Related papers: Wythoff polytopes and low-dimensional homology of …
We give an different proof of our result computing the stable homology of dihedral group Hurwitz spaces. This proof employs more elementary methods, instead of higher algebra.
The holomorph of a free group $F_n$ is the semidirect product $F_n \rtimes Aut(F_n)$. Using the methods of Hatcher and Vogtmann, we derive stability results and calculate the mod-$p$ homology of these holomorphs for odd primes $p$ in…
A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of…
The endpoint of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we…
This paper aims to use topological methods to compute $\mathrm{Ext}$ between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a…
Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…
In this article we give a unified treatment of the construction of all possible Weitzenb\"ock formulas for all irreducible, non--symmetric holonomy groups. The resulting classification is two--fold, we construct explicitly a basis of the…
We present a complete acyclic matching of the Hasse diagram associated with the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. We will then utilize this…
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…
The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the…
We compute the homotopy groups of $\mathrm{THH}(\mathrm{ku})$ as a $\mathrm{ku}_\ast$-module using the descent spectral sequence for the map $\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$, which is the motivic spectral…
We propose new tools based on basic lattice theory to calculate the integral cohomology of the quotient of a manifold by an automorphism group of prime order. As examples of applications, we provide the Beauville--Bogomolov forms of some…
Spin models like the Heisenberg Hamiltonian effectively describe the interactions of open-shell transition-metal ions on a lattice and can account for various properties of magnetic solids and molecules. Numerical methods are usually…
We use categorical description of the invariant 2-cohomology group of Hopf algebra to compute such cohomology for two finite dimensional Hopf algebras: the group ring of $Z_8\rtimes Aut(Z_8)$ and Kac-Paljutkin algebra. For the first of…
Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they…
This paper presents a discrete homotopy theory and a discrete homology theory for finite posets. In particular, the discrete and classical homotopy groups of finite posets are always isomorphic. Moreover, this discrete homology theory is…
We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In…
We study a certain family of simple fusion systems over finite $3$-groups, ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and $A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to the…
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.
Two results are presented for reduced Yang-Mills integrals with different symmetry groups and dimensions: the first is a compact integral representation in terms of the relevant variables of the integral, the second is a method to…