Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts – both geometric ones like Allen Hatcher's and algebraic-focused ones like the one by Rotman and more recently, the beautiful text by tom Dieck (which I'll be reviewing for MAA Online in 2 weeks, watch out for that!) – there are almost no texts which bring the reader even close to the frontiers of the subject.
GEOMETRIC topology has quite a few books that present its modern essentials to graduate student readers – the books by Thurston, Kirby and Vassiliev come to mind – but the vast majority of algebraic topology texts are mired in material that was old when Ronald Reagan was President of The United States. This is partly due to the youth of the subject, but I think it's more due to the sheer vastness of the subject now. Writing a cutting edge algebraic topology textbook – TEXTBOOK, not MONOGRAPH – is a little like trying to write one on algebra or analysis. The fields are so gigantic and growing, the task seems insurmountable.
There are only 2 "standard" advanced textbooks in algebraic topology and both of them are over 30 years old now: Robert Switzer's Algebraic Topology: Homology And Homotopy and George Whitehead's Elements of Homotopy Theory. Homotopy theory in particular has undergone a complete transformation and explosive expansion since Whitehead wrote his book. (That being said, the fact this classic is out of print is a crime.) There is a recent beautiful textbook that's a very good addition to the literature, Davis and Kirk's Lectures in Algebraic Topology – but most of the material in that book is pre-1980 and focuses on the geometric aspects of the subject.
We need a book that surveys the subject as it currently stands and prepares advanced students for the research literature and specialized monographs as well as makes the subject accessible to the nonexpert mathematician who wants to learn the state of the art but not drown in it. The man most qualified to write that text is the man to uttered the words I began this post with. His beautiful concise course is a classic for good reason; we so rarely have an expert give us his "take" on a field. It's too difficult for a first course, even for the best students, but it's "must have" supplementary reading. I wish Dr. May – perhaps when he retires – will find the time to write a truly comprehensive text on the subject he has had such a profound effect on. Anyone have any news on this front of future advanced texts in topology?
I'll close this box and throw it open to the floor by sharing what may be the first such textbook available as a massive set of online notes. I just discovered it tonight; it's by Garth Warner of The University Of Washington and available free for download at his website. I don't know if it's the answer, but it sure looks like a huge step in the right direction. Enjoy. And please comment here.
http://www.math.washington.edu/~warner/TTHT_Warner.pdf
Best Answer
At the moment I'm reading the book Introduction to homotopy theory by Paul Selick. It is quite short but covers topics like spectral sequences, Hopf algebras and spectra. This is the first place I've found explanations (that I understand) of things like Mayer-Vietoris sequences of homotopy groups, homotopy pushout and pullback squares etc.. The author writes in the preface that the book is inteded to bridge the gap which the OP talks about.