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Rational Homotopy Theory and Differential Forms

Phillip Griffths and John Morgan
Publication Date: 
Number of Pages: 
Progress in Mathematical Physics16
[Reviewed by
Michael Berg
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I would not touch this book until I’ve done a good bit of study from Bott-Tu, Differential Forms in Algebraic Topology.” Griffiths and Morgan state explicitly at the very start of their Introduction that “[t]he purpose of [their] course is to relate the \(C^\infty\)-differential forms on a manifold and algebro-topological invariants.” They proceed to cite, very naturally, the de Rham theorem to the effect that de Rham cohomology, quintessentially a Hodge theoretic beast as it concerns differential forms on a smooth manifold, is isomorphic to the manifold’s singular cohomology, where the manifold is of course regarded as a topological space per se. The de Rham complex engenders a differential graded complex, i.e. a differential graded algebra (DGA) when multiplication is given by the wedge product, and the authors state that “[t]he main theorem of this course will be that from the DGA of \(C^\infty\)-forms it is possible to calculate all the real algebro-topological invariants of the manifold.” The fact that this is not meant for raw beginners is even more emphatically illustrated by the next sentence: “More precisely, we shall be able to use the forms to obtain the Postnikov tower tensored with R, of the manifold.” Solid algebraic and differential geometry are really non-negotiable prerequisites.

The procedure Griffiths and Morgan use to get all this off the ground and cruising at 30,00 feet is due to Dennis Sullivan: start with piecewise linear de Rham (working over Q), then develop the homotopy theory of differential graded algebras replete with minimal models. Observe that over Q there is a duality between DGA minimal models and Postnikov towers, specifically when the DGA is the algebra of piecewise linear forms on a simply connected simplicial complex. Of course, a general \(C^\infty\)-manifold can be smoothly triangulated and such a triangulation provides a simplicial complex and all the players are on the field. After a bit of commutative algebra, one tensors over Q with R and wins the game. It all amounts to a wonderful example of some very beautiful algebraic topology, as one would expect form Sullivan.

Griffiths and Morgan present a ramified treatment of this exciting material, with the action in the story spread out through the middle chapters in which Chapter 12 is something of a lynchpin. They add marvelous discussions of various related or connected themes, including Daniel Quillen’s work on rational homotopy which they characterize as dual to the way Sullivan proceeded: “Instead of using differential forms as the basic model, Quillen uses differential graded Lie algebras.” See Chapter 17 in this connection: a fascinating analysis of the indicated dual approaches by two masters of modern algebraic topology.

In contrast, or rather (again) dually, to my recommendation to dig deeply in Bott-Tu’s book before even cracking the present book, Griffiths and Morgan recommend as prerequisite sources the well-known texts by Greenberg and by Spanier (on algebraic topology) and by Hu (on homotopy theory). Indeed.

Finally, the book under review is a repackaging of a forty-year old note set from a summer course taught at Instituto Matematico “Ulisse Dini” in Florence, but the age of the subject covered does not detract from its austerity, nor does it diminish its elegance. Quite the opposite, in fact; for instance, in the interim the present subject was applied to algebraic geometry. Beyond this the authors note that what they did in the Florence notes “represented an approach and point of view that does not appear in the literature.” Happily now it does.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

1 Introduction

2 Basic Concepts

3 CW Homology Theorem

4 The Whitehead Theorem and the Hurewicz Theorem

5 Spectral Sequence of a Fibration

6 Obstruction Theory

7 Eilenberg-MacLane Spaces, Cohomology, and Principal Fibrations

8 Postnikov Towers and Rational Homotopy Theory

9 deRham's theorem for simplicial complexes

10 Differential Graded Algebras

11 Homotopy Theory of DGAs

12 DGAs and Rational Homotopy Theory

13 The Fundamental Group

14 Examples and Computations

15 Functorality

16 The Hirsch Lemma

17 Quillen's work on Rational Homotopy Theory

18 A1-structures and C1-structures

19 Exercises