Pure braid groups are not residually free
We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.
math.GR↗
arXiv subjects
Richard Randell
Publications and source records attributed to Richard Randell.
Homotopy groups and twisted homology of arrangements
Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generalizing both results.
math.GT↗
Mobius transformations of polygons and partitions of 3-space
The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K. Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon. This may be a different knot type. For example, a certain 7-segment figure-8 knot can be transformed to a figure-8 knot, right and left handed trefoils, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, almost all n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Mobius group. The number of knot types arising from an n-vertex polygon is bounded by the number of complementary domains of a certain system of n(n-3)/2 round 2-spheres in R^3. We show the number of domains is at most polynomial in the number of spheres. In the analysis, we obtain an exact formula for the number of complementary domains. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.
math.GT↗
Morse theory, Milnor fibers and minimality of hyperplane arrangements
Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of a complex hyperplane arrangement has the homotopy type of a CW complex in which the number of p-cells equals the p-th betti number.
math.AT↗
On the homotopy theory of arrangements, II
In "On the homotopy theory of arrangements," published in 1986, the authors gave a comprehensive survey of the subject. This article updates and continues the earlier article, noting some key open problems.
math.CO↗