# Math 527 Fall 1999

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).

### Outline

Topology is a branch of geometry which studies those properties of figures which
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.

### Assessment

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

### Office Hours

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.

### Outline and Schedule of Lectures

Week 1 (to August 27th): Definition and first properties of topological spaces.  Ch 1.1 through 1.3
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