Date | Material covered |
---|---|

4/4 | No lecture. |

4/6 | Introduction. Naturality properties of cup
product and cap product. On a compact, oriented
n-manifold, the cup product pairing is non-degenerate
(induces an isomorphism from H modulo
torsion to Hom(^{k}H,Z)).
^{n-k} |

4/9 | Poincare duality for manifolds with boundary and some applications: non-degenerate pairing on the middle homology of an oriented even-dimensional compact manifold without boundary. |

4/11 | Algebraic structure of the cup product on the middle cohomology of an even dimensional manifold which is the boundary of another compact manifold. Technical results about (topological) manifolds: Existence of collar neighborhoods of the boundary. Any (compact) topological manifold is homeomorphic to a subspace of euclidean space, and is the retract of a neighborhood there. |

4/16 | Smooth manifolds. Motivation and basic definitions: Atlas, smooth atlas, smooth structure, smooth manifold. Smooth functions between such. Submanifolds and the implicit function theorem. (Also mentioned the Brieskorn spheres.) |

4/18 | Smooth vector bundles on a smooth
manifold. The tangent bundle of a smooth manifold.
Tangent vectors as differential operators on germs of
functions. Vector fields as differential operators on
functions. Transversality (in general for two smooth
functions with the same target; as a special case, a map
can be transverse to a submanifold; two submanifolds can
be transverse). If f
and _{0}f are transverse, then the
pullback is a smooth submanifold of _{1}M.
_{1} x
M_{2} |

4/23 | Whitney embedding theorem. Class taught by Ralph Cohen. |

4/25 | Sard's theorem. Whitney embedding theorem: bound on dimensions, and non-compact manifolds. |

4/30 | Thom's transversality theorem. |

5/2 | More on Thom's transversality theorem. |