Abstract:
A striking feature in the study of Riemannian manifolds of positive sectional curvature
is the narrowness of the collection of known examples. In this thesis, we examine the
structure of the cohomology rings of three families of compact simply connected seven dimensional
Riemannian manifolds that may contain new examples of positive curvature. An
explicit computation of these rings reveals that there are infinitely many homotopy types
represented in each family. In addition, it becomes possible to identify those manifolds
to which there are associated well-known topological invariants distinguishing homeomorphism
and diffeomorphism types.
All of these manifolds support an action by S³ × S³ with orbit space a closed
interval. Such manifolds are known to be diffeomorphic to the union of the total spaces
of two disk bundles. This structure is exploited in two long exact cohomology sequences,
which relate the cohomology of the manifold to that of the orbits of the S³ × S³-action.
These sequences, and lemmas derived from them, comprise the primary tools employed in
computing the cohomology rings.
