Related papers: Gottlieb groups of spheres

For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$3$-extension is a Schur $\sigma$-group. If this has Zassenhaus type $(3,3)$, there are 13 possibilities for the isomorphism class of the finite…

For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to\mathbb R^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge…

We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points.…

An algebraic system from a finite set $\Sigma$ of cardinality $k$ and an $n$-ary operation $f$ invertible in each argument is called an $n$-ary quasigroup of order $k$. An autotopy of an $n$-ary quasigroup $(\Sigma,f)$ is a collection…

Let $M_{l,m}$ be the total space of the $S^3$-bundle over $S^4$ classified by the element $l\sigma+m\rho\in{\pi_4(SO(4))}$, $l,m\in\mathbb Z$. In this paper we study the homotopy theory of gauge groups of principal $G$-bundles over…

In this note we present the Stiefel-Whitney classes (SWCs) for orthogonal representations of several finite groups of Lie type, namely for $G=\text{SL}(2,q),$ $\text{SL}(3,q),$ $\text{Sp}(4,q)$, and $\text{Sp}(6,q)$, with $q$ odd. We also…

We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry…

We compute the \v{C}ech homotopy groups of the $m$-dimensional infinite earring space $\mathbb{E}_m$, i.e. a shrinking wedge of $m$-spheres. In particular, for all $n,m\geq 2$, we prove that $\check{\pi}_n(\mathbb{E}_m)$ is isomorphic to a…

Let $n$ be a natural number greater or equal to $3$, $R$ a commutative ring and $\sigma\in GL_n(R)$. We show that $t_{kl}(\sigma_{ij})$ (resp. $t_{kl}(\sigma_{ii}-\sigma_{jj}))$ where $i\neq j$ and $k\neq l$ can be expressed as a product of…

For given spaces $X$ and $Y$, let $map(X,Y)$ and $map_\ast(X,Y)$ be the unbased and based mapping spaces from $X$ to $Y$, equipped with compact-open topology respectively. Then let $map(X,Y;f)$ and $map_\ast(X,Y;g)$ be the path component of…

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…

In this paper, we study the cohomology of the Morava stabilizer algebra $S(3)$. As an application, we show that for $p \geq 7$, if $s\not \equiv 0, \pm 1 \,\, mod \,p $, $n\not \equiv 1 \,\, mod\, 3$, $n>1$, then $\zeta_n\gamma_s$ is a…

We examine the topology of the clique complexes of the graphs of weakly and strongly separated subsets of the set $[n]=\{1,2,...,n\}$, which, after deleting all cone points, we denote by $\hat{\Delta}_{ws}(n)$ and $\hat{\Delta}_{ss}(n)$,…

Infinitely many large Schur sigma-groups G with non-elementary bicyclic commutator quotient G/G' = C(3^e) x C(3), e >= 2, are constructed as periodic sequences of vertices in descendant trees of finite 3-groups. A single root gives rise to…

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our…

The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $SL^+(2,\mathbb{C})$; specifically, $SO^+(Q,\mathbb{Z})$ for $Q$ some integral quadratic form of…

Let $S_n$ denote the symmetric group of order $n$. Say that two subsets $x, y\subseteq S_n$ are \emph{equivalent} if there exist permutations $g_1, g_2\in S_n$ such that $g_1xg_2=y$, where multiplication is understood elementwise. Recently,…

This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$ configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that $\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is odd and…

In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres.

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.