Related papers: On the group structure of $[\Omega \mathbb S^2, \O…

We introduce the notion of an $L_{\infty}$-bialgebra structure on a vector space. We show that the rational homotopy groups $\pi_{*}(\Omega \Sigma Y)\otimes \mathbb{Q}$ admit such a structure for the loop space $\Omega \Sigma Y$ of a…

Let $F_n(\Sigma_{g,1})$ denote the configuration space of $n$ ordered points on the surface $\Sigma_{g,1}$ and let $\Gamma_{g,1}$ denote the mapping class group of $\Sigma_{g,1}$. We prove that the action of $\Gamma_{g,1}$ on…

The goal of this thesis is to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory,…

For a closed connected manifold N, we establish the existence of geometric structures on various subgroups of the contactomorphism group of the standard contact jet space J^1N, as well as on the group of contactomorphisms of the standard…

We let the mapping class group $\Gamma_{g,1}$ of a genus $g$ surface $\Sigma_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(\Sigma_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove…

Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if $N(G)=\Omega\times \{1,n\}$ for specific set $\Omega$ of integers, then $G\simeq A\times B$ where $N(A)=\Omega$, $N(B)=\{1,n\}$, and $n$ is a power of prime.

In the first part of this paper we present a formalization in Agda of the James construction in homotopy type theory. We include several fragments of code to show what the Agda code looks like, and we explain several techniques that we used…

We describe the structure present in algebras over the little disks operads for various representations of a finite group $G$, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in…

The Johnson filtration of the mapping class group of a compact, oriented surface is the descending series consisting of the kernels of the actions on the nilpotent quotients of the fundamental group of the surface. Each term of the Johnson…

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let $G$ be a Lie group acting on a space…

Let $H$ be a group and $E$ a set such that $H \subseteq E$. We shall describe and classify up to an isomorphism of groups that stabilizes $H$ the set of all group structures that can be defined on $E$ such that $H$ is a subgroup of $E$. A…

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, not necessarily connected, with Lie algebra $g$, endowed with an adjoint action invariant scalar product, let $\xi \colon P \to \Sigma$ be a principal $G$-bundle, and pick a…

If $(A_n)_n$ is a decreasing filtration of a module $A$ and $\widehat{A} = \lim_n A/A_n$, then $\lim^1_n A_n$ is identified with the cokernel of the canonical map $A \longrightarrow \widehat{A}$. In this note, we show that any…

We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case…

We consider Yang-Mills theory with $N{=}1$ super translation group in eleven auxiliary dimensions as the structure group. The gauge theory is defined on a direct product manifold $\Sigma_3\times S^1$, where $\Sigma_3$ is a three-dimensional…

We calculate the structure of the finitely-generated groups H_2(SL_2(Z[1/m])) when m is a multiple of 6. We construct explicit homology classes which generate these groups and have prescribed orders. When n is at least 2 and m is the…

Given a simplicial complex $X$, we construct a simplicial complex $\Omega X$ that may be regarded as a combinatorial version of the based loop space of a topological space. Our construction explicitly describes the simplices of $\Omega X$…

We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has…

On the space ${\cal L}$, of loops in the group of Hamiltonian symplectomorphisms of a symplectic quantizable manifold, we define a closed ${\bf Z}$-valued 1-form $\Omega$. If $\Omega$ vanishes, the prequantization map can be extended to a…

Let $\Sigma$ be a closed orientable surface of genus at least two, and let $X, Y$ be distinct marked Riemann surface structures on $\Sigma$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably…