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Algebraic Combinatorics

Peter Orlik and Volkmar Welker
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
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This is really two books in one. The first half, written by Peter Orlik, is about hyperplane arrangements. It starts with a chapter in which hyperplane arrangements are discussed for their own sake. The first three sections contain the enumerative aspects of the topic. Then the discussion turns more algebraic and topological, covering the Orlik-Solomon Algebra, the NBC complex, and the Aomoto complex. The subject of the rest of the chapter is that of combinatorial equivalence of two simple arrangements. The second chapter of the first half covers applications to algebraic topology, such as local cohomology, and moduli spaces.

This reviewer thinks that this half of the book is too advanced for a graduate course. It could be used by a weekly reading group by faculty and the occasional very advanced, talented and motivated student.

The second half, written by Volkmar Welker, is about Discrete Morse theory and Free resolutions. It starts with a relatively easy-to-read introductory chapter on simplicial complexes that goes as far as discussing open problems. The second chapter defines the remaining necessary concepts, namely cellular resolutions and CW-complexes. The third chapter is devoted to many examples of cellular resolutions. The last chapter is on Discrete Morse theory, as developed by Robin Forman. This theory enables the user to construct a CW complex that is homotopy equivalent to a given complex, but has fewer cells. The author shows that this theory can be used to construct minimal free resolutions.

While the second half of the book is also very advanced, it could be possibly be used for a special topics graduate course, for students who already had a graduate class in topology. This is because of its numerous exercises and its reader-friendly style.

Finally, this reviewer believes that "Lectures on Hyperplane Arrangements and Cellular Resolutions" would have been a more accurate title for the book than the actual, not sufficiently specific title.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.
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