Medial/skeletal linking structures for multi-region configurations

We consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{\Omega_i\}$ in $\mathbb{R}^{n+1}$ which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal{B}_i$ in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal structures" on each of the regions, allowing us to significantly extend the mathematical methods introduced for single regions to the configuration. We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. This requires proving several transversality theorems for certain associated "multi-distance" and "height-distance" functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties. In addition to yielding geometric invariants which capture the shapes and geometry of individual regions, the linking structures are used to define invariants which measure positional properties of the configuration such as: measures of relative closeness of neighboring regions and relative significance of the individual regions for the configuration. These invariants, which are computed by formulas involving "skeletal linking integrals" on the internal skeletal structures, are then used to construct a "tiered linking graph," which identifies subconfigurations and provides a hierarchical ordering of the regions.Comment: 135 pages, 36 figures. Version to appear in Memoirs of the Amer. Math. So

Topology of Exceptional Orbit Hypersurfaces of Prehomogeneous Spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.Comment: to appear in the Journal of Topolog

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities II: Vanishing Topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups. Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix singularities. We compute the Milnor number of isolated Cohen--Macaulay surface singularities of this type in $\mathbb{C}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of this type in $\mathbb{C}^5$.Comment: 53 pages. To appear in Geometry & Topology. Changes in response to helpful referee: replace the erroneous Corollary 6.2 with a new version, specify that we consider 2x3 Cohen-Macaulay singularities, calculate more entries of Table 5, improve wording, format for publicatio

On the freeness of equisingular deformations of plane curve singularities

We consider surface singularities in arising as the total space of an equisingular deformation of an isolated curve singularity of the form f(x,y)+zg(x,y) with f and g weighted homogeneous. We give a criterion that such a surface is a free divisor in the sense of Saito. We deduce that the Hessian deformation defines a free divisor for nonsimple weighted homogeneous singularities, and that the failure of this property “almost” characterizes the simple singularities. The criterion also yields distinct deformations of the same curve singularity, exactly one of which is free, showing that freeness is not a topological property

Topological triviality of versal unfoldings of complete intersections

We obtain algebraic and geometric conditions for the topological triviality of versal unfoldings of weighted homogeneous complete intersections along subspaces corresponding to deformations of maximal weight. These results are applied: to infinite families of surface singularities in $C^4$ which begin with the exceptional unimodular singularities, to the intersection of pairs of generic quadrics, and to certain curve singularities

Swept regions and surfaces: Modeling and volumetric properties

We consider “swept regions” and “swept hypersurfaces” in (and especially ) which are a disjoint union of subspaces or obtained from a varying family of affine subspaces . We concentrate on the case where and are obtained from a skeletal structure . This generalizes the Blum medial axis of a region , which consists of the centers of interior spheres tangent to the boundary at two or more points, with denoting the vectors from the centers of the spheres to the points of tangency. We extend methods developed for skeletal structures so that they can be deduced from the properties of the individual intersections or and a relative shape operator , which we introduce to capture changes relative to the varying family . We use these results to deduce modeling properties of the global in terms of the individual , and determine volumetric properties of regions expressed as global integrals of functions on in terms of iterated integrals over the skeletal structure of which is then integrated over the parameter space