DescriptionDifferential Topology is the study of topological spaces with techniques coming from Analysis.One of the most basic classes of spaces in topology is the class of manifolds. These spaces, wich can be thought of as higher dimensional analogues of surfaces, appear in many branches of mathematics, for example, as solutions to algebraic equations or as configuration spaces of mechanical systems. Often these spaces come with what is called a differentiable structure, which appears to be complicated, but actually helps immensely in understanding their topology. Objects such as vector fields which you know from previous courses have their natural home in this setting, and we will see further generalisations which lead to vector bundles or deRham cohomology. Other topics include the mapping degree and the concept of framed cobordism, which leads to beautiful relations to Algebraic Topology. Some of the more advanced topics can be used to describe exotic spheres, which are manifolds homeomorphic, but not diffeomorphic to a standard sphere. This even has surprising connections to Number Theory. Overall this is a rich field, with many possible directions to explore. PrerequisitesTopology III.ResourcesA nice introduction to Differential Topology is in Topology from the Differentiable Viewpoint by John Milnor.A more comprehensive book is Topology and Geometry by Glen Bredon, and another standard reference is Differential Topology by Morris Hirsch. |
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