On the topology of moduli spaces of non-negatively curved Riemannian metrics

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

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 rational homotopy and minimal models

We prove a result that enables us to calculate the rational homotopy of a wide class of spaces by the theory of minimal models.

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.