Bott–Grove–Halperin conjecture

A round sphere has positive curvature, and every loop drawn on it can shrink to a point. Does nonnegative curvature force every smooth closed space with shrinkable loops to have only finitely many independent higher-dimensional deformation classes when counted with rational numbers? Those classes come from mapping spheres of dimension two, three and higher into the space and deforming the maps continuously. Nonnegative sectional curvature is a condition on every two-dimensional tangent direction. Rational counting ignores torsion, the classes that disappear after a finite multiple; it does not require every higher-dimensional map to shrink.[1] [2]

Does local nonnegative curvature prevent infinitely much independent rational deformation information in higher dimensions? A round two-sphere has shrinkable loops but its identity map from a two-sphere is not deformable to a constant; simple connectivity does not make all sphere maps trivial. For every closed simply connected Riemannian manifold $M$ with $\operatorname{sec}_M\ge0$, the target is $\dim_{\mathbb Q}\bigoplus_{i\ge2}(\pi_i(M)\otimes\mathbb Q)<\infty$. Here $\pi_i$ records maps from the i-sphere up to based homotopy. This is rational ellipticity: ordinary cohomology is already finite dimensional for a closed manifold. Tensoring with the rationals removes torsion, not every nontrivial homotopy class.[1] [2]

Round spheres are familiar members of the predicted rationally elliptic class. The published quotient result proves rational ellipticity under an additional comparison condition: a symmetry quotient must match that of a space with zero topological entropy, a dynamical condition. That theorem does not turn nonnegative curvature alone into the required finite collection of rational deformation information for every closed simply connected space.[1] [2]

The cited 2025 theorem gives rational ellipticity when the orbit space of a compact group action is isometric to a comparison orbit space whose ambient metric has zero topological entropy. That is an additional symmetry-quotient and dynamical hypothesis, not the curvature condition alone. The general closed simply connected nonnegatively curved case is retained as the conjecture in the selected survey. Only the published abstract of the quotient paper was accessible; no full proof is certified here.[1] [2]

Curvature is measured locally, but the conclusion would restrict global topology. For a space satisfying the prediction, the total number of independent rational homology classes is at most that of a torus of the same dimension, and its Euler characteristic is nonnegative. Euler characteristic can be counted from cells: a tetrahedral sphere gives four vertices minus six edges plus four faces, or two. These consequences would constrain the possible shapes, not merely describe their local bending. That elementary count alone does not establish rational ellipticity.[1] [2]

Rationally elliptic spaces have controlled homotopy growth and restrictive global counts: $\sum_i b_i(M)\le2^{\dim M}$ and $\chi(M)\ge0$, with Betti numbers taken over the rationals. A tetrahedral model of a two-sphere has $4-6+4=2$ as its Euler characteristic, illustrating one of those counts without computing all homotopy groups. The conjecture would force such rational-topological restrictions from measurements of sectional curvature at each point. It therefore relates differential geometry to algebraic topology across dimensions; neither the Euler inequality nor ordinary cohomology finiteness alone is the conjecture.[1] [2]