Toral Actions On 5- And 6- Dimensional Manifolds.
Hae Soo Oh · Deep Blue (University of Michigan) · 1980
Suppose M is a closed orientable smooth manifold of dimension (n + 2) on which an n-torus T('n) acts smoothly and effectively. Raymond classified this kind of manifold in the case of n = 1. Raymond, Orlik, and Pao succeeded in obtaining almost complete classification in the case of n = 2. The classification problem is open for n (GREATERTHEQ) 3. The main concern of this thesis is the study of the cases n = 3 and 4. We show that if the orbit space is a 2-disk and there exist no exceptional orbits, then the fundamental group is a finite abelian group with at most (n - 2) generators. Various examples are constructed showing that some of the published results in these dimensions were incorrect. We have a homology classification of 5-manifolds on which T('3) acts effectively and smoothly so that the orbit space is a 2-disk and there exist no exceptional orbits. By applying Barden's results to this we determine the simply connected 5-manifolds with effective T('3)-actions. Moreover, we also partially determine the non-simply connected 5-manifolds with effective T('3)-actions. A homology classification of 6-manifolds with effective T('4)-actions is obtained under the hypothesis that the orbit space is a 2-disk and there exist no exceptional orbits. Using Jupp's and Wall's results, we then obtain a classification theorem of simply connected 6-manifolds with effective T('4)-actions. Suppose T('n) acts smoothly and effectively on a closed orientable smooth (n + 3)-manifolds. Then the orbit space is a 3-manifold when the action is nice enough. In the appendices, we prove that if T('3) acts effectively on a simply connected 6-manifold M so that the orbit space M('*) is D('3) and the image of the orbits of type T('1) under the orbit map are disjoint circles, then M can be expressed as a connected sum of well known manifolds. By using a direct geometric argument, we also obtain a classification theorem of 5-manifolds with effective T('2)-actions in the case that the orbit space is a 3-disk.