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We consider the topology for a class of hypersurfaces with highly nonisolatedsingularites which arise as exceptional orbit varieties of a special class ofprehomogeneous vector spaces, which are representations of linear algebraicgroups with open orbits. These hypersurface singularities include bothdeterminantal hypersurfaces and linear free (and free*) divisors. Althoughthese hypersurfaces have highly nonisolated singularities, we determine thetopology of their Milnor fibers, complements and links. We do so by using theaction of linear algebraic groups beginning with the complement, instead ofusing Morse type arguments on the Milnor fibers. This includes replacing thelocal Milnor fiber by a global Milnor fiber which has a complex geometryresulting from a transitive action of an appropriate algebraic group, yieldinga compact model submanifold for the homotopy type of the Milnor fiber. Thetopology includes the (co)homology (in characteristic 0, and 2 torsion in onefamily) and homotopy groups, and we deduce the triviality of the monodromytransformations on rational (or complex) cohomology. The cohomology of theMilnor fibers and complements are isomorphic as algebras to exterior algebrasor for one family, modules over exterior algebras; and cohomology of the linkis, as a vector space, a truncated and shifted exterior algebra, for which thecohomology product structure is essentially trivial. We also deduce from Bott'speriodicity theorem, the homotopy groups of the Milnor fibers for determinantalhypersurfaces in the stable range as the stable homotopy groups of theassociated infinite dimensional symmetric spaces. Applying a Theorem of Oka weobtain a class of formal linear combinations of exceptional orbit hypersurfaceswhich have Milnor fibers which are homotopy equivalent to joins of the compactmodel submanifolds.

Topology of Exceptional Orbit Hypersurfaces of Prehomogeneous Spaces