“Topology: A Very Short Introduction” by Richard Earl

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Rating: 3/5 Stars

An introductory book that gives a look at topology: what it is, what is can be used for and some work being done in topology. The first chapter goes in gently by looking at Euler’s formula for polygons and showing how it applies to polygons in general. Later chapters rapidly become very mathematical and probably requires some level of mathematical education to appreciate properly, even if you have to skim through some mathematical relationships to get at the heart of topology.

Chapter One gives an introduction to the study of topology, which is concerned with the relationship of shapes, connections and relative positions of objects. It then introduces Euler’s formula, which relates the number of vertices (V), edges (E) and faces (F) of objects into a mathematical formula and shows that for standard, three-dimensional shapes, V - E + F = 2 always holds. The chapter then goes on to give a readable proof for why this equation is true of such shapes. Then, using the formula, it demonstrates why there can are only five Platonic solids in three dimensions.

Chapter Two looks at surfaces in general. Starting from a square plane, its edges are then deformed and glued together in certain procedures to give rise to shapes like a torus. It also shows how the value found by applying Euler’s formula changes for different kind of surfaces. However, Euler’s formula is not sufficient as the introduction of one-sided surfaces (the Möbius strip and Klein bottle) show. This leads to the Classification Theorem which is used to classify two-sided and one-sided surfaces. Complex numbers are then briefly introduced, leading to Riemann Surfaces, used to represent surfaces in higher dimensions.

Chapter Three looks at Continuous Functions and the issues with coming up with a definition for them that is mathematically rigorous. This chapter is probably ‘heavy going’ for non-mathematicians but gives an idea of why some ‘common sense’ definitions may not be rigorous enough for the needs of mathematicians.

Chapter Four continues the mathematical theme by looking at defining metrics and distances between functions, leading on to sets and subsets and connectivity. All of which would come to bear on more advanced topics on topology.

Chapter Five builds on the previous two chapters by covering different kinds of topology. From geometric topology introduced in Chapter One, other kinds of topology are introduced here: differential topology (dealing with curves), the ‘hairy ball theorem’ which deals with vectors on topological surfaces, and so on.

Chapter Six looks at the topic of knots (and unknots), showing how topology deals with how to define whether a loop is a knot and what kind of knot it is.

Book read from 2021/02/10 to 2021/02/21