By Gunnar Carlsson (auth.), Gunnar E. Carlsson, Ralph L. Cohen, Wu-Chung Hsiang, John D. S. Jones (eds.)

In 1989-90 the Mathematical Sciences study Institute performed a application on *Algebraic Topology and its **Applications*. the most parts of focus have been homotopy concept, K-theory, and purposes to geometric topology, gauge concept, and moduli areas. Workshops have been carried out in those 3 parts. This quantity contains invited, expository articles at the themes studied in this application. They describe contemporary advances and aspect to attainable new instructions. they need to end up to be priceless references for researchers in Algebraic Topology and comparable fields, in addition to to graduate students.

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In this section we will discuss these moduli spaces in the examples when lR,3 has both the Euclidean metric and X = lR, and when lR,3 has a hyperbolic metric and X = Sl. 26 RALPH L. COHEN AND R. JAMES MILGRAM By a theorem of Taubes [Tl] in both of these cases, the ambient manifold of based gauge equivalence classes of invariant connections is homotopy equivalent to the space of basepoint preserving, smooth self maps of the sphere S2, n2 S2. Via theorems of Atiyah and Donaldson, the relevant moduli spaces are given up to homeomorphism by the space of based holomorphic maps of the Riemann sphere to itself, or equivalently, the space of rational functions of one complex variable.

JAMES MILGRAM 24 where T(Ui) is the graded tensor algebra (over IQ) on a class Ui of dimension 2. Proof. 8. Since 1f'qSU(2) ~ 1f'qS3 is torsion for q > 3, we have that the map n3 SU(2)--MaPo(Mj BSU(2)) induces an isomorphism on 1f'o, and that each component of n 3SU(2) has the rational homotopy type of a point. Hence each component of MaPo(Mj BSU(2)) maps to I1g nSU(2) via a rational homotopy equivalence. The proposition follows since Bk ~ Map~(Mj BSU(2)). 5) is an isomorphism in this setting. We close this section by citing two recent results by [Ma, Mil on the cohomology groups H*(Bq) when the Lie group is SU(2).

Monk '----+ Mk 1· '----+ Holk(S2,nSU(2)) Holk(S2, S2) 1· in in s n 2k 2 4. E ---+ n3 INSTANTONS ON s3 • 84 We now return to the problem of studying the geometry of the pairs (Bk' Mk) where Bk is the space of based gauge equivalence classes of connections on the SU(2) bundle Pk -+ S4 of Chern class C2 = k, and Mk is the subspace of self dual connections ("instantons") with respect to the round metric on S4. Recall from section one that there is a homotopy equivalence Bk ~ n~s3. The first solutions of the self duality equations were given by 't Hooft [tH) , and were generalized in the work of Atiyah, Drinfeld, Hitchin and Manin, [ADHM), (see also Atiyah's Pisa lecture notes for a more detailed account), to give the following prescription for the space Mk.