  This is the first part of a series of papers where we compute Euler
characteristics, signatures, elliptic genera, and a number of other invariants
of smooth manifolds that admit Riemannian metrics with positive sectional
curvature and large torus symmetry. In the first part, the focus is on
even-dimensional manifolds in dimensions up to 16. Many of the calculations are
sharp and they require less symmetry than previous classifications. When
restricted to certain classes of manifolds that admit non-negative curvature,
these results imply diffeomorphism classifications. Also studied is a closely
related family of manifolds called positively elliptic manifolds, and we prove
the Halperin conjecture in this context for dimensions up to 16 or Euler
characteristics up to 16.
