Torus actions on rationally elliptic manifolds

Abstract

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.Date: June 15, 2018. 2010 Mathematics Subject Classification. 55P62,57R91,57S15.

Symmetries of Spaces with Lower Curvature Bounds

“…Escher and Searle in [7] generalized this result to all isotropy-maximal torus actions on closed, simplyconnected, non-negatively curved 𝑛-manifolds, showing that they are all equivariantly diffeomorphic to a quotient of a free linear torus action of 𝒵, as in the Maximal Symmetry Rank Conjecture. Indeed, if the Bott Conjecture holds, one can combine the isotropy-maximal classification result of Escher and Searle [7] with the work of Galaz-García, Kerin, and Radeschi [11] mentioned in Remark 3.3, to show that the Maximal Symmetry Rank Conjecture holds.…”

“…On the other hand, if the Bott Conjecture holds, then for an effective torus action of sufficiently large rank on a closed, simply-connected manifold of non-negative curvature, the torus action will have non-trivial isotropy. This is quantified in the work of Galaz-García, Kerin, and Radeschi [11], who show that if 𝑀 𝑛 , a rationally elliptic 𝑛-dimensional smooth manifold, admits a smooth and effective 𝑇 𝑘 -action, then 𝑘 ≤ ⌊ • Small quotient space, that is, dim(𝑀/𝐺) is small; • Large fixed point set, that is, dim(𝑀 𝐺 ) is large with respect to the dimension of the manifold 𝑀; and • Large rank, that is, we consider group actions 𝐺 for which rk(𝐺) is large with respect to the maximal possible rank of a group action on a manifold.…”

Symmetries of Spaces with Lower Curvature Bounds

“…Escher and Searle in [19] generalized this result to all isotropy-maximal torus actions on closed, simplyconnected, non-negatively curved n-manifolds, showing that they are all equivariantly diffeomorphic to a quotient of a free linear torus action of Z, as in the Maximal Symmetry Rank Conjecture. Indeed, if the Bott Conjecture holds, one can combine the isotropy-maximal classification result of Escher and Searle [19] with the work of Galaz-García, Kerin, and Radeschi [29] mentioned in Remark 3.3, to show that the Maximal Symmetry Rank Conjecture holds.…”

“…On the other hand, if the Bott Conjecture holds, then for an effective torus action of sufficiently large rank on a closed, simply-connected manifold of non-negative curvature, the torus action will have non-trivial isotropy. This is quantified in the work of Galaz-García, Kerin, and Radeschi [29], who show that if M n , a rationally elliptic n-dimensional smooth manifold, admits a smooth and effective T k -action, then k ď t 2n 3 u, and any subtorus acting freely on M n has rank bounded above by t n 3 u. 3.4.…”

Manifolds that admit a double disk-bundle decomposition

“…Borrowing heavily from [21], the basics of rational homotopy theory required in this work can be summarized as follows. For a full treatment, see [16,17,18].…”