Atiyah-Singer index theorem. An introduction (Q357889)

The author provides a detailed and efficient discussion of the results in [\textit{M. F. Atiyah} and \textit{I. M. Singer}, Ann. Math. (2) 87, 484--530 (1968; Zbl 0164.24001); \textit{M. F. Atiyah} and \textit{G. B. Segal}, Ann. Math. (2) 87, 531--545 (1968; Zbl 0164.24201); \textit{M. F. Atiyah} and \textit{I. M. Singer}, Ann. Math. (2) 87, 546--604 (1968; Zbl 0164.24301)], including the proofs of these results, and the definitions, theorems, and proofs required to understand the papers referenced above. This book will be useful for a reader who has a general understanding of the material but wants the details collected in a single source. Such a reader will not be troubled by isolated expository errors, such as a statement of the Leray-Hirsch theorem in which a clause that should be part of the hypotheses is presented as part of the conclusion. The reviewer would not recommend the book to a reader unfamiliar with the subject matter. Such a reader would benefit from more commentary and examples than appear in this book. The brief historical remarks in the book's introduction completely overlook the role played by \textit{I. M. Gel'fand} [Russ. Math. Surv. 15, No. 3, 113--123 (1960); translation from Usp. Mat. Nauk 15, No. (93; Zbl 0095.07804)] in the early development of the index theorem. See, for example, [\textit{M. Atiyah}, Collected works. Volume 3: Index theory: 1. Volume 4: Index theory: 2. Oxford: Clarendon Press (1988; Zbl 0724.53001)] for acknowledgment of the role of Gel'fand's paper.