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Classical topology and combinatorial group theory. 2nd ed. (English) Zbl 0774.57002
Graduate Texts in Mathematics. 72. New York: Springer-Verlag. xii, 334 p. (1993).
“This superbly written, elegantly illustrated textbook is all too modestly addressed to the senior undergraduate student. Unfolding the thesis that topology ’resulted from the visualization of problems from other parts of mathematics,’ the author composes an overture to the field that combines vivid geometrical imagery with precise, clear algebra.” Thus begins G. K. Francis’s review of the first edition (1980; Zbl 0453.57001). The second edition incorporates only minor corrections through Chapter 8. The primary difference between editions is the addition of a Chapter 9. Since I heartily agree with and cannot improve upon Francis’s review of the first edition, I refer the reader to that review, for a discussion of the text’s general nature and a description of Chapters 0 through 8. Chapter 0 includes proofs of results only mentioned in the first edition. It begins with Turing machines and HNN extensions and then moves to the proof of the unsolvability of the word problem for groups. This proof was made possible by a new approach to the word problem discovered by S. Aanderaa and D. E. Cohen [Word problems II, Stud. Logic Found. Math. Vol. 95, 1-16 (1980; Zbl 0445.20016)] around 1980. This naturally leads to the proof of the unsolvability of the homeomorphism problem, which has not previously appeared in a textbook. The addition of Chapter 9 achieves the incredible result of making the already remarkable first edition into an even better second edition.

##### MSC:
 57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes 55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology 20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory 57Mxx General low-dimensional topology 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010) 20Fxx Special aspects of infinite or finite groups