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Bott periodicity and the index of elliptic operators. (English) Zbl 0159.53501
This paper gives proofs of the Bott periodicity theorems (on the homotopy of the classical groups) using the index of elliptic operators. In view of the other connections between \(K\)-theory and the index developed by I. M. Singer and the author [Ann. Math. (2) 87, 484–530 (1968; Zbl 0164.24001)] there is a strong case for regarding the proofs presented here as being basic. The main ideas are best explained in the complex case. One has the natural “Bott map” \(\beta: K(X)\to K(R^2\times X)\) which one wants to prove is an isomorphism. The most fundamental and natural way to do this is to exhibit an explicit inverse \(\alpha\). Now by using an elliptic operator on \(S^2\) (essentially the \(\deltabar\) operator) one obtains a homomorphism \(K(S^2\times X)\to K(X)\) and \(\alpha\) is then obtained by composing this with the natural map \(K(R^2\times X)\to K(S^2\times X)\). This method of proof extends quite directly to all the various generalizations of Bott periodicity, notably to the equivariant case. These cases are dealt with in the paper, and the final sections examine the connection between the proofs presented here and the “elementary” proof given by R. Bott and the author [Acta Math. 112, 229–247 (1964; Zbl 0131.38201)].
Reviewer: Michael F. Atiyah

MSC:
55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity
58J20 Index theory and related fixed-point theorems on manifolds
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