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Moise's "Geometric Topology in Dimensions 2 and 3" was somewhat of an anachronism even when it was first published in 1977, containing no result from after 1960, and with much of it dating from decades earlier. This introductory text in low-dimensional PL topology is both inadequate as a PL topology book (the standard references are Rourke and Sanderson or Hudson for this now-disused subject) and hopelessly outdated as a 3-manifold topology book. But it does have one major saving grace: It contains just about the only modern and complete coverage of classical theorems such as the Hauptvermutung and triangularization theorem of Rado that are frequently cited but not proved.
The main topics in 2-dimensions are the Jordan Curve Theorem, the Schoenflies Theorem, Rado's triangularization theorem for 2-manifolds (i.e., topological 2-manifolds are PL), the Hauptvermutung (i.e., any 2 triangularizations are PL equivalent), and the well-known classification of compact 2-manifolds. There are also chapters on PL approximations of homeomorphisms, tame imbeddings, and homeomorphisms of Cantor sets. In 3 dimensions the highlights are the PL Schoenflies Theorem (the originally conjectured topological version is false), the Loop Theorem and the Dehn Lemma, PL approximations, triangularization of 3-manifolds, and the Hauptvermutung, the latter 2 being the culmination of the last 100 pages of the book. There's also an entertaining account of Antoine's wild sphere imbedding and Stallings's counterexample for a simpler version of the Loop Theorem.Read more ›
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Geometric topology deals with questions of the existence of homeomorphisms to paraphrase the author. The author is credited with the first proof of the existence of triangulations for 3-manifolds(3-manifold triangulation theorem). A topological space(subspace or possibly manifold) has a triangulation if a homeomorphism can be found which maps it onto a polyhedral or simplicial complex(possibly infinite). Though the proofs are detailed this is still I'd say a graduate level text. Just from this brief description you're already dealing with general topology and the topology of polyhedra or complexes, i.e., homology of complexes or simplicial homology. The author cites the Seifert/Thelfall text for much of this material but this is hard to find and/or pricey. Adequate coverage of general topology and the fundamental group(including the Seifert-Van Kampen theorem) can be found in Munkres'Topology (2nd Edition). Simplicial homology is covered in the first few chapters of Munkres'Elements Of Algebraic Topology. In fact Theorem 26.6 in chapter 3 of the Munkres topology text is frequently used in establishing the existence of a homeomorphism. This same theorem can be found in Rudin's Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) as Theorem 4.17 in chapter 4 though homeomorphism is not mentioned. Hopefully since you're reading this review, these books are already in your library.Read more ›
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