The Homology and Homotopy Groups of Hermitian Symmetric Spaces

A method for computing homotopy groups of SU(3) and G 2 from the cross sectioning of exact sequences for the fibrations SU(2) → SU(3) → S 5 and SU(3) → G 2 → S 6 , and the localizations at odd primes, is developed. The homol-ogy and homotopy groups of hermitian symmetric spaces are derived through exact sequences. It is found that the homotopy groups of the classical and exceptional coset manifolds with simple stability groups may be deduced generally from the known homotopy groups of unitary groups and spheres, When the stability group consists of a product of simple groups, the homology groups will receive an extra component by the Künneth formula. It follows that the second homology group and the first Chern characteristic class of the two exceptional Hermitian symmetric spaces will be nonvanishing. Bott periodicity of the stable homotopy groups of the classical Hermitian symmetric spaces is proven through the exact sequences for the cosets of orthogonal, unitary and symplectic groups. Homotopy groups of EIII = E 6 / (Spin(10)×U (1))/Z 4 , EVII = E 7 / (E 6 ×U (1))/Z 3 , EIX = E 8 / (E 7 ×SU(2))/Z 2 and the boundaries of noncompact Hermtian symmetic spaces in plance convex domains are computed. A geometrical representation for universal Teichmüller space as the limit of a sequence of Hermitian symmetric spaces is given, and the Diff(S 1 )/S 1 theory, formulated through vector bundles on this manifold is an integrable system. MSC: 55E05, 55E40, 57F10, 57F15