Subject: Re: text request: "triple combo.!" reply
Date: Tue, 3 Apr 2001 11:11:54 -0600 (MDT)
From: David Pengelley <davidp@nmsu.edu>
To: Don Davis <dmd1@lehigh.edu>
CC: David Pengelley <davidp@nmsu.edu>

On Tue, 3 Apr 2001, Don Davis wrote:

> Subject: algebraic text
> Date: Mon, 2 Apr 2001 19:49:27 -0500 (CDT)
> From: Brayton Gray <brayton@math.uic.edu>
>
> OK, how about suggestions for a text in algebraic topology. I am
> looking for a text that will not get too technical too early and will
> have lots of concrete examples. Content: homology and cohomology
> possibly with the fundamental group and covering spaces for starters.

In the last couple of years I've been applying to graduate topology all
the things I've learned over the years about getting undergrads. active as
learners in and out of the classroom.

I have been using the following triple combo. and teaching method very
successfully in a first year two semester graduate topology course that
assumes no previous topology, only some analysis and algebra.  My goal is
to get students to learn by reading several good points of view on the
subject, with just enough basics to begin algebraic topology quickly.  I
have them write lots of specific and general questions and analysis of the
reading for me to read and react briefly to before class.  I want them to
read closely and write very specific questions about things they don't
understand, and also place what they're reading into the big picture.  In
class we discuss their reading, questions, and examples, differing points
of view, and then they do (and redo!) lots of (often tough) exercises out
of class.  (I hold a "work session" weekly to help them with tough
exercises.)  In class we work through details of proofs ONLY if they have
questions about them, or if I think there is a special pedagogical point
to doing so.  A crucial part of the approach is that we don't spend most
of our time with me regurgitating proofs, etc., from the book, since they
have done really valuable advance reading, thinking, writing, and thus
reflection. I believe advance writing on reading is essential for this.
My students really get committed to this after I train them in it.  Our
class discussions are thus at a more valuable and advanced level.  The
idea with reading more than one resource is to gain breadth of view and
approaches.  Done right students really feel they learn a whole lot from
this, and I see that they do.

The triple combo. is

C. Kosniowski, A First Course Course in Algebraic Topology (out of print,
but don't let that stop you; fabulous book!)
H. Sato, Algebraic Topology: An Intuitive Approach (AMS translation,
cheap, delightful!)
W. Massey: A Basic Course in Algebraic Topology (GTM 127, Springer)

The first 9 sections of Kosniowski contain all the general topology
needed, with good concrete examples and wonderful exercises.  Then
surfaces get classified, with final assistance from the fundamental group
and Seifert-Van Kampen, and covering/orbit spaces appear.  (Massey
provides a contrasting approach to surfaces and their classification,
which we study for breadth of view; our goal is to come away with the
union of the treatments.  Also use Massey as comparison and contrast for
other things in Kosniowski.) Selected reading and discussion from sections
of Sato are liberally sprinkled throughout the whole year for a very
different, very intuitive, point of view, that brings in more advanced
topics for a bigger picture (e.g. fibre bundles as generalization of
coverings). At the end of Kosniowski he gives an intro. to singular
homology, from which we segue into Massey at Ch. 6 (background and
motivation for homology), fleshing out Kosniowski with Massey Ch. 7, 8, ..
on homology (cubical for contrast and similarities, the point of axioms,
Mayer-Vietoris, etc.).  Last year I ended with Clint McCrory's "Cellular
Homology by Degree", an old preprint from Brown U.  Then students are
ready for our Ph.D. qualifying exam.  (I think there are probably other
books which could be used instead of Massey in my scheme, but Massey is
about at the right level, detail, and comprehensiveness for me, and
becomes a reference book to keep for the future too.)

It's mix and match, which if done right, with students doing all the
advance reading, thinking, comparisons, and in class discussion of these
contrasts, enriches rather than obfuscates!  The different approaches can
be an asset to building the big picture in one's mind, not a hindrance!
The instructor must provide the guidance at every step to make the mix and
match a success.

Best,
David

David Pengelley (davidp@nmsu.edu)
Mathematics, New Mexico State University, Las Cruces, NM 88003 USA
Tel: 505-646-3901=dept., 505-646-2723=my office; Fax: 505-646-1064
http://math.nmsu.edu/~davidp