arXiv subjects
Publications and source records attributed to Emmanuel D. Farjoun.
K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space $K_ω.$ The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.
The purpose of this note is to observe that a homomorphism of discrete groups $f:Γ\to G$ arises as the induced map $π_0(\mathfrak{M})\to π_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by homotopically discrete topological groups.
Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have strong ties to topological constructions.
In this note, starting with any group homomorphism $f\colonΓ\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $Γ$ with the same surjective property, such that for any central extension $m\colon M\twoheadrightarrow G,$ under $f,$ there is a unique homomorphism $U\to M$ with the obvious commutation condition. The kernel of $u$ is the relative Schur multiplier group $H_2(G,Γ;\mathbb{Z})$ as defined in the paper. The case where $G$ is perfect corresponds to $Γ=1$. This yields homological obstructions to lifting solution of equations in $G.$ Upon repetition, for finite groups, this gives a universal hypercentral factorization of the map $f\colonΓ\to G$.
Let $φ\colonΓ\to G$ be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations $Γ\xrightarrowψ M\xrightarrow{n} G$ of $φ,$ with $n$ a subnormal map. We search for a universal such factorization. When $Γ$ and $G$ are finite we show that such universal factorization exists: $Γ\toΓ_{\infty}\to G,$ where $Γ_{\infty}$ is a hypercentral extension of the subnormal closure $\mathcal{C}$ of $φ(Γ)$ in $G$ (i.e.~the kernel of the extension $Γ_{\infty}\to {\mathcal C}$ is contained in the hypercenter of $Γ_{\infty}$). This is closely related to the a relative version of the Bousfield-Kan $\mathbb{Z}$-completion tower of a space. The group $Γ_{\infty}$ is the inverse limit of the normal closures tower of $φ$ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit $Γ_{\infty}$.
Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets or the category of chain complexes over a commutative ring.
In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel K of the cellular map G-> M . We show that in general a torsion free reduced abelian group M may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group M has certain ``freeness'' properties, then its cardinality must be bounded.