightarrow B_G$ be the universal ...
- 445
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266
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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
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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 ...
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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 ...
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2
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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
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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
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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
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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
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179
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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 ...
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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
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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 ...
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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 ...
- 2,995
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sufficient conditions for rational homotopy equivalence
Is it true that if a finite CW complex $X$ is simply connected, and $\tilde{H}_i(X, \mathbb{Q}) =0$ for $i \neq D$, then $X$ is rationally homotopy equivalent to a bouquet of $D$-dimensional spheres?
...
- 7,857
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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
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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
$$
F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j ...
- 1,424
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The image of the Hurewicz map for rational loop spaces
Let $K$ be the rationalization of a simply-connected finite CW complex. Then the Samelson product gives $\pi_*(\Omega K)$ the structure of a graded Lie algebra, and the Hurewicz map
$h: \pi_*(\Omega ...
- 12.5k
4
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Rational homotopy type of rational mapping spaces
I was interested in the question of figuring out the rational homotopy type of mapping spaces (regular or rational) between two algebraic varieties over $\mathbb{C}$. I encountered the following paper ...
- 5,901
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158
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Hochschild chain model for the evaluation map at half
Let M be a manifold and $\Lambda(M)$ its free loop space, $A= C^*(M)$ denotes the cochain algebra of $M$. We know that Hochschild chain model for the evaluation $ ev_0: \Lambda(M) \rightarrow M$ is ...
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Minimal Model for $\mathbb{CP}^2 \# \mathbb{CP}^2 \# \mathbb{CP}^2$
I'm a grad student studying non-negative curvature on simply-connected manifolds and the conjectured relationship with rational homotopy theory.
The rational homotopy groups (and so the number of ...
- 183
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113
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Does integration induce a Kan fibration between the mapping spaces of CDGAs and cochain complexes over $\mathbb{Q}$?
For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices
$$
\Phi : A \to B \otimes \Omega^*(\Delta^n)
$$
and simplices maps ...
- 285
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170
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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
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197
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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 ...
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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
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333
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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
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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 ...
- 1,147
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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
$$ \...
- 40.3k
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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
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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
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194
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Zero-divisors in a graded Lie algebra
Let $\mathfrak{g}$ be positively graded Lie algebra over $\mathbb{Q}$, concentrated in even degrees.
Question: If $\mathfrak{g}$ is not free, must there exist linearly independent elements $a,b\in\...
- 35.9k
3
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1
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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
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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-...
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110
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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 ...
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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
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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 ...
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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
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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 ...
- 273
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178
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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 ...
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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 $...
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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-...
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82
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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 ...
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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 ...
- 606
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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
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246
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Convergence of a sum with the ranks of homotopy groups
Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. ...
user62675
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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}}...
- 239
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1
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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
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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.
...
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