Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. Equivariant stable homotopy theory and the Kervaire invariant problem. (English) Zbl 1484.55001 New Mathematical Monographs 40. Cambridge: Cambridge University Press (ISBN 978-1-108-83144-4/hbk; 978-1-108-91727-8/ebook). ix, 870 p. (2021). The Kervaire Invariant One problem was one of the great outstanding conjectures in homotopy theory, until its solution was announced in 2009, with a full article by the three authors of this book following in [M. A. Hill et al., Ann. Math. (2) 184, No. 1, 1–262 (2016; Zbl 1366.55007)]. Its central question is whether the element \(h_j^2\) in the 2-line of the Adams spectral sequence for \(p=2\) is a permanent cycle for all \(j\) and thus detects an element \(\theta_j\) in the stable homotopy group \(\pi_{2^{j+1}-2}(S^0)\). For \(j=1,2,3\) the \(\theta_j\) have been constructed using framed manifolds and relate to the Hopf elements, while \(\theta_4\) and \(\theta_5\) were found using algebraic methods.This book contains “everything” one needs for the authors’ main result, namely that for \(j \ge 7\), \(\theta_j\) does not exist. The case of \(j=6\) remains open. More precisely, the goal is the construction of a spectrum \(\Xi\) with the following properties. ● (Detection Theorem) There is an Adams-Novikov spectral sequence for \(\pi_*(\Xi)\) which detects \(\theta_j\).● (Periodicity Theorem) There is a stable equivalence \(\Omega^{256}(\Xi) \cong \Xi\). In particular, the homotopy groups of \(\Xi\) are 256-periodic.● (Gap Theorem) \(\pi_k(\Xi)=0\) for \(-4 < k < 0.\) As a consequence, the \(\theta_j\) (\(j \ge 7\)) cannot exist for degree reasons, as \(|\theta_j| =2^{j+1}-2\) is congruent to \(-2\) mod 256.The first, introductory chapter explains the context of the main result from classical homotopy theory and geometry towards modern equivariant stable homotopy theory and algebraic calculations, which in itself is an interesting survey across much of algebraic topology. The authors pay respect to the beginnings of the subject, much like a hiker may look back at the distance already travelled after a long ascent.Part One (Chapters 2-6) is called “The Categorical Toolbox”. Unsurprisingly, the journey towards the main result starts with some background in category theory: “the object is to save the reader the trouble of looking up these concepts elsewhere”. Necessary tools are presented, with priority given to notions needed in later chapters. Some proofs are omitted in favour of references, technical details are included where needed. Examples are included throughout.The other chapters continue in a similar vein. Chapter 3 provides a foundation in enriched category theory, with the aim of discussing enrichments in \(G\)-spaces. Chapters 4 and 5 are dedicated to the background in model categories. The first deals with the basics, with a nod towards Quillen’s original notes. Chapter 5 continues with some more contemporary tools in model categories, concentrating on those methods needed to set up a good category of \(G\)-spectra later.Part One finishes with a short chapter on Bousfield localisation. It contains basic definitions, properties and some examples, presents the most commonly used existence results, but again leaves proofs to the references.Part Two (Chapters 7-10) is “Setting Up Equivariant Stable Homotopy Theory”. This is needed to construct and understand the spectrum \(\Xi\) of the main theorem(s), which carries a \(C_8\)-action. Chapter 7 starts with a comparison of different model categories of spectra before settling on a category of spectra suitable for the purposes of this book and studying its properties. The next chapter introduces some equivariant techniques, before moving on to orthogonal \(G\)-spectra in Chapter 9 and their monoidal properties in Chapter 10. Again, this part of the book is a useful and relatively self-contained point of reference for anyone working with \(G\)-spectra.With all of the above under our belt, the third and final part (Chapters 11-13) is “Proving the Kervaire Invariant Theorem”. Chapter 11 sets up the main computational tool, which is the slice spectral sequence. This spectral sequence has been a useful ingredient for many research articles of the last years, so the chapter provides a good overview for anyone who wishes to work on related topics. Chapter 12 deals with the \(C_2\)-spectrum \(MU_R\), which lays the foundation of the construction of the \(C_8\)-spectrum \(\Xi\) of the main theorem. Thus, the chapter is more specialised in nature than any of the previous ones. Chapter 13 then applies the methods of Chapter 12 to obtain the spectrum \(\Xi\), and the techniques of Chapter 11 to prove the three points of the main theorem, doing its best to present the proof in manageable pieces.In a philosophy very similar to the problem of calculating stable homotopy groups, rather than being entirely result-oriented, a lot of the work’s value lies in the journey to the main theorem. This book gives the methods employed more breathing space and provides a potentially friendlier entry point to the subject than the relevant research articles. Furthermore, the background chapters on category theory and equivariant stable homotopy theory have the scope to be solid references in their own right to graduate students and experts alike.All in all, the book succeeds in simultaneously being readable as well as presenting a complex result, in providing tools without being lost in details, and in showing an exciting journey from classical to (at the time of this review) modern stable homotopy theory. Thus, we can expect that it will find a home on many topologists’ bookshelves. Reviewer: Constanze Roitzheim (Canterbury) Cited in 1 Document MSC: 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 55P42 Stable homotopy theory, spectra Keywords:algebraic topology; homotopy theory Citations:Zbl 1366.55007 PDFBibTeX XMLCite \textit{M. A. Hill} et al., Equivariant stable homotopy theory and the Kervaire invariant problem. Cambridge: Cambridge University Press (2021; Zbl 1484.55001) Full Text: DOI