代数拓扑

We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent…

We improve the central stability ranges for H2(Torelli subgroup of Aut(Fn)'s) as GL_n(Z)-representations, H2(Torelli subgroup of mapping class groups) as Sp_2g(Z)-representations, Hk(congruence subgroups of GL_n(R)'s) as…

We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles. The former are Gibbons-Manton torus bundles over…

We study the structure of mod 2 cohomology rings of oriented Grassmannians $\tilde{\operatorname{Gr}}_k(n)$ of oriented $k$-planes in $\mathbb{R}^n$. Our main focus is on the structure of the cohomology ring ${\rm…

Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. Here we introduce a Hodge-theoretic approach to spectral theory and dimensionality reduction for…

We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our…

In 1970 Vasquez proved that to every finite group $G$ we can assign a natural number $n(G)$ with the property that every flat manifold with holonomy $G$ is a total space of a fiber bundle, with the fiber being a flat torus and the base…

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. Then there is a map $\Theta\colon\mathrm{Hom}(\pi,G)_0\to\mathrm{map}_*(B\pi,BG)_0$ between the null-components of the spaces of homomorphism and based maps, which…

Motivated by various developments in algebraic combinatorics and its applications, we investigate here the fine structure of a fundamental but little known theorem, the Gerstenhaber and Schack cohomology comparison theorem.The theorem…

Ando established an algebraic criterion for when a complex orientation for a Morava E-theory is an $H_\infty$-map. The criterion relates such an orientation to a specific property of the formal group associated to the E-theory, namely, a…

We determine $\pi_*(BDiff_\partial(D^{2n})) \otimes \mathbb{Q}$ for $2n \geq 6$ completely in degrees $* \leq 4n-10$, far beyond the pseudoisotopy stable range. Furthermore, above these degrees we discover a systematic structure in these…

We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in ${\mathbb C}^k$, $k \geq 3$, and carry out these calculations to obtain these groups of low dimensions.

A generalized-homology bordism-theory is constructed, such that for certain manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant geometric fundamental-classes exist. The construction combines three ideas: Firstly,…

We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on…

We apply persistent homology, the main method in topological data analysis, to the study of demographic data. Persistence diagrams efficiently summarize information about clusters or peaks in a region's demographic data. To illustrate how…

The realization theorem asserts that for a finitely presented group G, the D(2) property and the realization property are equivalent as long as G satisfies a certain finiteness condition. We show that the two properties are in fact…

Using the pro\'etale site, we construct models for the continuous actions of the Morava stabiliser group on Morava E-theory, its $\infty$-category of $K(n)$-local modules, and its Picard spectrum. For the two sheaves of spectra, we evaluate…

In this largely expository paper, we will present a list of En -operads and give complete, and in some cases new, proofs of the equivalences between these operads.

An analog of the Tits building is defined and studied for commutative rings. We prove a Solomon-Tits theorem when $R$ either satisfies a stable range condition, or is the ring of $S$-integers of a global field. We then define an analog of…

In 2020, Alexander Grigor'yan, Yong Lin and Shing-Tung Yau [4] introduced the Reidemeister torsion and the analytic torsion for digraphs by means of the path complex and the path homology theory. Based on the analytic torsion for digraphs…