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This is the first part of a series of papers where we compute Eulercharacteristics, signatures, elliptic genera, and a number of other invariantsof smooth manifolds that admit Riemannian metrics with positive sectionalcurvature and large torus symmetry. In the first part, the focus is oneven-dimensional manifolds in dimensions up to 16. Many of the calculations aresharp and they require less symmetry than previous classifications. Whenrestricted to certain classes of manifolds that admit non-negative curvature,these results imply diffeomorphism classifications. Also studied is a closelyrelated family of manifolds called positively elliptic manifolds, and we provethe Halperin conjecture in this context for dimensions up to 16 or Eulercharacteristics up to 16.

Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16