DescriptionMetric and topological spaces arise in many ways, and not always as subspaces of something simple like Euclidean space. For example the Mobius band and the Klein bottle are usually described as spaces obtained by "gluing up" simpler spaces - strips of paper, or squares. This project investigates the problem of knowing, for simple spaces, when you can find a given space as a subspace of a (low dimensional) Euclidean space. To continue our examples, the Mobius band and Klein bottle are both examples of non-orientable (one sided) surfaces. The first can be found as a subspace of R3, but you need R4 for the Klein bottle. However, if you allow "small" self-intersections (to be made precise), the Klein bottle sits inside R3. Finding a space X as a true subspace of Rn is called an embedding of X in Rn; allowing certain types of self intersection is called an immersion of X in Rn. The project can look at various tools to help decide when you can find an embedding or an immersion of a given space, and what the minimum dimension of the Euclidean spaces needed is. It can also look at particular examples of "nice"—i.e. particularly symmetric or good in other ways—embeddings or immersions. An important and classic example is the problem of embedding "n-dimensional projective space", RP(n), an object that has played an important role in many areas of geometry, both modern and classical. Many results can easily be obtained here, but the overall problem is still unsolved. PrerequisitesTopology III - MATH3281 is necessary. It would be helpful to have taken (or be taking as a corequisite) Algebraic Topology IV - MATH4161. ResourcesGood places to start for the techniques are the texts:
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