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 carried out a application on *Algebraic Topology and its **Applications*. the most parts of focus have been homotopy concept, K-theory, and functions to geometric topology, gauge conception, and moduli areas. Workshops have been carried out in those 3 parts. This quantity contains invited, expository articles at the subject matters studied in this application. They describe fresh advances and aspect to attainable new instructions. they need to end up to be valuable references for researchers in Algebraic Topology and similar fields, in addition to to graduate students.

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In this formula {3 : lR--+[O, 1] is a smooth cut - off function which is identically 0 if t ~ 1/2 and identically 1 if t ~ 3/4, and [,] is the Lie bracket in su2. (See [Tl] for details). The following was proved in [Tl]. 1. The map I: (l2S2--+A is a homotopy equivalence. We remark that the homotopy inverse to the map I is given by sending a pair (A, fjJ) to the restriction of the Higgs field fjJ to the "sphere at 00" which is to say the map ({>: S2--+S2 given by ((>(x) = lim fjJ(tx). 2) t-+oo Since the bundle over lR3 is trivial the based gauge group of bundle automorphisms in this setting is given by go = Mapo(lR 3,SU(2)) where here MaPo means smooth maps 9 : lR3--+SU(2) which satisfy the basepoint condition lim g(t, 0, 0) = 1 E SU(2).

Milgram, The topology of the space of rational maps into generaized flag manifolds, Preprint, U. of New Mexico. P. M. Mann, Homology operations on instantons, J. Diff. Geom. 28, 423-465. P. M. Mann, Monopoles, non-linear (1 models and two fold loop spaces, Comm. Math. Phys. 115, 571-594. P. M. Mann, The hyperkiihler geometry of the ADHM construction and quaternionic geometric invariant theory, Proc. , Diff. Geom. (to appear). R. L. M. J. Milgram, The topology of rational junctons and divisors of surfaces, Acta Math.

Donaldson, Nahm's equations and the classification of monopoles, Comm. Math. Phys. 100 (1984), 387-407. K. Donaldson, Anti-self-dual Yang Mills connections over complez algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50, 1-26. [DK] S. Donaldson and P. Kronheimer, The Geometry of Four- Manifolds, Oxford Univ. Press, 1990. [FU] D. Freed and K. Uhlenbeck, Instantons and four- manifolds, MSRI Publications, vol. 1, Springer Verlag, 1984. [F] M. K. B. Thomas, editors 150 (1990), Lond.