Differential Forms in Algebraic Topology (Graduate Texts in Mathematics)
“Bott and Tu give us an introduction to algebraic  topology via differential forms, imbued with the spirit of a master who  knew differential forms way back when, yet written from a mature point  of view which draws together the separate paths traversed by de Rham  theory and homotopy theory. Indeed they assume "an audience with prior  exposure to algebraicordifferential topology". It would be  interesting to use Bott and Tu as the text for a first graduate course  in algebraic topology; it would certainly be a wonderful supplement to a  standard text.“Bott and Tu write with a consistent  point of view and a style which is very readable, flowing smoothly from  topic to topic. Moreover, the differential forms and the general  homotopy theory are well integrated so that the whole is more than the  sum of its parts. "Not intended to be foundational", the book presents  most key ideas, at least in sketch form, from scratch, but does not  hesitate to quote as needed, without proof, major results of a technical  nature, e.g., Sard's Theorem, Whitney's Embedding Theorem and the Morse  Lemma on the form of a nondegenerate critical point.”—James D. Stasheff (Bulletin of the  American Mathematical Society) “This book is an excellent  presentation of algebraic topology via differential forms. The first  chapter contains the de Rham theory, with stress on computability. Thus,  the Mayer-Vietoris technique plays an important role in the exposition.  The force of this technique is demonstrated by the fact that the  authors at the end of this chapter arrive at a really comprehensive  exposition of Poincaré duality, the Euler and Thom classes and the Thom  isomorphism.“The second chapter develops and  generalizes the Mayer-Vietoris technique to obtain in a very natural way  the Čech-de Rham complex and the Čech cohomology for presheaves. The  third chapter on spectral sequences is the most difficult one, but also  the richest one by the various applications and digressions into other  topics of algebraic topology: singular homology and cohomology with  integer coefficients and an important part of homotopy theory, including  the Hopf invariant, the Postnikov approximation, the Whitehead tower  and Serre’s theorem on the homotopy of spheres. The last chapter is  devoted to a brief and comprehensive description of the Chern and  Pontryagin classes.“A book which covers such an  interesting and important subject deserves some remarks on the style: On  the back cover one can read “With its stress on concreteness,  motivation, and readability,Differential forms in algebraic topologyshould be suitable for self-study.” This must not be misunderstood  in the  ense that it is always easy to read the  book. The authors invite the reader to understand algebraic topology by  completing himself proofs and examples in the exercises. The reader who  seriously follows this invitation really learns a lot of algebraic  topology and mathematics in general.”—Hansklaus Rummler  (American Mathematical Society)