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Elements of Homology Theory

V. V. Prasolov
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 81
[Reviewed by
Michael Berg
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In my undergraduate days, many decades ago at UCLA, I managed to do a severe injustice to my own mathematical upbringing in that in that I cut far too many classes in the introductory course in topology (whence my performance left a great deal to be desired, too). I was dating a Music major at the time and her schedule demanded a lot of sacrifices from mine; being twitterpated, and pretty goofy besides, I was happy to accommodate…

Goofiness is not easy to overcome, of course, and some years later, in graduate school in La Jolla, I profoundly underappreciated what Max Karoubi was giving us in his seminars. This time it wasn’t a question of romance, just foolishness, I must admit. In any case, given this grim school-history in the area of topology and its ancillary subjects, my later (research-driven) dealings with (co)homology have generally been somewhat problematical for me, entailing a lot of picking and choosing of topics as a function of need — the usual story, I guess — and adding up to a discontinuous whole, so to speak.

But I think this kind of history in a field not one’s own (mine being number theory) is not atypical among mathematicians, and for many of us a wistfulness remains vis à vis the mathematics to which we did an injustice in past. For me this has long been true in the above topological connection, and so V. V. Prasolov’s Elements of Homology Theory appears on the scene as a most welcome surprise, even though this, for lack of a better word, remedial, aspect of the book appears to be far from what the author himself has in mind. Specifically, the book under review is intended to be a sequel to Prasolov’s Elements of Combinatorial and Differential Topology, and it is abundantly clear that the collective objective of these two books is to provide a sound grounding in general graduate level topology from soup to nuts, as it were.

However, while this objective is clearly met, Elements of Homology Theory is also written and structured in such a way as to perfectly meet the needs of a non-specialist like me, seeking to make up lost ground. The book is remarkably self-contained and the arrangement of topics is somewhat different from what one encounters in other books on introductory algebraic topology.

Most notably, Prasolov starts with simplicial rather than singular homology, postponing the latter till halfway through the book. Accordingly, simplicial complexes are the vehicle for a host of deep themes introduced relatively early on, including the Universal Coefficient Theorem, the Euler characteristic, and the Lefschetz Theorem. Prasolov then goes on to cover the Künneth Theorem (in the context of manifolds), characteristic classes, and Steenrod squares, all before getting to singular homology some 20 pages into the book. Then, after developing the latter topic, manifolds are quickly revisited, as are characteristic classes, and the according extensions and expansions are taken care of. It strikes me that this is a particularly effective pedagogical approach — one that I would very much like to try myself.

Thus, Elements of Homology Theory is poised to serve very well as both a classroom text for a second course in topology in graduate school and as a text for self-study (even if a smidgen of picking and choosing remains unavoidable). The book is exceptionally well written in a crisp no-nonsense style, and, as I already indicated, is largely self-contained. I am particularly taken with Prasolov’s almost Landauesque inclusion of proofs of everything he asserts, pretty much in the Satz–Beweis format. Beyond this there are 136 problems distributed throughout the text, replete with hints or solutions, so this invaluable aspect of active student participation is also more than adequately covered.

Finally, the last two chapters of Elements of Homology Theory are devoted to the all-but-obligatory feature of special topics, making for the author’s suggestion of where to go from here. Thus we encounter Čech and de Rham cohomology (properly singling out sheaf cohomology), as well as the topology of manifolds once again, including the topics of how to calculate Chern classes for complete intersections and “some homological properties of Lie groups and H-spaces.”

I think this is a very good and very useful book.

Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.