This paper is a greatly expanded account of the preliminary
material that appeared in Atiyah & Bott (1980). Part ofthe
reason for the long delay between that paper and this present
version was that new view-points emerged that added further
interest to the topic and enabled us to provide proofs for what had
earlier been conjectures. The length of this paper is due to our
desire to present the several different aspects of the problem. We
feel that this is justified because the main interest lies not so
much in the actual applications as in the methods employed and the
interaction between different approaches. Because ofits long
gestation period and its consequent size, we feel that we owe the
reader a substantial and leisurely introduction that puts the paper
into historical perspective. In fact, on a personal level, one of
the attractions of this subject to us is that it brings together
algebraic vector bundles and Morse theory, topics to which we
separately made contributions a quarter of a century ago (Atiyah
1955, 1957; Bott 1958, Bott &Samelson 1958). Even better, the
catalyst that produced this interaction came from a quite
unexpected quarter, namely that of theoretical physics in the form
of the Yang-Mills equations.
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