Richard Randell

The lower central series of a fiber-type arrangement

SummaryFor a certain class (“fiber-type”) of arrangements, including the supersolvable ones of Jambu and Terao [3], we prove a formula relating the Poincaré polynomial of the complement with the ranks of successive quotients in the lower central series of the fundamental group. Such a formula was proved by Kohno [5] for the single family of examplesAl.We also show that the formula doesnot hold for allK(π, 1) arrangements.

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 any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of p-cells equals the p-th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are Eilenberg-Mac Lane spaces.

AN ELEMENTARY INVARIANT OF KNOTS

We indicate how certain invariants can be computed for polygonal knots with few edges. This leads to conclusions regarding the minimal number of edges required to represent certain knots.

Homotopy and group cohomology of arrangements

Abstract It is well known that the complexification of the complement of the arrangement of reflecting hyperplanes for a finite Coxeter group is an Eilenberg-MacLane space. In general, the cohomology of the complement of a general complex arrangement is well behaved and well understood. In this paper we consider the homotopy theory of such spaces. In particular, we study the Hurewicz map connecting homotopy and homology. As a consequence we are able to derive understanding of the “obstructions” to such spaces being Eilenberg-MacLane spaces. In particular, in the case of arrangements in a three-dimensional vector space, we find that whether or not the complement is Eilenberg-MacLane depends solely on its fundamental group.

The Milnor fiber of a generic arrangement

and F f l ( 1 ) is the Milnor fiber of the map f . Let ~=exp(27~i/n). Let h*: H*(F) ---~H*(F) be the monodromy induced by h(zl,...,zl)=(~zl,...,~zz). Consider all homology and cohomology with complex coefficients and let bk=dim Hk(F) be the k-th Betti number of F. Since F is a Stein space of dimension ( / 1 ) we have H k ( F ) = 0 for k>_l. If f has an isolated singularity it is known from Milnors work that bk(F)=0 for l < k < l 2 and that bz_t(F)=(n-1) l. The characteristic polynomial of the automorphism induced by the monodromy on Hz-I(F) was computed in [7]. In [9] we gave an explicit basis of differential forms for the nonvanishing group HI-I(F). The classes are all represented as restrictions to F of differential forms qw where q is a homogeneous polynomial and

Invariants of piecewise-linear knots

We study numerical and polynomial invariants of piecewise-linear knots, with the goal of better understanding the space of all knots and links. For knots with small numbers of edges we are able to find limits on polynomial or Vassiliev invariants sufficient to determine an exact list of realizable knots. We thus obtain the minimal edge number for all knots with six or fewer crossings. For example, the only knot requiring exactly seven edges is the figure-8 knot.

Milnor fibrations of lattice-isotopic arrangements

We show that the associated Milnor fibrations are equiivalent in a smooth family of arrangements with constant intersection lattice.

Pure braid groups are not residually free

We show that the Artin pure braid group P n is not residually free for n ≥ 4. Our results also show that the corank of P n is equal to 2 for n ≥ 3.

MÖBIUS TRANSFORMATIONS OF POLYGONS AND PARTITIONS OF 3-SPACE

The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But 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-eight knot can be transformed to a figure-eight knot, a trefoil, 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, most 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 is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ3. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ3. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n. Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ3 and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ3 and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge.

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.