differential topology - Dundas
There are several excellent texts on di erential topology. Unfortunately none of them
proved to meet the particular criteria for the new course for the civil engineering students
at NTNU. These students have no prior background in point-set topology, and many have
no algebra beyond basic linear algebra. However, the obvious solutions to these problems
were unpalatable. Most elementary text books were not su ciently to-the-point, and it
was no space in our curriculum for the necessary background for more streamlined and
advanced texts.
The solutions to this has been to write a rather terse mathematical text, but provided with
an abundant supply of examples and exercises with hints. Through the many examples
and worked exercises the students have a better chance at getting used to the language and
spirit of the eld before trying themselves at it. This said, the exercises are an essential
part of the text, and the class has spent a substantial part of its time at them.
The appendix covering the bare essentials of point-set topology was covered at the be-ginning of the semester (parallel to the introduction and the smooth manifold chapters),
with the emphasis that point-set topology was a tool which we were going to use all the
time, but that it was NOT the subject of study (this emphasis was the reason to put this
material in an appendix rather at the opening of the book).
The text owes a lot to Bröcker and Jänich's book, both in style and choice of material. This
very good book (which at the time being unfortunately is out of print) would have been
the natural choice of textbook for our students had they had the necessary background
and mathematical maturity. Also Spivak, Hirsch and Milnor's books have been a source
of examples.
These notes came into being during the spring semester 2001. I want to thank the listeners
for their overbearance with an abundance of typographical errors, and for pointing them
out to me. Special thanks go to Håvard Berland and Elise Klaveness.
There are several excellent texts on di erential topology. Unfortunately none of them
proved to meet the particular criteria for the new course for the civil engineering students
at NTNU. These students have no prior background in point-set topology, and many have
no algebra beyond basic linear algebra. However, the obvious solutions to these problems
were unpalatable. Most elementary text books were not su ciently to-the-point, and it
was no space in our curriculum for the necessary background for more streamlined and
advanced texts.
The solutions to this has been to write a rather terse mathematical text, but provided with
an abundant supply of examples and exercises with hints. Through the many examples
and worked exercises the students have a better chance at getting used to the language and
spirit of the eld before trying themselves at it. This said, the exercises are an essential
part of the text, and the class has spent a substantial part of its time at them.
The appendix covering the bare essentials of point-set topology was covered at the be-ginning of the semester (parallel to the introduction and the smooth manifold chapters),
with the emphasis that point-set topology was a tool which we were going to use all the
time, but that it was NOT the subject of study (this emphasis was the reason to put this
material in an appendix rather at the opening of the book).
The text owes a lot to Bröcker and Jänich's book, both in style and choice of material. This
very good book (which at the time being unfortunately is out of print) would have been
the natural choice of textbook for our students had they had the necessary background
and mathematical maturity. Also Spivak, Hirsch and Milnor's books have been a source
of examples.
These notes came into being during the spring semester 2001. I want to thank the listeners
for their overbearance with an abundance of typographical errors, and for pointing them
out to me. Special thanks go to Håvard Berland and Elise Klaveness.
