Algebraic Topology Beyond the Basics: Any Texts Bridging The Gap?
Good lord, Charles, was the reposting of this an invitation for another advertisement from me? ``More concise algebraic topology. Localization, completion, and model categories'', by Kate Ponto and myself, is available for purchase and will be formally and officially published next month. I have a copy in my hand, and the final version is 514 pages (including Bibliography and Index). Still 65 dollars (and don't fall for pirate editions on the web). It is not perfect, of course. (I know of one careless mistake every reader will catch and one subtle mistake almost no reader will catch). I offer $10 to any reader finding a mistake I don't know about, even misprints. The book is intended to help fill the gap (and another, more calculational, follow up to Concise is planned). The first half covers localization and completion and is more technical than I hoped simply because so much detail was needed to fill out the theory as it was left in the great sources from the early 1970's (Bousfield-Kan, Sullivan, Hilton-Mislin-Roitberg, etc), especially about fracture theorems. The second half is an introduction to model category theory, and it has a number of idiosyncratic features, such as emphasis on the trichotomy of Quillen, Hurewicz, and mixed model structures on spaces and chain complexes. The order is deliberate: novices should see a worked example of serious homotopical algebra before starting on categorical homotopy theory. There is a bonus track on Hopf algebras for algebraic topologists and a brief primer on spectral sequences. There are example applications sprinkled around, although more might have been desirable. The book is quite long enough as it is. Merry Christmas all.
Homotopic Topology by Fuchs, Fomenko, and Gutenmacher, mentioned above by Ilya Grigoriev, is a wonderful book which is practically unknown here (english version was done by an obscure eastern european publisher and has been out of print for decades) and hard to get even via an interlibrary loan. It's now availaible in pdf at
http://www.math.columbia.edu/~khovanov/algtop2013/
although the files are pretty large.
I'm absolutely thrilled by the existence of Strom's "Modern Classical Homotopy Theory":
http://www.ams.org/bookstore-getitem/item=GSM-127
Makes for essential reading, I think. Warmly recommended.