In summary, the map between two unit circles is topologically stable because it cannot be transformed into a map where S1 is mapped onto a point of S1.
Marlon if you are interested in a great free online textbook on homotopy/homology/cohomology with lots of pictures (and in my opinion the best book bar none) try Allen Hatchers book.
If like me, you find Hatcher's book tedious, I suggest the more elementary book of Artin and Braun, or the differential topology book of Milnor, Topology from the differentiable veiwpoint, or the more detailed rewrite by Guillemin and Pollack, or the nice book on algebraic topology by William Fulton.
But Hatcher's book is free.
Another book I like a lot, is Differential forms in algebraic topology, by Bott / Tu.
My friends who are professional algebraic topologists also praise Hatcher, but I fail to see why. and I even liked Spanier.
Of course I have not read Hatcher closely, nor taught from it. maybe it grows on you. What do you like about it, Haelfix?
another excellent introduction for beginners is Andrew Wallace's book on the fundamental group, or any book by Wallace. Vick's Homology Theory is also easy to get into.
Ok I just looked at hatchers book, at least the first few paghes, and remember why I dislike it. He introduces the mapping cylinder on page 2. Thats ridiculous. No one can appreciate that before seeing a sphere or a torus. and the retraction illustrations are ugly computer generated ones.
You see now how impatient I am as well. No doubt this is just a poor choice of how to frame an introduction, and in the book proper he does better, but so far this is not my idea of pedagogy.
if you want an ideal example of good writing / teaching, see the first two pages of milnor's morse theory. with a few simple minded but attractive pictures he actually conveys almost as much morse theory as most people need, in the first two pages of the introduction.
but probably it is unfair to compare anyone's work to milnor's. even serre comes in second in my opinion. and maybe no one else comes close.
i will look further, but perhaps someone will kindly point me to a recommended well written section of hatcher to examine more closely.
Ok, it is indeed chapter zero which is so off putting, and chapter one is quite reasonable. In my view he should simply have begun with chapter one.
Well one thing I really like about Hatcher's book is the homework problems. Some of them are damn hard, and if you are able to tackle those, then you pretty much are well off and understand the topic at a high lvl. I remember spending many hours on a few of them.
I will agree some of the discussion gets a little complex, for instance he starts talking about non standard spaces (Eilenberg Maclane spaces) in the appendix of chapter 1. Thats very subtle material, and confused the hell out of me the first time I took the course.
Otoh hand his chapter on homology is very complete. I remember finding the answer to several problems I had (I had this fuzzy misunderstanding relating CW complexes and simplicial complexes), the latter I knew well but couldn't find many places where they talk about the relationship.
yes as i look further, the book looks useful. he gives a lot of information in a few well chosen words, at least in places.
i do not know that much algebraic topology, and never read any book on it thoroughly.
i did learn some things from greenberg's nice book, which in turn is based on the lovely elementary book of artin and braun.
spivak's differential geometry book, volume 1, has a wonderful chapter on de rham cohomology which is quite readable.
i also worked out the basic techniques of de rham theory for myself when i was teaching calculus of differential forms in order to give applications of stokes theorem to geometry for my students. i still remember how excited i was when i realized that stokes theorem implied the unit circle was not homotopic to a point in the complement of the origin in the plane, and that this implied the fundamental theorem of algebra.
i also took a course from ron stern in which we read great notes of griffiths and morgan on homotopy theory and postnikov towers. the postnikov towers came in very handy years later for me when i was proving some results on the homotopy and cohomology of the moduli space of abelian varieties.
and i had a course from ed brown. brown was a master of homotopy theory and made the material seem very intuitive.
fortunately brown discussed such things as eilenberg maclane spaces, or K(<pi>,n) spaces with only one homotopy group. he built them up by attaching spheres to generate the group and then attaching cells to kill the relations.
afterwards he used these spaces to show cohomology is a representable functor in the homotopy category, a nice technique that was imitated later in algebraic geometry and deformation theory by mike schlessinger and others.
i have just looked at hatcher's discussion of this topic on page 448, and am disenchanted with his book in this section again. i.e. he uses more sophisticated language than necessary, no doubt seking greater efficiency. but i can hardly understand his statement, whereas brown himself made it seem like the most natural thing in the world.
i.e. i do not even know what an omega spectrum is, but i do know what brown theorem says and if i think hard enough to remember the idea, possibly also essentially how to prove it.
i like easy clear versions of things, not sophisticated, technical versions.
as brown put it, his theorem says that the cohomology functor on finite cw complexes, apparently rather techical and contrived, actually "occurs in nature!"
one topic i still feel ill at ease with is characteristic classes, e.g. chern classes, although of course the classifying space approach makes them technically simple to define: just embed your manifold and its tangent spaces in euclidean space and then this defines a clasifying amap from the manifold to a grassmannian, and pull bcak the standard cohomology classes from the grassmannian.
does hatcher explain these too? maybe in his later books on vector bundles.