Math Conjectures

Bott rational-ellipticity conjecture

OPENMajorConjectureProposed c. 1982 · Full conjecture

Canonical statement

If MM is a closed simply connected smooth manifold that admits a Riemannian metric of nonnegative sectional curvature, then MM is rationally elliptic, meaning
∑i≥2dim⁡Q(πi(M)⊗Q)<∞. \sum_{i\ge2}\dim_{\mathbb Q}\bigl(\pi_i(M)\otimes\mathbb Q\bigr)<\infty.
View source LaTeX
If \(M\) is a closed simply connected smooth manifold that admits a Riemannian metric of nonnegative sectional curvature, then \(M\) is rationally elliptic, meaning \[ \sum_{i\ge2}\dim_{\mathbb Q}\bigl(\pi_i(M)\otimes\mathbb Q\bigr)<\infty. \]
GEO-TOP-010:[GroveHalperin1982Dupin][GroveWilkingYeager2019RationalEllipticity][GalazGarciaKerinRadeschi2020TorusActions]
Geometry and topology

Notes

Bott's conjecture links Riemannian geometry to rational homotopy theory: a closed simply connected smooth manifold admitting a metric of nonnegative sectional curvature should be rationally elliptic, meaning that ∑i≥2dim⁡Q(πi(M)⊗Q)\sum_{i\ge2}\dim_{\mathbb Q}\bigl(\pi_i(M)\otimes\mathbb Q\bigr) is finite. The conjecture took shape in Bott's early-1980s formulation within the nonnegative-curvature program, and it is discussed in work of Grove and Halperin [GroveHalperin1982Dupin].

The evidence is largely example-driven: every presently known closed simply connected manifold of nonnegative curvature, the classical sources being homogeneous spaces, biquotients, and constructions derived from them, is rationally elliptic. Beyond examples, the conjecture has been proved under symmetry hypotheses, for instance in cohomogeneity two even under almost nonnegative curvature [GroveWilkingYeager2019RationalEllipticity], and torus actions on rationally elliptic manifolds have been analyzed with the conjecture in view [GalazGarciaKerinRadeschi2020TorusActions].

What is missing is any general mechanism converting a sectional-curvature bound into finiteness of rational homotopy; no such link is known, and the conjecture remains open in full generality.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.