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 software on *Algebraic Topology and its **Applications*. the most parts of focus have been homotopy conception, K-theory, and functions to geometric topology, gauge idea, and moduli areas. Workshops have been performed in those 3 components. This quantity involves invited, expository articles at the issues studied in this software. They describe contemporary advances and aspect to attainable new instructions. they need to end up to be invaluable references for researchers in Algebraic Topology and similar fields, in addition to to graduate students.

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**Example text**

9 we have the following. 13. The limiting spaces Rat oo and K(f3oo, 1) are stably homotopy equivalent. The following is the main result of [CCMM]. 14. The spaces Ratk and K (f32k, 1) are stably homotopy equivalent. This theorem gives an effective description of the stable homotopy type of the rational function spaces Ratk and hence the monopole spaces Mk since quite a bit is known about the stable homotopy type of the braid groups. We remark that the unstable homotopy type of these rational functions has been studied in detail in [CS].

2. MONOPOLES, RATIONAL FUNCTIONS, AND BRAIDS In this section we consider connections on trivial principal bundles over four dimensional manifolds of the form M4 = lR,3 X X, where X is either the real line or the circle group. Viewing X as a one dimensional topological group, the corresponding space of connections is acted upon by X, and we will consider those connections which are invariant under this action. The resulting moduli spaces of self dual invariant connections are known as spaces of "monopoles".

Moreover since lR3 is contractible, the basepoint condition in go = Mapo(JR3, SU(2)) implies that go is a contractible topological group. Hence this bundle has contractible fibers. This implies the following. 3. The projection onto the orbit space A----+B = A/go is a homotopy equivalence. 3 we have the following theorem of Taubes proved in [Tl]. 4. There is a natural homotopy equivalence B ~ 0 28 2 • In view of this result the path components of B (and of A) are labelled by the integers Z corresponding to the degree of the map in 0 2 8 2 .