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Let $X$ be a rationally elliptic space. Utilizing the Gorenstein algebra structure of $X$, we present three algorithms that together induce a generating class of $Ext^N_{(\Lambda V,d)}(\mathbb{Q},(\Lambda V,d))$ with $N$ being the formal…

The bedrock of persistence theory over a single parameter is decomposition of persistence modules into intervals. In [HLM24], the authors leveraged interval decomposition to produce a cell decomposition of the minimal model of a simply…

Let $BP$ denote the Brown-Peterson spectrum at a prime $p$, whose homotopy groups are isomorphic to the polynomial algebra generated by elements $v_i$'s for $i\ge 1$. We consider the homotopy groups of the $v_n^{-1}BP$-localized sphere…

In this note we show that in the simplicial setting, the classifying space construction converts short exact sequences of groups not just to homotopy fibrations, but in fact to fibre bundles.

We review the Chang-Skjelbred lemma for torus-equivariant cohomology and discuss several generalizations of it: to other coefficients, other groups and also to syzygies in equivariant cohomology and the Atiyah-Bredon sequence.

Building on the recent computation of the cohomology rings of smooth toric varieties and partial quotients of moment-angle complexes, we investigate the naturality properties of the resulting isomorphism between the cohomology of such a…

We prove that Szczarba's twisting cochain is comultiplicative. In particular, the induced map from the cobar construction of the chains on a 1-reduced simplicial set X to the chains on the Kan loop group of X is a quasi-isomorphism of dg…

A dga model for the integral singular cochains on a moment-angle complex is given by the twisted tensor product of the corresponding Stanley-Reisner ring and an exterior algebra. We present a short proof of this fact and extend it to real…

We upgrade the Cauchy--Frobenius Lemma (`Burnside's Lemma') to a homotopy equivalence of $\infty$-groupoids, essentially given by double counting/Fubini in the free loop space of the quotient.

We give down-to-earth proofs of the structure theorems for persistence modules.

Visual paradoxes like the Penrose staircase present a fundamental tension: locally coherent geometric relationships that cannot be realized globally. Inspired by Penrose's observations connecting such paradoxes to cohomology, we develop a…

The synthetic analogue functor $\nu$ from spectra to synthetic spectra does not preserve all limits. In this paper, we give a necessary and sufficient criterion for $\nu$ to preserve the global sections of a derived stack. Even when these…

In this article we generalize the main structure theorems of rational homotopy theory to the persistent setting. Our main motivation is the computation of an explicit finite, cellular presentation of the persistent minimal model that…

J. Milnor introduced a specific class of codimension-$1$ submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides…

Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, there are still very few examples of racks whose homology has…

In this paper, we present an explicit method to identify equivariant suboperads of coinduced operads that contain only fixed points associated to any desired transfer system. Our method works for a class of operads that we call intersection…

Defining cellular sheaves beyond graph structures, such as on simplicial complexes containing higher-dimensional simplices, is an essential and intriguing topic in topological data analysis (TDA) and the development of sheaf neural…

Building on ideas of Kohno, we develop a framework for the construction of higher holonomy functors via the transport of formal power series connections. Using these techniques, we obtain functors from the path groupoid, the path…

We construct and study a morphism of spectra implementing the Anderson duality of topological modular forms ($\mathrm{TMF}$). Its differential version will then be introduced, allowing us to pair elements of $\pi_d\mathrm{TMF}$ with spin…

Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. Using the Serre spectral sequence associated to the fibration $BU_n\to BPU_n\to K(\mathbb{Z},3)$, we compute the integral cohomology…