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We study cobordisms of nested manifolds, which are manifolds together with embedded submanifolds, which can themselves have embedded submanifolds, etc. We identify a nested analog of the Pontryagin-Thom construction. Moreover, when the…

We provide an axiomatic treatment of Quillen's construction of the model structure on topological spaces to make it applicable to a wider range of settings, including $\Delta$-generated spaces and pseudotopological spaces. We use this…

The diagonal lemma asserts that if a map of bisimplicial sets is a levelwise weak equivalence in the Kan-Quillen model structure, then it induces a weak equivalence of the diagonal simplicial sets. In this short note, we observe that the…

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric…

In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain…

We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for ($n$-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for…

We propose a new model for the theory of $(\infty,n)$-categories (including the case $n=\infty$) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our…

We characterise the profinite Grothendieck-Teichm\"uller group $\widehat{\mathsf{GT}}$ as the group of automorphisms of the profinite completion of a cyclic operad of parenthesised ribbon braids. This operad generates a symmetric monoidal…

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…

Using configuration space level Pontryagin--Thom constructions, we construct a simple Koszul self duality map for the little disks operad $\Sigma^\infty_+ E_n$. For a framed $n$-manifold $M$, we show that a compatible self duality map…

Given a (colored) operad and a set of unary operations, we can form an associated $\infty$-operad via localization. We show that localization determines an equivalence of homotopy theories of relative operads and $\infty$-operads. As an…

Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the…

In this paper, we study the Hochschild cohomology of diagrams of algebras introduced by Gerstenhaber and Schack and provide computations for filtrations of incidence algebras. Our aims are threefold: firstly, we revisit and explore the…

Real toric manifolds are the real loci of nonsingular complete toric varieties. In this paper, we calculate the integral cohomology groups of real toric manifolds in terms of the combinatorial data contained in the underlying simplicial…

Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where $m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $\mathcal{Z}_K$, and a canonical action of $\mathbb{Z}_2^m$ on…

For an integer $r\ge 2$, the space of $r$-immersions of $M$ in $\R^n$ is defined to be the space of immersions of $M$ in $\R^n$ such that at most $r-1$ points of $M$ are mapped to the same point in $\R^n$. The space of $r$-immersions lies…

We discuss filtrations arising from de Rham-type cohomology theories for $E_\infty$ rings and $E_n$ rings. Examples include the HKR filtration on relative topological Hochschild homology, the Hodge filtration on $E_\infty$ infinitesimal…

Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $\pi_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these…

We introduce and study structured enhancement of the notion of a crossed simplicial group, which we call an operadic crossed simplicial group. We show that with each operadic crossed simplicial group one can associate a certain operad in…

We will use the tools developed in [Rie24] to give a Morse-theoretic description of a string topology product on the homology of the space of paths in a manifold Y with endpoints in a submanifold X and a module structure on this homology…