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Cohomology and Differential Forms

Izu Vaisman
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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One of the mainstays of differential geometry is the theory of differential forms on a smooth manifold, i.e., the players in what is generally called the de Rham complex for the manifold. The habitat for these differential forms is the graded vector space of \(k\)-fold wedge products of differentials, with coefficients being smooth functions on the manifold; the differentials themselves are the natural generators of the manifold’s cotangent spaces (working locally, i.e. in terms of local coordinates), and we vary \(k\) from \(0\) to the dimension of the manifold. Thus, still working locally, the differentials on the manifold constitute the dual basis to the natural basis for the tangent spaces. One then obtains the de Rham complex by virtue of the differential operator \(d\) taking \(k\)-forms to \((k+1)\)-forms, and one gets that \(d^2=0\) as an immediate consequence of a straightforward calculation in the context of exterior algebra, i.e. the yoga of the wedge product. So we get, right off, two titanic themes, namely, the cohomology of the manifold, aka de Rham cohomology, entirely as a consequence of having a differential complex as just indicated, and the general version of Stokes’ Theorem, which one can think of as the most evolved form of the Fundamental Theorem of Calculus and Green’s Theorem: it’s all ultimately quite the same.

In this setting we also encounter the Theorem of de Rham properly so-called, asserting essentially that different flavors of cohomology are the same. In the book under review these important results occur thrice: on p. 70 Vaisman shows that the de Rham cohomology groups can be realized via presheaves, on p. 82 he shows that for paracompact spaces the de Rham cohomology groups can be realized in terms of sheaves, and on p. 203 he derives the indicated theorem for real cohomology from that for sheaves. What is at issue here is, of course, that cohomology is taken with varying coefficients, from presheaves on the underlying space, to sheaves, and then to the real numbers, realized (in case the underlying space is a manifold) as a constant sheaf.

So in a very noteworthy sense the story is told backwards, as it were, given that presheaves and sheaves are more sophisticated entities that the real numbers and the latter situation, i.e. de Rham cohomology, with real coefficients, for a smooth manifold, is manifestly what is of most use and interest to, say, differential geometers — or so one would imagine. But Vaisman’s approach makes it clear that the largely French revolution of the 1950s, led by H. Cartan, Serre, and Grothendieck, has indeed resulted in more liberty: by going at things with category theory, presheaves and sheaves, and so on, a very general framework is created from the outset in which major results like de Rham’s theorem, can be presented most effectively and holistically. Bourbaki vindicatus.

Vaisman’s book is accordingly a testament to what is a most fecund way of doing geometry, be it algebraic or differential, although clearly the general methodology of sheaf theory is most closely associated with the former. But this is not to minimize the role played by the indicated methods in differential geometry, as Vaisman’s treatment amply illustrates. It is interesting to note, for example, how heavily certain developments in its pages involve Chern’s results: differential geometry par excellence.

Yes, this is an excellent book, and will serve, even now, over forty-three years after its first appearance (kudos to Dover, as always, for reissuing the book), as an excellent introduction to not just sheaf cohomology (and ipso facto the category theory everyone needs to know) but also to differential geometry proper, the theory of fiber and vector bundles and characteristic classes (well, let’s just say Chern classes), and even certain themes in Riemannian geometry. 

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

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