How Surfaces Intersect In Space: An Introduction To Topology (2nd Edition)
This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced.In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space.Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface.Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view.This is a guide book to a wide variety of topics. 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.
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How Surfaces Intersect in Space: An Introduction to Topology
J. Scott Carter
No preview available - 1995
2-handle 3-dimensional space 3-sphere annular annulus arcs of double ball basket shaped thingy boundary circles boundary components bounds a disk Boy's surface branch points Cartesian product Chapter closed surface consider constructed coordinate disks critical points cross cap cut-line deformation dimension disk bounded disk in 4-space double arc double points equator equatorial Euler number example Gauss word genus given glued handle decomposition handle slides higher dimensional homeomorphism Hopf link illustrates image surface indicates inside intersecting disks intrinsic surface invariant Jordan Curve Theorem Klein bottle knot diagram lens space loop manifold Möbius band movie moves n-manifold neighborhood non-orientable surface number of boundary orientable surfaces pair of mates planar surface preimage projective plane projective space punctured torus quotient rank Reidemeister moves Roman surface rotation separate sequence simple closed curve slice smoothing ſº solid torus substantial arc substantial circles substantial set surface with boundary topological triangles triple point unknot vertex