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We define $\mathcal{H}$-fibration sequences as fibrations where the holonomy action of the fundamental group of the base on the fiber lies in a given subgroup $\mathcal{H}$ of $\mathcal{E}(F)$, where $\mathcal{E}(F)$ is the homotopy…

The tautness for a cohomology theory is formulated and studied by various authors. However, the analogous property is not considered for a homology theory. In this paper, we will define and study this very property for the Massey homology…

Partial groups are a natural generalization of discrete groups recently introduced by Chermak in connection with the theory of fusion systems. In this paper we develop an extension theory for partial groups based in the classical theory of…

To a direct sum of holomorphic line bundles, we can associate two fibrations, whose fibers are, respectively, the corresponding full flag manifold and the corresponding projective space. Iterating these procedures gives, respectively, a…

We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over…

We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal $A_\infty$-algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on the…

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-equivariant algebraic vector bundle over $X$. A section of $E$ is regular if it is transversal to the zero…

Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply-connected space $X$ there is a dg Lie algebra $\Lambda_X$ and a (coassociative…

We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be…

We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric…

In this article, we study the magnitude homology of geodesic metric spaces of curvature $\leq \kappa$, especially ${\rm CAT}(\kappa)$ spaces. We will show that the magnitude homology $MH^{l}_{n}(X)$ of such a meric space $X$ vanishes for…

We prove that the topological complexity of every symplectically atoroidal manifold is equal to twice its dimension. This is the analogue for topological complexity of a result of Rudyak and Oprea, who showed that the…

Let $G$ be a compact connected Lie group. We show that the category $\mathbf{Loc}_{\infty}(BG)$ of $\infty$-local systems on the classifying space of $G$, can be described infinitesimally as the category…

We construct a model structure on the category $\mathrm{DblCat}$ of double categories and double functors. Unlike previous model structures for double categories, it recovers the homotopy theory of 2-categories through the horizontal…

The motivic Mahowald invariant was introduced in \cite{Qui19a} and \cite{Qui19b} to study periodicity in the $\mathbb{C}$- and $\mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of…

D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the…

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability…

We produce a direct Quillen equivalence between two models of $(\infty,2)$-categories: the complete Segal $\Theta_2$-spaces due to Rezk and the $2$-complicial sets due to Verity.

We show that the associated Lie algebra of the Malcev $\mathbb{Q}$-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. We also prove that the homology of this completed Lie algebra is of…

This paper lays the foundations of triangulated persistence categories (TPC), which brings together persistence modules with the theory of triangulated categories. As a result we introduce several measurements and metrics on the set of…