Victor Turchin

On the rational homology of high-dimensional analogues of spaces of long knots

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of

The rational homology of spaces of long knots in codimension > 2

\mathbb{R}^m

Homotopy graph-complex for configuration and knot spaces

into

Context-free manifold calculus and the Fulton–MacPherson operad

\mathbb{R}^n

Two-loop part of the rational homotopy of spaces of long embeddings

. We view the space of embeddings as the value of a certain functor at

Commutative hairy graphs and representations of Out (Fr)

\mathbb{R}^m

The homotopy theory of operad subcategories

, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when

Embedding Calculus and the Little Discs Operads

2m+1<n

Hodge-type decomposition in the homology of long knots

, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.

Real homotopy theory of semi-algebraic sets

We determine the rational homology of the space of long knots in R for d 4 . Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree d 1 , which can be obtained as the homology of an explicit graph complex and is in theory completely computable.