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Geometry / Topology

Course material: Mathematics 225ABC. The basic concepts of metric space point-set topology will be presumed known but will not be covered explicitly in the examination. For students unfamiliar with point-set topology, Mathematics 121 is suggested, although the topics covered in the analysis part of the Basic Examination are nearly sufficient. Geometry/Topology Area Exams given prior to September 2009 will cover the older syllabus which can be found here.

Topics:

The topics covered fall naturally into three categories , corresponding to the three terms of Math. 225. However, the examination itself will be unified, and questions can involve combinations of topics from different areas.

  • 1) Differential topology: manifolds, tangent vectors, smooth maps, tangent bundle and vector bundles in general, vector fields and integral curves, Sard’s Theorem on the measure of critical values, embedding theorem, transversality, degree theory, the Lefshetz Fixed Point Theorem, Euler characteristic, Ehresmann’s theorem that proper submersions are locally trivial fibrations
    2) Differential geometry: Lie derivatives, integrable distributions and the Frobenius Theorem, differential forms, integration and  Stokes’ Theorem, deRham cohomology, including the Mayer-Vietoris sequence, Poincare duality, Thom classes, degree theory and Euler characteristic revisited from the viewpoint of deRham cohomology, Riemannian metrics, gradients, volume forms,  and the interpretation of the classical integral theorems as aspects of Stokes’ Theorem for differential forms
    3) Algebraic topology: Basic concepts of homotopy theory, fundamental group and covering spaces, singular homology and cohomology theory, axioms of homology theory, Mayer-Vietoris sequence, calculation of homology and cohomology of standard spaces, cell complexes and cellular homology, deRham’s theorem on the isomorphism of deRham differential –form cohomology and singular cohomology with real coefficient

Main References:

Additional References:

  • Abraham, R., Marsden, J., and Ratiu T. (1988). Manifolds, Tensor Analysis and Applications, New York: Springer Verlag.
  • Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry
  • Greenberg, M. & Harper, J. (1981). Algebraic Topology, A Course, Reading, Mass.: Benjamin/Cummings Pub. Co.
  • Milnor, J. (1965). Topology from the Differential Viewpoint, Charlottesville, University Press of Virginia.
  • Warner, F., (1983). Foundations of differentiable manifolds and Lie groups, Springer.