Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item Immersions of surfaces(Montana State University - Bozeman, College of Letters & Science, 2022) Howard, Adam Jacob; Co-chairs, Graduate Committee: David Ayala and Ryan GradyTo determine the existence of a regular homotopy between two immersions, f, g : M --> N, is equivalent to showing that they lie in the same path component of the space Imm(M, N). We identify the connected components, pi 0 Imm(W g, M), of the space of immersions from a closed, orientable, genus-g surface W g into a parallelizable manifold M. We also identify the higher homotopy groups of Imm(W g, M) in terms of the homotopy groups of M and the Stiefel space V 2 (n). We then use this work to characterize immersions from tori into hyperbolic manifolds as self covers of a tubular neighborhood of a closed geodesic up to regular homotopy. Finally, we identify the homotopy-type of the space of framed immersions from the torus to itself.Item Rotation sets of flows on higher dimensional tori(Montana State University - Bozeman, College of Letters & Science, 2001) Dumonceaux, Doreen NormaItem L-cuts for genus two translation surfaces(Montana State University - Bozeman, College of Letters & Science, 2013) Bouwman, Andrew Kevin; Chairperson, Graduate Committee: Jaroslaw KwapiszA connected sum is a topological way of joining two Riemann surfaces which results in another surface. It is used in the classification of all connected closed orientable surfaces as being homeomorphic to either the sphere, or a connected sum of tori. The reverse operation, here referred to as a splitting, decomposes a surface as a connected sum. It was recently shown by Curtis McMullen that any translation surface of genus two can be written in infinitely many ways as a connected sum of two flat tori. His method was to find a certain straight saddle connection J, and perform a splitting along J U eta(J), where eta is the hyperelliptic involution (the unique degree-two automorphism on the surface which fixes exactly six points). In this dissertation, we give an elementary argument for existence of such J and show that for all surfaces of genus two on which the vertical flow is minimal, the same kind of splitting is possible along a parallel pair of paths with the straight saddle connection replaced by an L-cut: a broken line with one horizontal and one vertical segment. As a direct consequence of this L-cut splitting, it is shown that a homeomorphism on a genus-two surface which is conjugated to a hyperbolic toral automorphism restricted to an invariant subset (if any such situation even exists) can only be pseudo-Anosov with non-orientable foliations. This makes progress toward addressing an old question of Stephen Smale about the existence of an invariant set of a hyperbolic toral automorphism which is itself a compact surface.