## Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

## Prerequisite

The prerequisites for the course are basic point set topology, such as Introduction to Topology (18.901 or equivalent) and algebra, such as Algebra I (18.701) or Modern Algebra (18.703) or equivalent.

## Course Overview

This course is intended as a graduate-level introduction to the machinery of algebraic topology. Specifically, we will focus on singular homology and the dual theory of singular cohomology.

## Text

The textbook for this course is:

Hatcher, Allen. *Algebraic Topology*. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521795401.

The text is freely available online, but paperback copies are also available. We will be focusing on chapters 2 and 3.

Other texts you might find interesting or useful include the following:

Massey, William S. *A Basic Course in Algebraic Topology*. New York, NY: Springer-Verlag, 1997. ISBN: 9780387974309.

Rotman, Joseph J. *An Introduction to Algebraic Topology*. New York, NY: Springer-Verlag, 1998. ISBN: 9780387966786.

Munkres, James R. *Elements of Algebraic Topology*. Boulder, CO: Westview Press, 1993. ISBN: 9780201627282.

## Exams and Assignments

There are no exams for this course. Your grade for this course will be based on weekly assignments due each Wednesday in-class. There are 12 assignments in all.

## Grading

Your lowest assignment grade will be not be counted towards your final grade. The remaining assignments will be given equal weight.

## Calendar

WEEK # | TOPICS |
---|---|

1 | CW-complexes, delta-complexes, simplicial homology, exact sequences, diagram chasing |

2 | Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms |

3 | Excision, computations for spheres, equivalence of simplicial and singular homology |

4 | Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficients |

5 | Tensor products, Tor, universal coefficient theorem for homology, products of simplices |

6 | The Eilenberg-Zilber shuffle "product" map, diagonal approximations, the Alexander-Whitney map, method of acyclic models, Kunneth formula |

7 | Duality, cohomology, Ext, universal coefficients for cohomology |

8 | Projective spaces and Grassmannians, cup products, relative cup products |

9 | Dual Kunneth formula, field coefficients, cup products in cohomology of projective spaces |

10 | Manifolds, local orientations, global orientations |

11 | Cap products and choices of appropriate sign conventions, statement of Poincare duality, limits |

12 | Compactly supported cohomology, proof of Poincare duality |

13 | Finish proof of Poincare duality Intersection pairing and cup product |

14 | Lefschetz fixed point theorem |

15 | Finish proof of Lefschetz theorem Assorted further topics |