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Formal Geometry and Bordism Operations

Eric Peterson
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics
[Reviewed by
Michael Berg
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The book under review sports a Foreword by Matthew Ando that starts off: “This book does a remarkable job of introducing some of the interactions between algebraic topology and algebraic geometry, which … goes by the name ‘chromatic stable homotopy theory.’”  However, even before we get to this Foreword, the book’s frontispiece has the following quote by none other than the originator of stable homotopy theory, J. F. Adams: “Let us be glad we don’t work in algebraic geometry” --- nice irony. We’re in for quite a ride, it seems.
All right, then. Some of us have a notion of what homotopy means; one particularly forgiving way of characterizing it is that it’s about smooth deformation of functions (or mappings) into each other, in the setting of topological spaces. But what does “stable” signify here?  It turns out (cf. Wikipedia!) that we’re talking about successive applications of the suspension functor and the phenomenon that at some point we might start (and keep) getting the same thing: this is stabilization (and somewhere Emmy Noether is smiling). That just leaves the word “chromatic.”  For a first pass at grasping this notion, we find on p. 158 of the present book the following characterization: “… a useful fact about coherent sheaves [on a certain moduli space] is that they are completely determined by their restrictions to all of the open submoduli. The analogous fact about finite spectra is referred to as chromatic convergence” (italics in the original). So we’re getting closer, but a natural question is “What about spectra?”  To get at this we go first to Peterson’s Introduction: “… we will define a homology theory called bordism homology [which is denoted MO(X) for a topological space X],” and then proceed to “Case Study 1” (p. 10 ff.), where the focus falls on the notion of unoriented bordism. Says Peterson: 
… we will require a definition of bordism spectrum [!] that we can manipulate computationally, using just the tools of abstract homotopy theory. Once that is established, we … bring [in] algebraic geometry: The main idea is that the cohomology ring of a space is better viewed as a scheme … and the homology groups of a spectrum [in the corresponding algebraic geometric sense where (ring) spectra attend schemes by definition] are better viewed as a representation for a certain elaborate algebraic group. This data … finds familiar expression in homotopy theory: … a form of group cohomology for this representation forms the input for the classical Adams spectral sequence …
So we have hit on quite a treasure, haven’t we?  The notion of spectrum not only arises with something of a double meaning, it in fact comprises the lynch pin for a deep interplay between, yes, algebraic topology (in the form of stable homotopy theory) and fundamental aspects of Grothendieck-style algebraic geometry, putting some more spin on the words by Adams quoted above.
All this having been said, it’s obvious that Peterson’s book is by no means safe for rookies. To get into this (I think very beautiful) material seriously, the reader needs to know a serious amount of algebraic topology, including bordism theory and probably stable homotopy theory a la Frank Adams, and should be well-versed in the mainstays of modern algebraic geometry. Hartshorne’s book Algebraic Geometry is clearly a sine qua non, here. Ando, in his Foreword, notes too, that “chromatic homotopy theory had its origins in the work of Novikov and Quillen, who first investigated the relationship between complex cobordism and formal groups and perceived its potential for investigating the stable homotopy group of spheres.”  So, to be sure, we’re deep in the forest, and had better come prepared if we travel this path.
Finally, a few comments about the style of this book, again through the agency of Ando’s remarks in the Foreword: “I had the good fortune to meet Eric [Peterson] as an undergraduate and convince him to work on some problems I was interested in. The things that make Eric fun to work with are well reflected in this book. It has a down-to-earth and inviting style (no small achievement in a book about functorial algebraic geometry). It is elegant, precise, and incisive, and it is strong on both theory and calculation. An important feature of the book is that it takes the time to give elegant proofs of some ‘theory-external’ results: theorems you might care about even if chromatic stable homotopy isn’t your subject.”
So, it all looks very good, and, as Ando puts it, inviting.


Michael Berg is a Professor of Mathematics at Loyola Marymount University.  He is also an instructor in judo.