Combinatorial topology and algebra. Selected papers based on the presentations at the instructional conference on combinatorial topology and algebra (ICCTA-93), Bombay, India, December 5--24, 1993 (Q2922486)

The Instructional Conference on Combinatorial Topology and Algebra (ICCTA-93) was held at the Indian Institute of Technology (IIT) in Mumbai during December 5--24, 1993, with the aim to explain some of the central aspects in these areas to young researchers in mathematics. Nine renowned Indian scholars had prepared lecture notes for the participants of ICCTA-93, the key feature of which was that all the relevant background material from commutative algebra, combinatorics, and topology was developed from scratch in order to keep the level as elementary (and accessible) as the thematic selection permitted.NEWLINENEWLINE Although about twenty years have elapsed since these lecture notes were written, their contents have maintained their timeless mathematical and educational relevance until now, and that is why the Ramanujan Mathematical Society has decided to publish them in its Lecture Notes Series even after two decades.NEWLINENEWLINE The volume under review contains these notes, together with two additional appendices and an epilogue attempting to briefly provide an update on some of the developments since 1993.NEWLINENEWLINE As to the structure of the book, the main text is organized in three major parts, each of which comes several chapter and sections, respectively.NEWLINENEWLINE Part I contains the first seven chapters and gives an introduction to those concepts and results from commutative algebra that are necessary to understand the combinatorial approach to the study of Cohen-Macaulay rings by \textit{G. A. Reisner} (1976), \textit{R. Stanley} (1977), and their successors. Chapter 1 recalls some basic material from algebra, including Noetherian rings and modules, integral ring extensions, and Hilbert's Nullstellensatz, whereas Chapter 2 discusses the primary decomposition of modules, with particular emphasis on graded rings and modules.NEWLINENEWLINE Chapter 3 is devoted to the dimension theory of rings, were the focus is on the Hilbert function of a graded module, the Hilbert-Samuel polynomial of a local ring, and the analysis of the concept of dimension for local rings, affine algebras, and graded rings, respectively. Chapter 4 treats the notion of depth in module theory, introduces then Cohen-Macaulay modules, and finally analyzes graded Cohen-Macaulay modules and the graded Noether normalization of a graded ring.NEWLINENEWLINE Chapter 5 gives a brief exposition of local cohomology theory for graded modules over standard \(k\)-algebras, thereby using the Čech cochain complex, illustrating the relation with the concept of depth, and proving the Grothendieck-Serre formula which characterizes the difference between the Hilbert function and the Hilbert polynomial of a graded module in terms of its local cohomology. Chapter 6 begins the study of face rings (or Stanley-Reisner rings) of simplicial complexes and their Hilbert series, culminating in the proof of the result that so-called shellable simplicial complexes have Cohen-Macaulay face rings. A more detailed account can be found in the standard text [\textit{W. Bruns} and \textit{J. Herzog}, Cohen-Macaulay Rings, Cambridge: Cambridge University Press (1993; Zbl 0788.13005)]. Finally, Chapter 7 presents the proof of Reisner's theorem on the characterization of the Cohen-Macaulay property of the face ring of a simplicial complex on the one hand, and R. Stanley's proof of the so-called ``Upper Bound Conjecture'' for those complexes which are triangulations of spheres, on the other.NEWLINENEWLINE Part II is titled ``Combinatorics'' and comprises the subsequent two chapters. The introduction to some aspects of partially ordered sets is the aim of Chapter 8, where posets, the Möbius inversion formula, the concrete computation of Möbius functions, Eulerian posets, lexicographically shellable posets, and the Schur-Macaulay property of poset rings are the main topics. Chapter 9 explains rotations and triangulations of graphs, with a view toward the combinatorial and the topological genus of a graph, Euler's combinatorial equation, and the problem of minimal triangulations of topological surfaces.NEWLINENEWLINE Part III, simply superscribed ``Topology'', contains the remaining five chapters and deals with combinatorial aspects of convex polytopes and topological surfaces, respectively. Chapter 10 gives an introduction to convex sets and convex polytopes in affine geometry, then derives Euler's formula for a \(d\)-dimensional convex polytope, turns over to shellable simplicial complexes and their homological properties, and finally illuminates the shellability property for boundary complexes of convex polytopes and a related ``Upper Bound Theorem'' in this context. Chapter 11 is a brief digest of simplicial topology, including simplicial complexes and their geometric realizations, barycentric subdivisions, simplicial approximations, and operations like links and stars for simplicial subcomplexes. Chapter 12 introduces the concept of homology, where the focus is on singular homology groups, simplicial homology groups, and the topological invariance of Reisner's condition for the Cohen-Macaulay property of the face ring of a simplicial complex. Chapter 13 addresses the problem of triangulation and classification of connected topological manifolds, especially with a view towards compact surfaces and the classical Jordan-Schönflies theorem. Chapter 14, the last chapter, gives an outlook to the problem of minimal triangulations of simplicial manifolds, with the case of surfaces serving as an illustrating example. Appendix A (by V. Srinivas) provides some topical facts on the combinatorics of simplicial convex polytopes and of toric varieties, whereas Appendix B (by H. Janwa) gives a short survey of applications of Gröbner bases to combinatorial problems. Finally, a very short epilogue by the editors suggest some further reading with regard to a few more recent books and research articles.NEWLINENEWLINE Summing up, the remarkably delayed publication of these lecture notes is still a welcome enhancement of the existing literature in combinatorial topology and algebra, as students and young researchers can profit a great deal from this tailor-made textbook also today and in the future.