代数拓扑

Koszul duality is a fundamental correspondence between algebras for an operad $\mathcal{O}$ and coalgebras for its dual cooperad $B\mathcal{O}$, built from $\mathcal{O}$ using the bar construction. Francis-Gaitsgory proposed a conjecture…

In this paper we generalize the discrete r-homotopy to the discrete (s, r)-homotopy. Then by this notion, we introduce the discrete motion planning for robots which can move discreetly. Moreover, in this case the number of motion planning,…

We prove a connectivity bound for maps of $\infty$-operads of the form $\mathbb{A}_{k_1} \otimes \cdots \otimes \mathbb{A}_{k_n} \to \mathbb{E}_n$, and as a consequence, give an inductive way to construct $\mathbb{E}_n$-algebras in…

We prove that the moment-angle complex $\mathcal Z_K$ corresponding to a 3-dimensional simplicial sphere $K$ has the cohomology ring isomorphic to the cohomology ring of a connected sum of products of spheres if and only if either (a) $K$…

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various…

Let $G$ be a compact simple Lie group equipped with the left invariant framing $L$. It is known that there are several groups $G$ such that $(G, L)$ is non-null framed bordant. Previously we gave an alternative proof of these results using…

In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $\infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a…

In this note, we give a self-contained account on a construction for a directed homology theory based on modules over algebras, linking it to both persistence homology and natural homology. We study its first properties, among which some…

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…

We study the Picard group of Franke's category of quasi-periodic $E_0E$-comodules for $E$ a 2-periodic Landweber exact cohomology theory of height $n$ such as Morava $E$-theory, showing that for $2p-2 > n^2+n$, this group is infinite…

We prove that if an $n$-dimensional space $X$ satisfies certain topological conditions then any triangulation of $X$ as well as any its representation as a simplicial set with contractible faces has at least $2^n$ faces of dimension $n$.…

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which…

Thin homotopies, introduced by Caetano-Picken, serve to axiomatize the holonomy of connections on principal bundles. This approach has been generalized to higher non-abelian bundles with connection through transport functors and higher…

We calculate the classical Iwasawa invariants of the Iwasawa modules associated to the $p$-adic topological $K$-theory of finite spectra. We show that the graded average of the orders of $n$ consecutive $K(1)$-local homotopy groups of a…

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension…

Following an outline of Rezk, we give a construction of complex-analytic $G$-equivariant elliptic cohomology for an arbitrary compact Lie group $G$ and we prove some of its fundamental properties. The construction is parametrised over the…

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors…

The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules…

We study the $\mathbb{F}_2$-synthetic Adams spectral sequence. We obtain new computational information about $\mathbb{C}$-motivic and classical stable homotopy groups.

We define a motivic Greenlees spectral sequence by characterising an associated $t$-structure. We then examine a motivic version of topological Hochschild homology for the motivic cohomology spectrum modulo a prime number $p$. Finally, we…