Topological embeddings (Pure and applied mathematics; a series of monographs and textbooks) by T. Benny Rushing (PDF)

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⭐With few prerequisites, mainly general or point set topology, linear algebra, and group theory, this text develops tools that are of use in dimensions 3 and above. The first chapter gives a brief but self contained presentation of the basics of PL topology like polyhedra, join, neighborhood star, link, general position (related to linear independence), etc. Where proofs are not given, enough is presented so the reader can do them as exercises. References like

⭐are given, if you’d rather not. In depth introductions to handlebody theory and collaring are also given here. M.Brown’s proof of the Generalized Schoenflie’s Theorem rounds out this introduction.Coverage is quite broad. Even the classic theorem on embedding a k-complex into 2k+1 dimensional space is given along with a worked out example of why this is the best possible dimension. Various embeddings are discussed along with their fundamental groups and these groups are presented through generators and relations (free groups initially). If you’ve done work on knots you’ve seen some of this before. By the way an embedding is a homeomorphism of one space onto a subset of another. A knot is an embedding of a circle or loop into 3-space. Note 3-space is needed for this to be interesting, if we just follow the loop we come back to where we started but if we consider a very narrow tube of which points of the loop form centers-here’s the knottedness.Another highlight of the text is a proof of the Poincare Conjecture for dimensions above 4 via Stalling’s Engulfing Theorem rather than the techniques of Smale. Maybe PL topology methods will be used in 4 dimensional triangulation and Hauptvermutung after all.Excellent reference book for geometric topologists.

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