The book under review completely solves an old and monumental problem in topology and was awarded the Ferran Sunyer i Balaguer Prize in 2020. Here’s the backstory.

A bit more than half-a-century ago, for various reasons, topologists came to believe that if torsion could be removed from their beloved algebraic invariants (e.g. homotopy and homology groups), then the homotopy types of spaces could be described completely algebraically. As a first step, a topological process called *rationalization* was developed that took a space and transformed it into an associated space with homotopy and homology groups the original space’s tensored with the rational numbers. Well, there was a problem with this process when the space wasn’t “nice” (i.e. if it had a nontrivial fundamental group which acted non-nilpotently on higher homotopy groups). To overcome this problem, rationalization was extended to a notion of rational (or $\mathbb{Q}$)-completion. The rational completion of a space was much harder to understand however and algebraic invariants did not behave as nicely as for rationalization. So, when a fundamental group is present, some extra hypotheses must be put in place in order to have a rational homotopy theory where algebraic invariants transform in a simple fashion.

In spite of this fundamental group problem, this belief in complete algebraization persisted and came to fruition in the late 1960’s with Quillen’s proof that, for simply connected spaces, rational homotopy types could be characterized in terms of, say, differential graded Lie algebras (dgla’s). (In fact, the dgla models could only be defined in the simply connected case, so the question of what to do with a fundamental group was still open.) A competing “differential form” approach due to Sullivan arose that had its genesis in the work of de Rham and Whitney and produced a commutative differential graded algebra (cdga) that could be defined for any space and could be realized as the \( \mathbb{Q} \)-completion. Therefore, in the simply connected (or even nilpotent) case, this cdga was a completely algebraic model for the rationalization. Voila, it is finished!

Well, not quite. Over the past 50 years, rational homotopy theory has been applied to a host of problems in topology and geometry and topologists have seen that, while some problems are highly amenable to the Sullivan approach, there are others that are more suited to Quillen’s methods. Of course, there are clever ways to get around such difficulties, but the dream has always been to have a complete Quillen theory (even for nonconnected spaces) dual to Sullivan’s. This is the purpose of the book under review. Not so long after Sullivan’s approach appeared, Bousfield and Guggenheim recast the theory using simplicial sets and showed that taking a cdga model of a simplicial set and spatially realizing a cdga by a simplicial set were adjoint functors that produced a correspondence on respective homotopy theories. In particular, spatially realizing a special type of cdga model of a space (called a minimal model) produced the \( \mathbb{Q}\)-completion of the space. This process of spatial realization required a cdga model on n-simplices for each \( n \geq 0 \). For a Quillen approach, a dgla analogue is needed and the first step to achieving this was made by Lawrence and Sullivan when they proposed a certain dgla model for the 1-simplex.

The book under review shows how this model for the 1-simplex can be extended to dgla models for all simplices. Putting these into a single structure allows for a definition of spatial realization of dgla’s that also produces the \( \mathbb{Q} \)-completion. The key here is to use not just dgla’s, but completions (in the algebraic sense) of dgla’s with respect to a filtration by bracket length in the dgla as well as a special multiplication on the degree 0 elements given by the Baker-Campbell-Hausdorff formula. As might be expected, there are many technical details which take up the bulk of the book. In particular, in order to obtain their *global Lie model* \( \mathfrak{L}_{X} \) for a simplicial set \( X \), they need to re-define the usual closed model category structure (i.e. fibrations and weak equivalences) of **cdgl**, the category of complete differential graded Lie algebras. After having done this and obtaining a global cdgl model \( \mathfrak{L}_{X} \), they show the connections between this model and all other previously defined models (in the simply connected or nilpotent cases) and, most importantly, they show how \( \mathfrak{L}_{X} \) (or rather, a “component of it) gives the spatial realization \( \mathbb{Q}_{\infty} X \), the \( \mathbb{Q} \)-completion of the simplicial set \( X \). The final chapter shows how their model gives information about various spaces including showing that the \( \mathbb{Q} \)-completion of a 2-complex obtained by attaching a cell to a wedge of circles is an Eilenberg-Mac Lane space with fundamental group the Malcev completion of the original fundamental group. Probably the most interesting application, however, is to mapping spaces. For simplicial sets \( X \) and \( Y \), the authors describe a tractable Lie model whose spatial realization gives \( \mbox{Map}(X, \mathbb{Q}_{\infty}, Y) \). This is an important extension of much

research that has appeared over the years.

So all of the above has been rather technical — perhaps too technical for the casual reader. But this is a very technical book (whose Introduction, however, can be read by all and gives a nice overview of the whole book) and it requires quite a bit of background to fully understand proofs etc. Nevertheless, it is likely that modern topology graduate students have the background needed to appreciate the efficacy and sheer beauty of this whole approach to Lie models, so it would certainly be appropriate (and recommended!) for advanced courses and seminars.

John Oprea (

jfoprea@gmail.com) is Professor Emeritus at Cleveland State University. His interests are in geometry and topology.