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

Extremal Configuration of Robot Arms in Three Dimensions

We define a volume function for a robot arms in 3-dimensional Euclidean space and give geometric conditions for its critical points. For 3-arms this volume function is an exact topological Morse function on the 3-sphere.Comment: 13 pages; Updated version of sections 6-9 of Oberwolfach preprint 2011-2

Cross Ratios of Quadrilateral Linkages

We discuss the cross-ratio map of planar quadrilateral linkages, also in the case when one of the links is telescopic. Most of our results are valid for a planar quadrilateral linkage with generic lengths of the sides. In particular, we describe the image of cross-ratio map for quadrilateral linkage and planar robot 3-arm.Comment: 12 pages,9 figures. March 2014: Proofs are added, typo's corrected; March 2015: Typo's correcte

Milnor Fibre Homology via Deformation

In case of one-dimensional singular locus, we use deformations in order to get refined information about the Betti numbers of the Milnor fibre.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1411.264

Curvature and Gauss-Bonnet defect of global affine hypersurfaces

The total curvature of complex hypersurfaces in \bC^{n+1} and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect in terms of singularities and tangencies at infinity.Comment: 15 p., some changes in editin