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In this note we define and study free global spectra: global spectra with non-trivial geometric fixed points only at the trivial group. We show that free global spectra often do not exist, and when they do, their homotopy groups satisfy…

The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an $E_2$-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne…

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the $\mathbb Z_2$-cup-length of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-planes in $\mathbb R^n$. Along…

Let $p \geq 5$ be an odd prime. Using the correspondence between secondary Adams differentials and secondary algebraic Novikov differentials, we compute four families of nontrivial secondary differentials on the fourth line of the Adams…

The Cartan formula relates the cup product and the action of the Steenrod algebra on mod~$p$ cohomology. For any pair of mod $p$ cocycles in a simplicial set, where $p$ is an odd prime, we effectively construct a natural coboundary…

The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance for which sublevel-set filtrations…

Parsummable categories were introduced by Schwede as input for his global algebraic $K$-theory construction. We prove that their whole homotopy theory with respect to the so-called global equivalences can already be modelled by the more…

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to…

In this work, we study those differential graded algebras (DGAs) that arise from ring spectra through the extension of scalars functor. Namely, we study DGAs whose corresponding Eilenberg-Mac Lane ring spectrum is equivalent to $H\mathbb{Z}…

We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G$…

A linear constraint system is specified by linear equations over the group $\ZZ_d$ of integers modulo $d$. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the…

We analyze a model for the homotopy theory of complete filtered $L_\infty$-algebras intended for applications in algebraic and algebro-geometric deformation theory. We provide an explicit proof of an unpublished result of E.\ Getzler which…

The purpose of this note is to give a concise account of some fundamental properties of the exponential group and the Maurer-Cartan space associated to a complete dg Lie algebra. In particular, we give a direct elementary proof that the…

Consider the configuration spaces of manifolds. We give a precise formula for the integral cohomological dimension (the degree of top non-trivial integral cohomology group) of unordered configuration spaces of manifolds with non-trivial…

Framed combinatorial topology is a recent approach to tame geometry which expresses higher-dimensional stratified spaces via tractable combinatorial data. The resulting theory of spaces is well-behaved and computable. In this paper we…

Integrally oriented normally nonsingular maps between singular spaces have associated transfer homomorphisms on KO-homology at odd primes. We prove that such transfers preserve Siegel-Sullivan orientations, defined when the singular spaces…

We define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of $\mu_p$-orientation analogous to a $C_2$-equivariant Real orientation. The definition hinges on a $C_p$-space $\mathbb{CP}^{\infty}_{\mu_p}$, which we prove to be…

For a path connected, locally path connected and semilocally simply connected space $X$, let $\Pi_1(X)$ denote its topologised fundamental groupoid as established in the first article of this series. Let $\mathcal{E}$ be the category of…

The notation of torus manifolds were introduced by A. Hattori and M. Masuda. Toric manifolds, quasitoric manifolds, topological toric manifolds, toric origami manifolds and $b$-symplectic toric manifolds are typical examples of torus…

These notes represent the transcript of three, 90 minute lectures given by the second author at the CRM in Barcelona in 2021 as part of the "Higher Structures and Operadic Calculus" workshop. The goal of the series was to introduce and…