Positive curvature and symmetry in small dimensions

Abstract

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 16.2010 Mathematics Subject Classification. 53C20 (Primary), 57N65 (Secondary).

Positive curvature and discrete abelian symmetry

“…Similarly, replacing Z 5 2 by Z4 2 , and the upper bound on the dimension by m + 2 ≤ 7, the same conclusion holds.…”

“…In the first three cases, Conclusion (1) holds immediately. In the last case, it follows that F 4 3 is 3-connected and hence is a cohomology S 4 , so again Conclusion (1) holds. Suppose then that m 3 = 6.…”

“…So we assume now that m 3 = 4 or m 3 = 6. In the former case, F 4 3 ⊆ F 8 4 is 4-connected by Observation 3.7 and hence F 4 is a cohomology S 8 , RP 8 , CP 4 , or HP 2 by the Codimension Four Lemma. In the first three cases, Conclusion (1) holds immediately.…”

“…We prove both statements at once. Suppose N m = M ι x , and consider the induced Z 4 2 -action on N m . By Parts 1 and 2 of Lemma 1.4, there exists ι 2 ∈ ι 1 such that the inclusion M ι 2…”

Bounds on Sectional Curvature and Interactions with Topology

“…Examples of such "inductive results" are the following: The classification of positively curved manifolds M n admitting an effective isometric action of a torus T (n+1)/2 respectively T (n−1)/2 by Grove-Searle (see [36]) respectively Fang-Rong (see [22]), homotopy and cohomology classification results by Wilking under effective isometric T n/4+1 and T n/6+1 actions (see [65]), as well as Euler characteristic computations by Amann-Kennard under isometric torus actions-here the rank of the torus is either linear or logarithmic in the dimension of the manifold (see [3], [6]).…”