This is a book of marvelous pictures that illustrates standard examples in low dimensional topology. The text starts at the most basic level (the intersection of coordinate planes) and gives hands on constructions of the most beautiful examples in topology: the projective plane, Poincare's example of a homology sphere, lens spaces, knotted surfaces, 2-sphere eversions, and higher dimensional manifolds. The text carefully explains the importance of the examples and the techniques without being bogged down in a morass of technicalities.Chapter 1 opens with the classification of orientable surfaces, and the meaning of space. Chapter 2 discusses examples of non-orientable surfaces including models of the projective plane and the Klein bottle. Chapter 3 discusses how curves fit on surfaces and gives a general discussion of knotted strings in space. In Chapter 4, some examples of other 3-dimensional spaces are described. These include the 3-dimensional sphere, lens spaces, and the quaternionic projective space. In Chapter 5, the author reviews the movie techniques of studying surfaces in 4-dimensions. He shows how to move among the standard examples of Klein bottles, and he gives a “movie move” decomposition of turning the 2-sphere inside out. In the final Chapter, higher dimensional spaces are examined from the same elementary point of view.The book is a guide book to a wide variety of topics. It will be of value to undergraduates who want to learn geometric topology and to graduate students who want examples with which they can make computations and who need an elementary description of topological spaces. Finally, the book should be interesting to other scientists and mathematicians who want to learn some examples of topological spaces.

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.

When surfaces intersect, one may desire to highlight the intersection curve in order to make the shape of the penetrating surfaces more visible. Highlighting the intersection is especially helpful when the surfaces become transparent, because transparency makes the intersections less evident. A technique for locating intersections in screen space using only the information locally available to a pixel is discussed. The technique is designed to exploit parallelism at the pixel level and was implemented on the Pixel-Planes 5 graphics supercomputer. Banks, David C. Unspecified Center NASA-CR-191498, NAS 1.26:191498, ICASE-93-41, AD-A269019 NAS1-19480; RTOP 505-90-52-01...

In this book the authors develop the theory of knotted surfaces in analogy with the classical case of knotted curves in 3-dimensional space. In the first chapter knotted surface diagrams are defined and exemplified; these are generic surfaces in 3-space with crossing information given. The diagrams are further enhanced to give alternative descriptions. A knotted surface can be described as a movie, as a kind of labeled planar graph, or as a sequence of words in which successive words are related by grammatical changes. In the second chapter, the theory of Reidemeister moves is developed in the various contexts. The authors show how to unknot intricate examples using these moves. The third chapter reviews the braid theory of knotted surfaces. Examples of the Alexander isotopy are given, and the braid movie moves are presented. In the fourth chapter, properties of the projections of knotted surfaces are studied. Oriented surfaces in 4-space are shown to have planar projections without cusps and without branch points. Signs of triple points are studied. Applications of triple-point smoothing that include proofs of triple-point formulas and a proof of Whitney's congruence on normal Euler classes are presented. The fifth chapter indicates how to obtain presentations for the fundamental group and the Alexander modules. Key examples are worked in detail. The Seifert algorithm for knotted surfaces is presented and exemplified. The sixth chapter relates knotted surfaces and diagrammatic techniques to 2-categories. Solutions to the Zamolodchikov equations that are diagrammatically obtained are presented. The book contains over 200 illustrations that illuminate the text. Examples are worked out in detail, and readers have the opportunity to learn first-hand a series of remarkable geometric techniques.

Abstract: "When surfaces intersect, one may desire to highlight the intersection curve in order to make the shape of the penetrating surfaces more visible. Highlighting the intersection is especially helpful when the surfaces become transparent, because transparency makes the intersections less evident. This paper discusses a technique for locating intersections in screen space using only the information locally available to a pixel. The technique is designed to exploit paralleism at the pixel level and has been implemented on the Pixel-Planes 5 graphics supercomputer."

The aim of this book is to give as detailed a description as is possible of one of the most beautiful and complicated examples in low-dimensional topology. This example is a gateway to a new idea of higher dimensional algebra in which diagrams replace algebraic expressions and relationships between diagrams represent algebraic relations. The reader may examine the changes in the illustrations in a leisurely fashion; or with scrutiny, the reader will become familiar and develop a facility for these diagrammatic computations. The text describes the essential topological ideas through metaphors that are experienced in everyday life: shadows, the human form, the intersections between walls, and the creases in a shirt or a pair of trousers. Mathematically informed reader will benefit from the informal introduction of ideas. This volume will also appeal to scientifically literate individuals who appreciate mathematical beauty.

The book constitutes the refereed proceedings of the International Workshop on Algebraic Frames for the Perception-Action Cycle, AFPAC '97, held in Kiel, Germany, in September 1997. The volume presents 12 revised full papers carefully reviewed and selected for inclusion in the book. Also included are 10 full invited papers by leading researchers in the area providing a representative state-of-the-art assessment of this rapidly growing field. The papers are organized in topical sections on PAC systems, low level and early vision, recognition of visual structure, processing of 3D visual space, representation and shape perception, inference and action, and visual and motor neurocomputation.