An Invitation to Computational Homotopy

There is a long tradition in mathematical software of writing a textbook that simultaneously acts as a User's Guide for the software itself. We can see examples in a wide range of fields - commutative algebra (A Singular Introduction to Commutative Algebra), spatial statistics (Spatial Point Patterns: Methodology and Applications with R; Applied Spatial Data Analysis with R), to mention a few fields of my own personal interest. This book continues that tradition, focusing An Invitation to Computational Homotopy on the GAP library HAP. HAP was written by Graham Ellis and is among the foremost systems for computational group cohomology in the current computational algebraic systems landscape.

 

The book sets out to teach algebraic topology leading into homological algebra and especially group cohomology. In doing so, it consistently and frequently brings up snippets of GAP code or HAP code that showcase how to perform computations directly related to the material covered. It takes a route I personally find quite pleasant, introducing CW-complexes very quickly, and building the theory of algebraic topology specifically over cell decompositions of spaces, rather than a more general but far more arduous route over singular simplices in more arbitrary topological spaces. Along the way, several powerful techniques are introduced at - what seems to me to be - quite early stages: discrete morse theory shows up very early in the book, as does - to my initial surprise - persistent homology and several other techniques from Topological Data Analysis (TDA).

 

Persistent homology, in this book, shows up in more contexts than a reader from the TDA camp may be prepared for. In their article Persistent Homology of Groups [Ellis-King, Journal of Group Theory (2011)], persistent homology was shown by Graham Ellis and Simon King to be useful for studying group homology; it should come as no surprise that in the years since then, Graham Ellis has found more uses for persistent homology in pure mathematics. Notably, at one point a spectral sequence page was proven to have trivial boundary maps by inspecting a persistence barcode.

 

The book is well equipped with exercises - several pages at each chapter end. These range from the purely theoretical (Prove that...) to quite data-driven (Using this dataset, calculate and interpret...), providing a widespread of approaches to the material in the book.

 

Which brings me to the first of my couple of annoyances with the book; the data used for data-driven exercises tend to not be provided. Instead, references to websites or even just to government agencies are handed out, expecting readers to find and retrieve the intended dataset on their own. This dates the book, and could - depending on the reliability of government agencies around the world - quickly lead to unusable exercises from sheer link rot.

 

The book dates itself in more ways than this - and unfortunately, some of the ways are pretty glaring and long out of date. When discussing the available software for computational homology, Ellis chooses to cite the deprecated JPlex - and with a URL that no longer works - rather than the more updated JavaPlex. Even when citing JPlex, Ellis uses a less common citation that obscures the authors of JPlex in favor of just citing Stanford University as a source.

 

While the book has plenty of code examples, there is no guidance for anyone new to the GAP computer algebra system to be found. For a book that I would expect to function better as an invitation to GAP, showcasing attractive functionality, the lack of even an appendix going over the basics of GAP syntax and the most common quirks to watch out for when coming from a different language makes the book more difficult to engage with.

 

As for the content itself, I would be tempted to call the title almost a misnomer. Homotopy is certainly studied, but the book spends far, far more time and energy on homology and cohomology in different settings. Even when cohomology is tightly connected to homotopy theory, calculations happen on the cohomology side of the connection, with few algorithms available in the homotopy world.

 

All in all, this book is definitely one I appreciate having read, and one that I can see myself recommending to students who already wish to go into something related to group theory. As such it balances on a pretty thin line; the perfect reader is a student who has heard enough about group theory to want to dig quite a bit deeper into it and with a computational approach, but who has yet to hear anything about algebraic topology. For these students, however, the book should function quite well.

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Mikael Vejdemo-Johansson is an Assistant Professor of Data Science at CUNY College of Staten Island.