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We study toric orbifolds of real dimension four with vanishing odd-degree cohomology and obtain a basis for its degree-two equivariant cohomology with integral coefficients by identifying it with the intersection of certain lattices. As…
We produce five 192-periodic infinite families of simple $\eta$-torsion elements in the stable homotopy groups of spheres with trivial image under the tmf-Hurewicz homomorphism. We also establish that several other 192-periodic families in…
We provide a multiplicative classification of polynomial endofunctors on spectra in terms of their Mackey functors of cross--effects. More precisely, we prove that various categories of multivariable excisive functors from spectra to…
We compute all the Chern, Milnor and Pontryagin numbers for canonical toric manifolds associated with abstract simplicial complexes and the Stiefel-Whitney numbers for their real counterparts. Applications include combinatorial…
We construct a category $\OrdFor$ as an arboreal extension of $\Delta_{\mathrm{epi}}\subseteq\Delta$, whose morphisms are ordered forests composed by grafting. We define a full functor $\pi\colon \OrdFor\to\Delta_{\mathrm{epi}}^{op}$…
We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and…
We define a generalization of (coloured) operads based on double lax functors and we construct a model structure on the associated category of generalized simplicial (coloured) operads. In particular, we obtain a model structure on the…
We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the…
Suppose $G$ is a finite group. In this paper, we construct an equivalence between the $\infty$-category of algebras over an $N_{\infty}$-operad $\mathcal{O}$ associated to a $G$-indexing system $\mathcal{I}$ and the corresponding…
We prove that the forgetful functor from algebras over an $N_{\infty}$-operad to equivariant spaces preserves sifted homotopy colimits
We show that, for a principal ideal domain R, the homotopy category of Bousfield R-local spaces injects fully faithful into a homotopy category of simplicial pointed flat coalgebras.
Let $m > n \ge 2$, and let $N$ be an $(n-1)$-connected $2n$-Poincar\'e complex. In this paper, we establish sufficient conditions under which the loop space of the total space $M$ of the sphere bundle $S^{m-1} \to M \to N$ (associated to a…
Transfer systems are a combinatorial model for $N_{\infty}$-operads, which encode commutative structures in equivariant homotopy theory. Blumberg--Hill and Chan gave criteria for when two transfer systems are a compatible pair, meaning they…
We systematically study L^2-Betti numbers in zero and prime characteristic and apply them to a conjecture of Wise stating that all towers of a finite 2-complex are non-positive if and only if the second L^2-Betti number vanishes.
Given a functor $\varphi : \mathcal{C} \to \mathcal{D}$ between two small categories, there is a homotopy equivalence $\kappa: hocolim _{\mathcal{D}} N(\varphi /-) \to N\mathcal{C}$ where $N(\varphi/-)$ is the functor which sends every…
The classical bar-cobar adjunction between dg algebras and dg coalgebras goes back to the origins of differential homological algebra as developed by Cartan, Eilenberg, Moore, and many others, and is part of the broader framework of Koszul…
An orientation theory for flow categories without bubbling is determined by a functor of $\infty$-categories $\mu \colon \mathcal{C} \to U/O$. For any such functor, we construct a stable $\infty$-category $\mathcal{F}low^{\mu}$ of…
We generalise Nielsen theory to coincidences of pairs $(f,g)$ where $f:X\multimap Y$ is $n$-valued multimap and $g:X\to Y$ is a single-valued map, for $X$ and $Y$ closed oriented triangulable manifolds of equal dimension. We prove a Wecken…
We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify…
Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of…