Graduate Thesis Or Dissertation


On the Structure of the Orbit Spaces of Almost Torus Manifolds with Non-negative Curvature Public Deposited

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  • An almost torus manifold $M$ is a closed $(2n+1)$-dimensional orientable Riemannian manifold with an effective, isometric $n$-torus action such that the fixed point set $M^T$ is non-empty. Almost torus manifolds are analogues of torus manifolds in odd dimension and share many of the characteristics of torus manifolds. For example, both almost torus manifolds and torus manifolds are $S^1$-fixed point homogenous. Just as torus manifolds are important examples of manifolds admitting so called isotropy-maximal actions, almost torus manifolds are important if one hopes to understand manifolds admitting almost isotropy-maximal actions. Recently Wiemeler classified simply connected torus manifolds with non-negative sectional curvature. To obtain this result, he proved a structure theorem for the quotient spaces of torus manifolds of non-negative curvature. In particular, he showed that non-negatively curved torus manifolds are locally standard, with orbit spaces homeomorphic to convex polytopes with acyclic lower dimensional faces. In this thesis, we obtain an analogous structure theorem for the orbit spaces of almost torus manifolds. Namely, we analyze the orbit spaces $M/T$ of simply connected almost torus manifolds with non-negative sectional curvature. The main result we obtain is that the action of $T$ is locally standard and although $M/T$ is homeomorphic to a disk and shares the combinatorial structure of a polytope, lower dimensional faces are in general not acyclic. Unlike locally standard torus manifolds, orbit spaces of locally standard almost torus manifolds need not be manifolds with corners. Nevertheless, the analysis of the orbit space structure of $M/T$ shows that almost torus manifolds are similar enough to torus manifolds, so that the result of this thesis can then be combined with results about extending isometric almost isotropy maximal $T^k$ actions to smooth $T^{k+1}$ actions to obtain a classification of non-negatively curved, simply connected almost torus manifolds.
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