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

In 1989-90 the Mathematical Sciences learn Institute carried out a application on Algebraic Topology and its Applications. the most components of focus have been homotopy concept, K-theory, and purposes to geometric topology, gauge conception, and moduli areas. Workshops have been carried out in those 3 parts. This quantity comprises invited, expository articles at the subject matters studied in this software. They describe contemporary advances and element to attainable new instructions. they need to turn out to be priceless references for researchers in Algebraic Topology and similar fields, in addition to to graduate students.

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Next consider the varieties Tn,k with k > O. It turns out that Tn,k is a singular variety with singularity set exactly Tn,Hl. ,k is to use Poincare duality and study instead the pairs (Tn,k, T n ,k+1). A spectral sequence is constructed which converges to H*(Tn,k; IF) for any field and the El-term and d1-differential are completely determined. The higher differentials are not determined in general but the results are sufficient to determine the rational cohomology groups. 2. The rational cohomology groups of the projective Toeplitz varieties are given as follows, H*(Tn,k; Q) ~ H*(8 2 ; Q), H*(Tn,o; Q) ~ k> 1 H*(pt; Q).

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. 166, 163-221. L. K. B. Thomas, editors 150 (1990), Lond. Math. Soc. Lect. Notes. L. S Jones, Monopoles, Braid Groups, and the Dirac operator" to appear. L. Cohen and D. Shimamoto, Rational junctions, labelled configurations, and Hilbert schemes, Jour. of Lond. Math. Soc.

We now have the following diagram. 11, was proved in [CCMM] using stable splitting theory in loop spaces, as well as homological calculations. 15. For suitably large N there is a map of suspension spaces so that the composition is a homotopy equivalence. 3. 8 1 - INVARIANT INSTANTONS ON 84 AND HYPERBOLIC MONOPOLES As we saw in section one, the minima of the Yang - Mills functional on a four dimensional manifold satisfy the self duality equations If we consider this equation for connections on the trivial 8U(2) - bundle over ]R4 and require that these connections have finite Yang - Mills energy YM(A) = JR,iFAI 2 < 00, then we are essentially considering connections that have a (unique) extension to a self dual connection on a principal 8U(2) - bundle over the sphere 8 4 • We will denote by Mn = Mn(8 4 ) the moduli space of gauge equivalence classes of anti - self dual connections on the principal 8U(2) - bundle classified by its second Chern class C2 = n.

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