In his 1951 review of Chevalley’s Introduction to the Theory of Algebraic Functions of One Variable (which carried a price-tag of $4.00 in those halcyon times) Weil famously observed: “Here is algebra with a vengeance; algebraic austerity could go no further.” Noting that one should only steal from the best, we might paraphrase Weil and observe in regard to the book under review, The Novikov Conjecture: Geometry and Algebra, by Matthias Kreck and Wolfgang Lück, that here is algebraic topology with a vengeance! And it’s perhaps proper to note that here, too, we encounter no small measure of algebraic austerity: a geometry book with no pictures. Of course this isn’t really a criticism per se, given that Kreck and Lück generally work in dimension at least five; besides, trumping everything with algebra is a virtuous act.
But naturally it’s not just algebra. The Novikov conjecture, as well as the other, related, conjectures the book is concerned with (e.g. the Borel conjecture, the Baum-Connes conjecture) are truly geometrical and topological assertions. Already in the Introduction the authors place their emphasis on questions surrounding the classification of manifolds, playing off the Poincaré conjecture (which may presently acquire theorem-status, if Perelman is right). Regarding the specific focus of what lies ahead Kreck and Lück note that “[t]he Borel Conjecture, which is closely related to the Novikov Conjecture, implies that the fundamental group determines the homeomorphism type of an aspherical closed manifold.” Thus, the prevailing context is that of topological invariants (e.g. homotopy, (co)homology, characteristic classes).
Well, then, what does the Novikov conjecture assert? The answer to this question is a bit involved: G is any discrete group, BG its classifying space, M a closed, oriented, smooth manifold, and u: M → BG. Let L(M) be the L-class of M, so that we get that L(M) is a sum of homogeneous polynomials in rational Pontjagin classes. If x is an element of the direct product of the non-negative dimensional cohomology groups of BG taking values in Q, define the signature sign(x, M, u) := , where <.,.> is the Kronecker product, [M] is M’s fundamental class in the dim(M)-dimensional homology group of M taking values in Z, and otherwise we’re dealing with the intersection pairing. The Novikov conjecture says that this signature is a homotopy invariant for every x.
By contrast, the Borel conjecture is a lot easier to state. To wit: given a pair of closed, aspherical manifolds, M, N. Then, first, each homotopy equivalence M ® N is homotopic to a homeomorphism; and, second, M and N are homeomorphic if and only if they have isomorphic fundamental groups. Kreck and Lück go on to tantalize the reader by observing that e. g. if u is a homotopy equivalence then Borel ⇒ Novikov, and that a certain instance of Borel implies nothing less than the 3-dimensional Poincaré conjecture. Exciting stuff.
Now for the question of the proper audience for this book. The propaganda on the book-cover warns that “[t]he prerequisites consist of a solid knowledge of the basics of manifolds, vector bundles, (co-)homology, and characteristic classes.” And we might couple this somewhat understated appraisal to the observation that the book springs from the authors’ Oberwolfach seminar on the indicated material, evidently involving an audience of initiates if not aficionados.
Thus, The Novikov Conjecture: Geometry and Algebra is not for the timid, the dabbler, or the dilettante. It is serious business and deals with a wealth of interesting and deep material from modern algebraic topology and differential geometry; here is a short sampling of topics: bordism, the signature, the Whitehead group, Whitehead torsion, s-cobordism, surgery, Poincaré duality, equivariant homology. Twelve chapters are taken care of by Kreck, eleven by Lück, one by Lück and Varisco. There are 71 exercises at the end of the book, with solution hints, and, apparently for good measure, a copy of the Oberwolfach schedule for the whole affair. The reader, if he is prepared to work hard (and is well-prepared to begin with), will learn a lot of wonderful contemporary mathematics by following the path Kreck and Lück lay out.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.