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For a $C_2$-commutative ring spectrum $R$, a twisted $R$-algebra is an $R$-module with a multiplication whose order is switched by the $C_2$-action. In this paper, we construct various quotients of $R$ as twisted $R$-algebras, when $R$ is…

Classical equivariant bordism theories require computing the full collection of equivariant characteristic numbers to detect whether an equivariant manifold bounds equivariantly or not. This paper establishes simplified equivariant bordism…

Classically, the Pontryagin-Thom isomorphism asserts that the multiplicative cohomology theory given by (structured) geometric cobordism is isomorphic to the cohomology theory determined by an associated Thom spectrum. We construct a…

In this article, we investigate the relationship between the fundamental group of a space and its continuous transformations. To be more precise, we show that if a continuous action of a Lie group on a space admits a simply connected…

We study homology manifolds through the eyes of the six functor formalism of spectral sheaves on locally compact Hausdorff spaces. As main results, we characterize cohomologically smooth objects by adapting an argument of Scholze, deduce…

We give algebraic models for the tame homotopy type of the configuration spaces of certain algebraic varieties of Tate type. Such tame models carry information on the l-adic homotopy type. Our method uses the theory of weights in \'etale…

We construct closed manifolds with vanishing L^2-Betti numbers over every field) which do not virtually fibre over the circle. The class of fundamental groups that occurs is the largest possible, and in many cases the dimension may be taken…

Zigzag persistence tracks topological features in spatio-temporal data through combinatorial invariants called barcodes. For binary videos, existing methods are bottlenecked by the construction of prohibitively large cubical complexes and…

We determine the continuous mod $p$ homology of the topological periodic homology $TP(MU)$ of the complex cobordism spectrum, as a graded algebra with Steenrod operations. The answer is given in terms of an explicit and purely algebraic…

Let $G$ be a compact Lie group, $M$ be a smooth manifold with a $G$ action, then all the data of this model is contained in the action groupoid $G\ltimes M$. If $U_y$ is a small enough neighbourhood of $y\in M/G$, the slice theorem says…

This paper characterizes the stabilized second James-Hopf invariant by means of three axioms. Specifically, we show that it is the unique natural transformation satisfying the Cartan formula, vanishing on suspensions, and a metastable EHP…

We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street--Roberts conjecture, presenting $(\infty,\infty)$-categories as sheaves on…

We propose a model of higher homotopy theory of $L_{\infty}[1]$-morphisms as a natural generalization of the $A_{\infty}$-homotopies defined by Fukaya-Oh-Ohta-Ono \cite{FOOO1}. Within this framework, we show that a filling condition holds…

We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations.…

We show that the category of lax involutive $n$-Frobenius algebras is Quillen equivalent to the category of right comodules of the $n$-Poisson cooperad. It follows in particular, that the cohomology of a parallelized $n$-manifold is…

We study a family of generalizations of the notion of Euler characteristic of discrete groups (or of orbifolds, depending on one's perspective) indexed on the natural numbers. For $n=0$, this is the classical orbifold Euler characteristic…

Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the…

Let $\Gamma_{2n}^\omega(p)$ be the level-$p$ principal congruence subgroup of $\text{Sp}_{2n}(\mathbb{Z})$ for all prime $p$. Borel--Serre demonstrated that the cohomology of $\Gamma_{2n}^\omega(p)$ vanishes above degree $n^2$. We prove…

We introduce the wreath product for a class of operadic categories and use it to construct an explicit isomorphism between the Boardman-Vogt tensor product of two colored operads in Set and an operad induced by the wreath product of…

Let $\mathcal{G}_k$ denote the gauge group of the principal $G_2$--bundle over $S^4$ classified by $k\in \pi_4(BG_2)\cong \mathbb Z$. Motivated by the $p$--local homotopy classification of these gauge groups, due to…