Surfaces in 4-space
type: Item
instance of: scholarly article
title: Surfaces in 4-space
zbMATH Open document ID: 1078.57001
author: J. Scott Carter
author: Seiichi Kamada
author: Masahico Saito
published in: Encyclopaedia of Mathematical Sciences
publication date: 2004-12-02T00:00:00Z
review text: The low-dimensional topology sky is now richer for one more book treating surfaces in 4-space. The book treats the theory of knotting of surfaces in 4-space presenting up to date results and research that has been flourishing during the last decades. It is a joint work of three authors who are already experienced in writing books about this subject: see e.g. [\textit{J. S. Carter} and \textit{M. Saito}, Knotted surfaces and their diagrams, Math. Surveys and Monographs 55 (1998; Zbl 0904.57010); \textit{S. Kamada}, Braid and knot theory in dimension four, Math. Surveys and Monographs 95 (2002; Zbl 0993.57012)]. Let us scan the titles of the chapters to get an idea of the contents. The book is divided into four chapters entitled: 1. Diagrams of Knotted Surfaces 2. Constructions of Knotted Surfaces 3. Topological Invariants 4. Quandle Cocycle Invariants. An appendix brings the list of quandles of order 3, 4, 5, 6 and their homology groups. All maps in the book, and consequently the knots, are assumed to be smooth (of class \(C^\infty\)) or PL locally flat. Each notion is precisely defined with a short historical account included. The results are gradually introduced, illustrated by examples, and original references are always cited. The reader is advised if a result has a higher dimensional counterpart. The book contains an exhaustive list of references and the index. It represents a nice, useful and reliable encyclopaedic presentation of the above mentioned subject up to the date it was put in print.
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57Q35
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 2121101
zbMATH Keywords: knot
zbMATH Keywords: surfaces
zbMATH Keywords: braid
zbMATH Keywords: link
zbMATH Keywords: moves
zbMATH Keywords: Roseman moves
zbMATH Keywords: cabling
zbMATH Keywords: quandle
zbMATH Keywords: quandle cocycle
zbMATH Keywords: Wirtinger presentation
zbMATH Keywords: Alexander modules
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2121101