The extended rational homotopy theory of operads

In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book [B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups. Part 2. The Applications of (Rational) Homotopy Theory Methods, Math. Surveys Monogr. 217, American Mathematical Society, Providence, 2017]. In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite-type rational cohomology groups).

A zoo of growth functions of mapping class sets

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which can be realized by [Formula: see text]-Lipschitz maps grows asymptotically as [Formula: see text], where [Formula: see text] is an integer determined by the rational homotopy type of [Formula: see text] and the rational cohomology of [Formula: see text]. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is [Formula: see text] but the true growth is [Formula: see text]. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number [Formula: see text], there is a pair [Formula: see text] for which the growth of [Formula: see text] is essentially [Formula: see text].

Rational Models of the Complement of a Subpolyhedron in a Manifold with Boundary

Let W be a compact simply connected triangulated manifold with boundary and let K ⊂ W be a subpolyhedron. We construct an algebraic model of the rational homotopy type of W\K out of a model of the map of pairs (K, K⋂∂W) ↪ (W, ∂W) under some high codimension hypothesis.We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.