Almost isotropy-maximal manifolds of non-negative curvature (Q6582242)

The authors present new classification results for manifolds that admit torus actions. The classification is done via the rank of the largest possible isotropy group of the action. This paper extends previous work by the second and third author [J. Reine Angew. Math. 780, 221--264 (2021; Zbl 1493.53049)], in which a classification was obtained under effective, isometric isotropy-maximal torus actions.\N\NAn action on a manifold \(M\) by a torus \(T^k\) is locally standard if each \(T^k\)-orbit has an invariant neighborhood that is equivariantly diffeomorphic to \(T^{r}\times W\times \mathbb{R}^m\), for some \(r<k\) that depends on the action and the dimension \(\dim(M)=n\). Here \(W\) is a faitful representation of even dimension. The action is called isotropy-maximal if the rank of the largest isotropy subgroup equals the codimension of the action. In this case, \(n-k\) is the largest possible dimension of any isotropy subgroup. In contrast, an almost isotropy-maximal torus action has an isotropy subgroup of rank one less than the codimension of the action, that is, of dimension \(n-k-1\). An almost isotropy-maximal action that is not isotropy-maximal is called strictly almost isotropy-maximal.\N\NIn the main results of this paper, it is assumed that the underlying smooth manifold \(M\), on which \(T^k\) acts, is closed, and that the \(T^k\)-action is smooth (or isometric), and effective.\N\NIn Theorem A, it is further assumed that \(M\) is a simply connected, non-negatively curved Riemannian manifold. Provided the \(T^k\)-action is by isometries, the theorem states that it can be extended to a smooth locally standard and isotropy-maximal action by \(T^{k+1}\). The quotient of this action and all of its faces are diffeomorphic to disks, after smoothing the corners.\N\NIn Theorem B, it is assumed that \(M\) is rationally elliptic, meaning that both the total dimensions of its rational cohomology and rational homotopy are finite. It is then proved that if the action is smooth and locally standard, \(M\) is equivariantly diffeomorphic to a quotient of a free linear torus action of a product of spheres.\N\NAdding to the assumptions of the Theorem B, if it is also assumed that the \(4\)-dimensional faces of the quotient are diffeomorphic to disks, after smoothing the corners, then \(M\) is equivariantly diffeomorphic to a quotient of free linear quotient by a torus on a product of spheres.\N\NUnder the same hypothesis of Theorem A, but the action being almost isotropy-maximal, the authors provide dimensional bounds for \(k\), depending on whether the action is isotropy-maximal or striclty almost isotropy-maximal. Moreover, they obtain the same equivariant diffeomorphism as in Theorem C.