This is an introductory course on the methods and results of algebraic
topology. The sequel (Math 528) continues this study but with a greater
focus on the geometry of smooth manifolds.

The text is *Topology and Geometry* by Glen Bredon (Springer-Verlag
Graduate Texts in Mathematics #139).

are not changed by homeomorphisms --- that is, continuous maps with

continuous inverses. In an introductory course in topology, such as Math 429

which is a prerequisite for this one, one needs to make precise the sense of the

word `continuous' in this statement. For this purpose one introduces abstract

theories of metric spaces, or perhaps of topological spaces, and their

fundamental properties such as completeness, compactness, or connectedness. As

well as its original geometric motivation, this material is of great value in

analysis and even as far afield as abstract algebra.

Our course will begin by consolidating and extending our knowledge of
the

`general' topology of the previous paragraph. Soon, however,
we will take a

more geometric direction, singling out for particular attention those

topological spaces which are `assembled' from simple pieces.
This point of

view directs our attention to the combinatorics of the way a complicated
space

is put together from simple pieces, and to a purely algebraic tool
(homological

algebra) which distils out the significant information from the plethora
of

available combinatorial data. We will study several machines
(`functors' is

the official name) for converting topology into algebra. The
first one is the

fundamental group, invented by Poincare. This gadget builds on

your experience with `winding numbers' (the kind that arose in Complex

Analysis) to construct a powerful invariant of topological spaces that
suffices

(for example) to prove the notorious Jordan curve theorem (every simple
closed

curve in the plane has an inside and an outside. Despite its
apparent

obviousness, this theorem is a real pain to prove rigorously.
The problem is

that continuous curves can be pretty weird; the theorem really is

quite easy for (say) polygonal curves.) Following our study of
the fundamental

group we will introduce some more recondite invariants, the homology

groups. These are harder to define than the fundamental group, but
a good deal

easier to calculate; and they enable us to solve high-dimensional problems

whereas the applications of the fundamental group are limited
to dimensions up

to two or three. Using the homology groups we will, for
instance, be able to

prove the Brouwer fixed point theorem, which says that any continuous

map from a closed n-cube to itself has a fixed point; this has applications
to

existence proofs for solutions of nonlinear problems in may areas of

mathematics.

The course will be assessed on the basis of 10 homework assignments
and 3 in-class tests. Each homework assignment will be graded out
of 55 points, and each in-class test out of 150 points, for a total of
1000 points. I will fix the grading curve in the light of experience
with the course, and let you know what it is midway through the semester;
but you may assume for the time being that roughly 875 points will secure
an A, 825 will secure an A-, and so on.

All tests will take place, and homework be due, on Fridays (except
Sep 24 and Nov 26). Homework will be assigned the previous Friday.
Here is a schedule for the assessments

August 27 | No homework due |

September 3 | Homework 1.1 due (1.2.2, 1.3.1, 1.3.8, 1.4.1, 1.4.5) |

September 10 | Homework 1.2 due (1.5.4, 1.5.6, 1.5.9, 1.7.2, 1.9.3) |

September 17 | Homework 1.3 due (1.8.3, 1.8.5, 1.8.8(c), 1.9.1, 1.9.2) |

September 27 | Test on Chapter 1 |

October 1 | No homework due |

October 8 | Homework 3.1 due (1.14.1, 1.14.3, 1.14.9, 3.3.1, 3.3.2) |

October 15 | Homework 3.2 due |

October 22 | Homework 3.3 due (3.3.4, 3.5.1, 3.5.2, 3.8.2, 3.8.6) |

October 29 | Test on Chapter 3 |

November 5 | No homework due |

November 12 | Homework 4.1 due (3.9.5, 3.9.6, 3.9.1, 3.9.10, 3.9.11) |

November 19 | Homework 4.2 due |

November 24 | Homework 4.3 due |

December 6 | Homework 4.4 due (4.13.3, 4.21.1, 4.21.2, 4.22.1, 4.23.1) |

December 10 | Test on Chapter 4 |

(Added December 15th, 1999: Final course grades may be accessed
here.)

My office is 427 McAllister. I will be available for discussion
of the course in my office from 11:00-12:15 on Fridays. (I am also
available at the same time on Tuesdays, but you may have to share that
hour with my calculus students) If you want to contact me about the
course at any other time please send email to roe@math.psu.edu
and I will be happy to make an appointment to talk with you.

Week 2 (to Sep 3rd): More about topologies, connectedness, sequences in metric spaces. Ch 1.3, 1.4, 1.9

Week 3 (to Sep 10th): Separation axioms, compactness Ch 1.5, 1.7

Week 4 (to Sep 17th): Products, Tychonoff's Theorem, continuous real-valued functions, metrization: Ch 1.8, 1.9, 1.10

Week 5 (to Sep 24th): Quotient spaces, partitions of unity, Baire Category Theorem. Ch 1.13, 1.12, 1.17

Assigned reading: Local compactness (1.11)

Suggested reading: Nets (1.6), Paracompactness (1.12), Topological
groups (1.15)