Bar Complexes and Formality of Pull-backs

Formality on rationalizations of simply connected CW complexes

In this paper, we show that for a simply connected CW complex $Y$ with $H^{*}(Y;\mathbb{Q})$ of finite dimension, if $H^{*}(Y;\mathbb{Q})$ is concentrated in degrees $\leq 3$, then the rationalization $Y_\mathbb{Q}$ is formal. As an application, we show that the spatial realizations of simply connected Sullivan algebras with rational homology concentrated in degrees $\leq 3$ are formal.

Formality of certain CW complexes

Let $X$ be a simply connected path connected topological space which is formal in the sense of rational homotopy theory. Let $Y=X\cup_α\mathbb{D}^{n}$ where $α:\mathbb{S}^{n-1}\to X$ is a non-torsion element. Then we obtain a condition on $α$ for the formality of $Y$. We give several illustrative examples concerning the formality of a finite CW complex having only even dimensional cells. This is the corrected version of the earlier version which contained a serious error in Theorem 1.4. This theorem, which now Theorem 1.1 of this version, has now been corrected. The proofs of Theorems 1.1, 1.2…

Rational formality of mapping spaces

Let X and Y be finite nilpotent CW complexes with dimension of X less than the connectivity of Y. Generalizing results of Vigué-Poirrier and Yamaguchi, we prove that the mapping space Map(X,Y) is rationally formal if and only if Y has the rational homotopy type of a finite product of odd dimensional spheres.

Diagram complexes, formality, and configuration space integrals for braids

We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long l…

On Davis-Januszkiewicz homotopy types I; formality and rationalisation

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we ext…

球面纤维化在连通和上的拉回

Pullbacks of Sphere Fibrations over Connected Sums

研究球面纤维化在连通和上的拉回的同调等价条件，基于同调理论提出新方法，扩展至庞加莱对偶复形和局部情形。