# Book Review: Essential Topology

Topology is an area of mathematics with which I have no prior experience. I guess it was felt historically that engineers don’t really need topology, and this has filtered down the generations of study. Yet I’ve always found the ideas of topology intriguing, even if I have not deeply understood them. They seem to pop up in the most wide variety of places. While running a math circle for primary school kids, we ended up discussing Euler’s Polyhedron Formula, and I realised that if I wanted to explore these ideas more deeply, I would need a proper grounding in topology. From the wonderful lectures of Tadashi Tokieda which I watch with my son, to the more theoretical end of computer science I bump up against in my day-to-day research, topology seems to be everywhere. I found this book, Essential Topology by Martin D. Crossley, while browsing in a bookshop and decided to take it on holiday as holiday reading.

As presented, the book naturally falls into three parts: an introductory section, a section on basic topology, and a section on more advanced algebraic topology. Although short, the book is written as a sequence of a good number of chapters, 11 in total, which make the material more digestible. Moreover, I found the two “interludes” between sections of the book to be a great innovation – invaluable as a mechanism for orienting my reading, providing the appropriate informal guidance about the journey on which the text was taking me.

The introductory section begins with looking at the question of continuity from a rigorous perspective, bringing in both the epsilon-delta approach to continuity with which I am familiar, and an approach based on open sets with which I was not – but which is easily generalised to functions other than from $\mathbb{R}$ to $\mathbb{R}$. It then moves on to axiomatically defines topological spaces, some standard topologies, and bases for topologies.

The second section begins to explore some of the properties that can be proved using the equipment set up in the previous chapters: connectedness, compactness, the Hausdorff property, homeomorphisms, disjoint unions, product and quotient spaces. I enjoyed this section greatly.

The third section then goes on to discuss various more advanced topics, looking at various topological invariants that have been well studied. Some highlight for me included:

Chapter 6 discusses the idea of homotopy as an equivalence between maps: two maps $f,g: S \rightarrow T$ are homotopic iff there is a continuous function $F: S \times [0,1] \rightarrow T$ allowing for a kind of continuous deformation of one function into the other, i.e. with $F(s,0) = f(s)$ and $F(s,1) = g(s)$. This is shown to be rather a broad equivalence, for example all continuous functions on $\mathbb{R}$ are homotopic. However, working with other topological spaces, things get quite interesting. A big part of the chapter is given over to working with circles $\mathbb{S}^1$, where it is shown that there is a countable set of homotopy classes of functions from $\mathbb{S}^1$ to $\mathbb{S}^1$. This is very clearly described, and I seem to remember through the mists of time some of these issues cropping up informally in the context of my control theory courses as an undergraduate (or, for that matter, fathoming ${\tt unwrap}$ in Matlab as an undergraduate!) The chapter ends with a proof of the fantastically named “Hairy ball theorem,” from which – amongst other things – it follows that at any given point in time, there’s a part of the world with zero wind speed!

Chapter 7 discusses the Euler characteristic as a topological invariant and introduces some interesting theorems. After introducing the idea of a ‘triangulable space’, it is stated that if two such spaces are homotopy equivalent then they have the same Euler number. More remarkable (to me!) is that when restricted to surfaces, it is sufficient for two such surfaces to have the same Euler number and the same orientability in order for them to be homotopy equivalent. Unfortunately, the proofs of both these results are omitted – apparently they are rather involved – references are given. I certainly appreciated the results collected in this chapter, but I found the exercises quite hard compared to other chapters, possibly partly because of the proof issue, but also because I found it hard to visualise some aspects, e.g. triangulation of a surface of genus 2, and since such surfaces had only been defined informally in the preceding chapters I could not (easily) fall back on a purely algebraic approach to the geometry. I did find it particularly interesting to see the Euler characteristic rigorously defined for a general class of spaces – the very definition in the primary math circle that had brought me to this book in the first place!

Chapter 8 discusses homotopy groups. The basic idea is that one can work with homotopy classes of maps from ${\mathbb S}^n$, the $n$-dimensional sphere, to a space $X$ (actually pointed homotopy classes of pointed maps, but lets keep things simple) and it turns out that these classes form a group structure: they have an identity element (constant maps), an addition operation, etc. I guess the purpose here is to bring out the topological structure of $X$, though the special role played by ${\mathbb S}^n$ was not totally apparent to me – why is this particular space so fruitful to study? I wonder if any readers can comment on this point for me.

Chapter 9 provides an introduction to Simplicial Homology, the idea – roughly – that those combinations of simplices within a simplicial complex which ‘could’ form boundaries of other simplices but ‘do’ not, tell us something about the topology of the simplicial complex. Homology is then introduced in a different form (Singular Homology) in Chapter 10, and it is stated that the two approaches coincide for triangulable spaces.

In these latter chapters of the book, some theorems tend to be stated without proof. The author is always careful to refer to other textbooks for proof, and I guess this is a necessary aspect of trying to introduce a very large subject in a brief textbook, but for me this made them somewhat less appealing than those in the first two sections. Nevertheless, I now feel suitably armed to begin looking at the world through a more topological lens. I wonder what I will see. For now I’m off to eat a coffee cup.