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We prove that the classical map comparing Adams' cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal…

Given a combinatorial (semi-)model category $M$ and a set of morphisms $C$, we establish the existence of a semi-model category $L_C M$ satisfying the universal property of the left Bousfield localization in the category of semi-model…

Let $R$ be a unital ring satisfying the invariant basis number property, that every stably free $R$-module is free, and that the complex of partial bases of every finite rank free module is Cohen--Macaulay. This class of rings includes…

We exploit the theory of $\infty$-stacks to provide some basic definitions and calculational tools regarding stratified homotopy theory of stratified topological stacks.

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is…

Inspired by the ideas and techniques used in the study of cluster algebras we construct a new class of algebras, called bistellar cluster algebras, from closed oriented triangulated even-dimensional manifolds by performing…

We introduce a notion of formally \'etale $\mathbb{E}_{\infty}$-coalgebras and show that they admit essentially unique, functorial lifts along square zero extensions of $\mathbb{E}_{\infty}$-rings. Using this, we show that for a perfect…

The combinatorial interpretation of the persistence diagram as a M\"obius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a…

Kadeishvili's minimal model theorem establishes the existence of an $A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg associative algebra, which captures its homotopy type. In this note we prove the existence of…

After 1-point compactification, the collection of all unordered configuration spaces of a manifold admits a commutative multiplication by superposition of configurations. We explain a simple (derived) presentation for this commutative…

For any field $\Bbb F$ and positive integers $m,n,d$ with $(m,n)\not= (1,1)$, Farb and Wolfson defined the certain affine varieties ${\rm Poly}^{d,m}_n(\Bbb F)$ as generalizations of spaces first studied by Arnold, Vassiliev, Segal and…

We prove that for finitely generated abelian groups $A$ and $B$, the space of $\mathbb{E}_\infty$-ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A \to…

For an arbitrary symmetric monoidal $\infty$-category $\mathcal{V}$, we define the factorization homology of $\mathcal{V}$-enriched $(\infty,1)$-categories over (possibly stratified) 1-manifolds and study some of its basic properties. In…

In this paper, we establish a theorem that proves a condition when an inclusion morphism between simplicial sets becomes a weak homotopy equivalence. Additionally, we present two applications of this result. The first application…

We give a characterization of locally standard, $\mathbb{Z}$-equivariantly formal manifolds in general position. In particular, we show that for dimension $2n$ at least $10$, to every such manifold with labeled GKM graph $\Gamma$ there is…

Let $G$ be a finite group and $V$ a finite dimensional (non-zero) orthogonal $G$-module such that, for each prime $p$ dividing the order of $G$, the subspace of $V$ fixed by a Sylow $p$-subgroup of $G$ is non-zero and, if the dimension of…

We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action…

For a fixed integer $d\geq 1$, we show that two quasitoric manifolds over a product of $d$-simplices are homotopy equivalent after appropriate localization, provided that their integral cohomology rings are isomorphic.

When $p$ is an odd prime, we prove that the $\mathbb F_p$-cohomology of $\mathrm{BP}\langle n\rangle$ as a module over the Steenrod algebra determines the $p$-local spectrum $\mathrm{BP}\langle n\rangle$. In particular, we prove that the…

For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ associated to a point cloud $\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of closed subspaces…