The Milnor-Moore theorem for $$L_\infty $$ algebras in rational homotopy theory

We give a construction of the universal enveloping $$A_\infty $$ A ∞ algebra of a given $$L_\infty $$ L ∞ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new $$A_\infty $$ A ∞ model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

Goodwillie Approximations to Higher Categories

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

Bounds on Sectional Curvature and Interactions with Topology

In this survey article we exemplarily illustrate implications of curvature assumptions on the topology of the underlying manifold. We shall mainly focus on sectional curvature and three different kinds of restrictions, namely on non-negative respectively on positive sectional curvature, as well as on two-sided curvature bounds.We shall see that there are various implications on the side of topology, namely, for example, geometry having an impact on elementary invariants like the Euler characteristic or Betti numbers as well as on concepts from rational homotopy theory or index theory, and that there are connections to K-theory.On our way of making these connections we shall draw on certain simplifications and tools like group actions or metrics with additional properties like geometric formality.

On a DGL-map between derivations of Sullivan minimal models

For a map $$f:X\rightarrow Y$$ f : X → Y , there is the relative model $$M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)\simeq M(X)$$ M ( Y ) = ( Λ V , d ) → ( Λ V ⊗ Λ W , D ) ≃ M ( X ) by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let $$\mathrm{Baut}_1X$$ Baut 1 X be the Dold–Lashof classifying space of orientable fibrations with fiber X (Dold and Lashof, Ill J Math 3:285–305, 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations $$\mathrm{Der}M(X)$$ Der M ( X ) of the Sullivan minimal model M(X) of X. Then we consider the condition that the restriction $$b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d) $$ b f : Der ( Λ V ⊗ Λ W , D ) → Der ( Λ V , d ) is a DGL-map and the related topics.

Cyclicity in homotopy algebras and rational homotopy theory

Kadeishvili proposes a minimal{C_{\infty}}-algebra as a rational homotopy model of a space. We discuss a cyclic version of this Kadeishvili{C_{\infty}}-model and apply it to classifying rational Poincaré duality spaces. We classify 1-connected minimal cyclic{C_{\infty}}-algebras whose cohomology algebras are those of{(S^{p}\times S^{p+2q-1})\sharp(S^{q}\times S^{2p+q-1})}, where{2\leq p\leq q}. We also include a proof of the decomposition theorem for cyclic{A_{\infty}}and{C_{\infty}}-algebras.

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).