Student Topology

We will state the Atiyah-Singer index theorem in the language of K-theory and sketch the proof. In short, this is done by characterizing the index function using some natural axioms, and proving the index of elliptic operators satisfies these axioms. If time permits, we will say something about…

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A link L is called a universal link, if every closed 3-manifold can be presented as the covering space of the 3-sphere with branch set L. We will discuss some techniques for defining branched coverings by link diagrams with colored arcs, and show that the Borromean rings is a universal…

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We will discuss how to exploit the higher-dimensional moduli spaces coming from Morse/Floer theory in order to get more structure than just Floer homology. The data extracted will take the form of a (stable) homotopy type, usually as a module over some bordism theory…

We will discuss the celebrated result of Kervaire and Milnor in the 1960s: the n-dimensional homotopy cobordism group is finite unless n=3. Further, we will investigate the interesting theorem of Gonzalez-Acuna in the 1970s: the n-dimensional homotopy and homology cobordism groups are isomorphic…