Non-formality of the odd dimensional framed little balls operads

We prove that the chain operad of the framed little balls (or disks) operad is not formal as a non-symmetric operad over the rationals if the dimension of their balls is odd and greater than 4.Comment: 10 pages, presentation improved, errors collected, references adde

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.Comment: 28pages, revised version, title changed, to appear in JPA

A note on non-simply connected rational homotopy models

In their generalization of the rational homotopy theory to non-simply connected spaces, G\'omez-Tato--Halperin--Tanr\'e adopted local systems of commutative differential graded algebras (CDGA's) as algebraic models. As another non-simply connected rational model, a $\pi_1$-equivariant CDGA with a semi-simple action was introduced by Hain and studied by Katzarkov-Pantev-To\"en and Pridham. These models have their own advantages and direct comparison might be important. In this note, we present an example of local system of CDGA's which is a non-nilpotent version of an example given by G\'omez-Tato et al., and compute the corresponding $\pi_1$-equivariant CDGA. We also see how some homotopy invariants such as homotopy groups and twisted cohomology are algebraically recovered from the equivariant CDGA with this example. We also describe a general procedure to obtain the corresponding equivariant CDGA model from a local system model in the case of $\pi_1=\mathbb{Z}\times \mathbb{Z}$.Comment: 10 pages, 1 figure, corrected references on the equivariant CDGA mode

The space of short ropes and the classifying space of the space of long knots

ArticleAlgebraic & Geometric Topology. 18: 2859–2873 (2018)journal articl

The de Rham homotopy theory and differential graded category

This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations and closed dg-categories with subsidiary data.Comment: 47 pages. final version. The final publication is available at http://www.springerlink.co