arXiv subjects

F. R. Cohen

Publications and source records attributed to F. R. Cohen.

22 records · Page 2Linked to original sources

arXiv2006

Configurations, and parallelograms associated to centers of mass

The purpose of this article is to \begin{enumerate} \item define $M(t,k)$ the $t$-fold center of mass arrangement for $k$ points in the plane, \item give elementary properties of $M(t,k)$ and \item give consequences concerning the space $M(2,k)$ of $k$ distinct points in the plane, no four of which are the vertices of a parallelogram. \end{enumerate} The main result proven in this article is that the classical unordered configuration of $k$ points in the plane is not a retract up to homotopy of the space of $k$ unordered distinct points in the plane, no four of which are the vertices of a parallelogram. The proof below is homotopy theoretic without an explicit computation of the homology of these spaces. In addition, a second, speculative part of this article arises from the failure of these methods in the case of odd primes $p$. This failure gives rise to a candidate for the localization at odd primes $p$ of the double loop space of an odd sphere obtained from the $p$-fold center of mass arrangement. Potential consequences are listed.

math.AT↗

arXiv2005

On the degree 2 map for a sphere

The purpose of this article is to compare the two self-maps of $Ω^kS^{2n+1}$ given by $Ω^k[2]$ the $k$-fold looping of a degree 2 map and $Ψ^k(2)$ the H-space squaring map. The main results give that in case $2n+1 \neq 2^j-1$, these maps are frequently not homotopic and also that their homotopy theoretic fibres are not homotopy equivalent. The methods are a computation of an unstable secondary operation constructed by Brown and Peterson in the first case and the Nishida relations in the second case. One question left unanswered here is whether the maps $Ω^{2n+1}[2]$ and $Ψ^{2n+1}(2)$ are homotopic on the level of $Ω^{2n+1}_0S^{2n+1}$. A natural conjecture is that these two maps are homotopic.

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arXiv2005

On injective homomorphisms for pure braid groups, and associated Lie algebras

The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie algebraic condition which guarantees that a homomorphism out of the classical pure braid group is faithful. However, it is unclear whether the methods here can be applied to any open cases such as the Gassner representation.

math.GR↗

arXiv2004

Braid groups, free groups, and the loop space of the 2-sphere

The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno, and provides connections between these various topics. Two consequences are as follows: 1) the homotopy groups of spheres are identified as "natural" sub-quotients of free products of pure braid groups, and 2) an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected $CW$-complexes.

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