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The Atiyah-Singer theorem and elementary number theory. (English) Zbl 0288.10001
Mathematics Lecture Series. 3. Boston, Mass.: Publish or Perish, Inc. xii, 262 p. $ 10.50 (1974).
The point of view of this book is so potentially revolutionary that the book can be adequately reviewed and the point of view can be adequately assessed only by judging future developments. Basically, the role of topology in number theory has progressed beyond the local methods such as \(p\)-adic theory to global methods such as intersection numbers of homology classes. The highest form of the art is the theorem of M. F. Atiyah and I. M. Singer [Ann. Math. (2) 87, 546–604 (1968; Zbl 0164.24301)].
The introduction makes the case by first noting that on an elementary level, analytic number theory must be topological since its commonest tool, the Cauchy residue theorem, is a special case of the Atiyah-Singer theorem. In a more sophisticated form, the theorem of M. F. Atiyah, V. K. Patodi and I. M. Singer [Bull. Lond. Math. Soc. 5, 229–234 (1973; Zbl 0268.58010)] equates two invariants of a \((4k-1)\) dimensional differentiable manifold \(M\) \((nM = \partial Y)\), namely the alpha-invariant which comes from the Pontryagin forms and the signature function of \(Y\) and the beta-invariant which comes from the \(L\)-series for a differential operator on \(M\). (Incidentally, a short, although less finalized, version of this approach is found in [F. Hirzebruch, Prospects Math., Ann. Math. Stud. 70, 3–31 (1971; Zbl 0252.58009)].)
Chapter I (Topological Preliminaries) contains the topics: Background on complex manifolds, signature theorems, \(L\)-class of a rational homology manifold, alpha-invariant of Atiyah and Singer. In Chapter II, the alpha-invariant leads directly to trigonometric sums which are related to Dedekind sums. C. Meyer had shown the number theoretic part of the connection earlier [J. Reine Angew. Math. 198, 143–203 (1957; Zbl 0079.10303)] and W. Meyer had shown the topological part [Math. Ann. 201, 239–264 (1973; Zbl 0241.55019)]. The signature function is applied to Brieskorn varieties to show that when exponents are essentially Markoff triples the signature is 0. This matter relates Dedekind sums and Markoff triples, compare H. Rademacher and E. Grosswald [Dedekind sums. Carus Math. Monographs No. 16 (1972; Zbl 0251.10020)] and the reviewer’s paper [Math. Ann. 196, 8–22 (1972; Zbl 0227.10018)].
Chapter II is certainly the core of the thesis. There may, however, be a first impression on the part of the number-theoretic reader that the topics are not at the very heart of number theory, demanding desperately to be resolved, but rather hapless victims of a topological ”overkill”. Nevertheless, the case for the circumstantial relevance of the methods is likely to grow stronger as more investigations are made. Even now, a good case might be found in Hirzebruch’s work on the Hilbert modular function and related results in quadratic class number [see Enseign. Math., II. Sér. 19, 183–281 (1973; Zbl 0285.14007) and Lect. Notes Math. 412, 75–93 (1974; Zbl 0301.14010)].
Chapter III seems to be primarily topological calculations. The subheadings are: signature theorem on low-dimensional manifolds, action of \(T^{n+1}\) on \(\mathbb P_n(\mathbb C)\), Brieskorn manifolds, Browder-Livesay invariant of lens spaces.
In short, this book is recommended to all number-theorists as a description of an area of activity of which they can no longer afford to be uninformed.

MSC:
11-02 Research exposition (monographs, survey articles) pertaining to number theory
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58J20 Index theory and related fixed-point theorems on manifolds
11Fxx Discontinuous groups and automorphic forms
11F20 Dedekind eta function, Dedekind sums