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 $D$-d …

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 in …

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 …

Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certai …

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?

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 $ …

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 $\ma …

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, …

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- …

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 expresse …

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 $ …

If $X$ is a topological space then the rational cohomology of $X$ carries a canonical $A_\infty$ structure (in fact $C_\infty$) with differential $m_1: H^*(X) \to H^{*+1}(X)$ vanis …

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 …

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 o …

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 grad …