Looking for good "Topology" text I'm seeking a good text on Topology to suplement my study of geometrical methods of mathematical physics. For those of you who are learned in this field please post your favorite Topology text. Thanks. Pete
I am certainly no expert, but I have a few books on topology on my shelf. One is Topology by Klaus Janich, which is in the Springer Undergraduate Texts in Mathematics series.
I haven't read that many topology books, but I liked James Munkres' "Topology" (which includes lot's of exercises).
what do you want to know about topology? there are general books on things like metric spaces, and general topological spaces. then there are books that try to introduce the detaield study of specific geometric objects from a topological standpoint, like classifying 2 manifolds, as spheres with handles. then there are introductinos to fundamental concepts like covering maps, and their relation with the homotopy clases of loops in the space, i.e. which study spaces essentially up to deformation. then there are global studies of manifolds of higher dimensons, and the tools that requires like homomology and cohomology. It is common to introduce differentil calculus here and study smooth manifolds, and de rham cohomology, and chern cohomology classes, to study obstruction to embeding manifolds in euclidean and projective spaces in small codimension. then there is knot theory, and I guess I am running out of thigns i know about. Oh yes, the general theory of bundles. I recommend Milnor's Topology from the differentiabkle viewpoint, and Bott- Tu's Differential Forms in algebraic topology, for a beautiful blend of algebraic and differential tools in topology, used to prove deep theorems. Guillemin pollack lso have an expanded version of MILNORS BOOK, WHICH IS BEAUTIFULLY DECEPTIVE IN ITS IMPRESSION THAT THe STUFF IT IS DOING IS EASY. Milnor is much better and more substantive, (and shorter and cheaper)but their book reads easier. andrew wallace always wrotes with the student in mind and is always readable. for trivial point set topology, and formlaism I myself started out on kelley's general topology, but that is boring and sort of vacuous.
anything by david massey is always carefully written for the student/learner. when you say geometrical methods in hpysics, you seem to need global topology of manifolds, a big topic. it is unfortunately hard to recommend books with that scope that are still introductory. i would suggest starting with the clasification of 2 manifolds and fundmental groups, as in massey's intro to topology. that way you have the study of low diemnsional amnifodls asa model for the higher diemnsional stuff. you also have the basic and always crucial study of th behavior of curves and homotopy, which eprsists in usefulness even in high dimensons. the next step is the theory of cohomology i guess. one could look at milnors topology from th differentiable viewpint for a beautiful succinct treatment of homotopy and cobordism. and spivaks diff geom tome, volume one is really an intro to manifodls and theoir topology and differentiabkle strudture. spivak is too long and mathematically detaield for a pohysicists taste i think, but the lakte chapters of volume one, on de rham cohomology and poincare duality are excellent and can perhaps be read without slogging throught the abstarct treatement of tangent bundles. just go straight there, skipoing the first few hundred pages, and see how it goes. bill fulton also has an intro to alg top and cohomology that should be good as all his books are well written. massey also wrote some intros to alg top and homologya nd cohomology. but start it t fundamental group i think, and 2 manifodls. i hope this helps.