Gropes, Towers, and Skyscrapers

Gropes, towers, and skyscrapers are carefully defined. These are the objects that the rest of Part II studies and seeks to construct. All three are 4-manifolds with boundary, obtained from stacking thickened surfaces on top of one another. Gropes are constructed from thickened orientable surfaces with positive genus, each stage attached to a symplectic basis of curves for the homology of the previous stage. Towers have an additional type of stage obtained from plumbed thickened discs. A skyscraper is the endpoint compactification of an infinite tower. An introduction to endpoint compactifications is included. The notion of a good group is also defined.

ON THE CONSTRUCTION OF TRANSFORMATIONS OF LINEAR HAMILTONIAN SYSTEMS TO REAL NORMAL FORMS

A technique for constructing the transformations to real Hamiltonian normal forms of linear Hamiltonian systems via permutation matrices is presented. In particular, a method is shown for obtaining the symplectomorphism between the symplectic basis of the real Jordan form to the standard symplectic basis in which the real Hamiltonian normal form resides. All possible degeneracies are accounted for since the algebraic and geometric multiplicities of nonsemisimple eigenvalues are not restricted, including the "difficult" cases of zero and imaginary eigenvalues. Since the normal forms are not unique, several possible arrangements of the suggested transformations are given which result in the various normal forms derived previously as well as in a few new ones for degenerate cases which have not appeared before.