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We introduce the notion of an E_k-ring with prelogarithmic structure, define logarithmic topological Hochschild homology and logarithmic topological cyclic homology in this context, and establish localization sequences for these theories.…

In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices.…

We investigate forms of the Hopf invariant one problem in motivic homotopy theory over arbitrary base fields of characteristic not equal to $2$. Maps of Hopf invariant one classically arise from unital products on spheres, and one…

We compare spaces of non-singular algebraic sections of ample vector bundles to spaces of continuous sections of jet bundles. Under some conditions, we provide an isomorphism in homology in a range of degrees growing with the jet ampleness.…

The homotopy theory of the blow up construction in algebraic and symplectic geometry is investigated via two approaches. The first approach introduces and develops fibrewise surgery theory, for which the fibrewise framing is characterized…

Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have previously constructed a Gysin sequence relating the cohomology of the manifold $M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an exotic…

Let $\mathbb{K}$ be a number field with ring of integers $\mathfrak{O}$ and let $\mathcal{G}$ be a Chevalley group scheme not of type $\mathtt{E}_8$, $\mathtt{F}_4$ or $\mathtt{G}_2$. We use the theory of Tits buildings and a result of…

In this paper, we introduce the GKM theoretical counterpart of the equivariant complex vector bundles as the "leg bundle". We also provide a definition for the projectivization of a leg bundle and prove the Borel-Hirzebruch type formula for…

This paper provides conditions for Morava $K$-theory to commute with certain homotopy limits. These conditions extend previous work on this question by allowing for homotopy limits of sequences of spectra that are not uniformly bounded…

Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is…

We use sheaf theory and the six operations to define and study the (equivariant) homology of stacks. The construction makes sense in the algebraic, complex-analytic, or even topological categories.

We give a complete description of which non-torsion generators are not in the image of the Thom morphism from complex cobordism to integral cohomology for the classifying space of exceptional Lie groups except for E_8. We then show that the…

We prove that if $G=(\mathbb{Z}/2)^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\mathbb{R}P ^{n_1} \times \cdots \times \mathbb{R} P ^{n_k}$ with trivial action on the mod-$2$ cohomology, then…

We prove injectivity of the canonical map from singular homology to measure homology for certain ``mildly wild" spaces, that is, certain spaces not having the homotopy type of a CW-complex, but having countable fundamental groups.

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric…

Given a compact Lie group $G$ and its finite subgroup $H$ we prove that the $\infty$-category of $G/H$-framed $G$-disc algebras taking values in a $G$-symmetric monoidal category $\underline{\mathcal{C}}^{\otimes}$ is equivalent to the…

We study the homotopy theory of polyhedral products associated to a combinatorial generalisation of manifolds known as pseudomanifolds. As special cases, we show that loop spaces of moment-angle manifolds associated to triangulations of…

Let $A$ be a real soft function algebra. In arXiv:2208.11431 we have obtained a canonical splitting $\mathrm{H}^* (\Omega ^\bullet _{A|\mathrm{R}}) \cong \mathrm{H} ^* (X,\mathrm{R})\oplus \text{(something)}$ via the canonical maps…

Dualising the construction of a polyhedral product, we introduce the notion of a polyhedral coproduct as a certain homotopy limit over the face poset of a simplicial complex. We begin a study of the basic properties of polyhedral…

We prove a homological stability theorem for families of discrete groups (e.g. mapping class groups, automorphism groups of free groups, braid groups) with coefficients in a sequence of irreducible algebraic representations of arithmetic…