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We prove that the loop space of the moment-angle complex associated to the $k$-skeleton of a flag complex belongs to the class $\mathcal{P}$ of spaces homotopy equivalent to a finite type product of spheres and loops on simply connected…

In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a…

In this paper, we calculate characteristic classes for certain quotients of real Stiefel manifolds $V_{n,k}$ and then derive results on certain numerical invariants, such as characteristic rank and skew embedding dimension, for those…

We show that global vanishing of Massey products on a commutative differential graded algebra is not invariant under field extension. Non-vanishing triple Massey products remain non-vanishing upon field extension, while higher Massey…

We prove that if a complex genus $\varphi \colon \varOmega^U \to R$ is rigid on $SU$-manifolds with a torus action then $\varphi$ is the elliptic Krichever genus.

The $S$-model category structures on filtered chain complexes and bicomplexes were introduced by Cirici, Egas Santander, Livernet and Whitehouse and later generalised by this author. In this paper we show they are left proper, cellular and…

This paper determines tangential structure set of $\#_k\mathbb{C}P^{n}$, for $3\leq n \leq 7$, by analyzing their stable cohomotopy groups and $KO$-groups. As a consequence, it establishes the existence of manifolds with tangential homotopy…

We establish monoidal model structures on model categories of filtered chain complexes constructed by Cirici, Egas Santander, Livernet and Whitehouse whose weak equivalences are the quasi-isomorphisms on the $r$-page of the associated…

Let $PU_n$ denote the projective unitary group of rank $n$, and let $BPU_n$ be its classifying space. We extend our previous results to a description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+9$ by showing that $p$-primary subgroups of…

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the $G$-Moore space problem. Specifically, given a finite group $G$, we consider a collection…

The rational Borel equivariant cohomology for actions of a compact connected Lie group is determined by restriction of the action to a maximal torus. We show that a similar reduction holds for any compact Lie group $G$ when there is a…

We describe the Goodwillie calculus of polyhedral products in the case that the fat wedge filtration on the associated real moment-angle complex is trivial. We do this by analysing the behaviour on calculus of the Denham-Suciu fibre…

While branching network structures abound in nature, their objective analysis is more difficult than expected because existing quantitative methods often rely on the subjective judgment of branch structures. This problem is particularly…

We introduce higher-order Massey products for algebras over algebraic operads. This extends the work of Fernando Muro on secondary ones. We study their basic properties and behavior with respect to morphisms of algebras and operads and give…

We show that the spaces of holomorphic and continuous maps from a smooth complex projective variety to a projective space have the same homology in a range depending on the degree of the maps.

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free…

We introduce $\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of Euclidean vector bundle when $\varepsilon = 0$. In particular, we study \v{C}ech…

It is well-known that the stable model structure on symmetric spectra cannot be transferred from the one on sequential spectra through the forgetful functor. We use the fibrant transfer theorem of Guetta--Moser--Sarazola--Verdugo to show it…

We abstract and generalize homotopical monadicity statements, placing in a single conceptual framework a range of old and recent recognition and characterization principles in iterated loop space theory in classical, equivariant, and…

If $E$ is a connective ring spectrum, then Pstragowski's category $Syn_E$ of $E$-synthetic spectra is generated by the bigraded spheres $S^{i,j}$. In particular, it is equivalent to the category of modules over a filtered ring spectrum.