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

ISBN-10: 1461395267

ISBN-13: 9781461395263

ISBN-10: 1461395283

ISBN-13: 9781461395287

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 functions 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 fresh advances and element to attainable new instructions. they need to end up to be beneficial references for researchers in Algebraic Topology and similar fields, in addition to to graduate students.

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**Example text**

The advantage of this replacement is that the fibers decompose e and we can study the original bundle E by restricting to the fiber Cpl 'so For Cpl the classification of holomorphic vector bundles is completely understood. Such bundles have no deformation space, and the classification is given by the Birkhoff-Grothendieck theorem [PS]. 3. Let ct -+ E -+Cpl be a holomorphic vector bundle. Then E is holomorphically isomorphic to a direct sum of line bundles efl EB· .. EBef t , 0:1 ~ 0:2 ~ ... ~ O:t, where is the Iml-fold tensor product of the Hopf bundle for m ~ 0 and of the conjugate Hopf bundle 1 for m < O.

We will give a proof shortly. But for now, consider the Serre spectral sequence of the fibration HOMOTOPY OF GAUGE THEORETIC MODULI SPACES 49 The d 4 -differential on [80(3)J* = h4' and so for>. , and, since Mk is finite dimensional, hV = 0 for some finite t. If h~-l >. =I- 0 then [80(3)J*h~-1 >. is an infinite cycle, and we see that, when we write H*(Mk;Q) as a Q[h4 J-module, then the Q[hrJ-indecomposables survive to H*(Mk;Q), and for each such indecomposable, there is also a second class as described above.

The most complete of these describe the spaces of projective equivalence classes of non-singular Toeplitz matrices. 1. The space of all projective equivalence classes of nonsingular n x n-Toeplitz matrices, Tn,o, is homeomorphic to the orbit space under the action of G L2 (q on the space of pairs (Pl (z), P2 (z)) of coprime polynomials with max(deg(Pl(z)), deg(p2(z))) = n where the action is given as Remark. The set of projective equivalence classes of such coprime pairs, HOMOTOPY OF GAUGE THEORETIC MODULI SPACES 47 with the degree condition can be identified with the space of unbased holomorphic maps ¢: C1P1---+C1P1 of degree n which we denote Rat n (ClP1), and the quotient above is the same as the quotient of the projective equivalence classes of these pairs by the quotient group P8L 2 (C) (acting by composition with linear fractional transformations).

### Algebraic Topology and Its Applications by Gunnar Carlsson (auth.), Gunnar E. Carlsson, Ralph L. Cohen, Wu-Chung Hsiang, John D. S. Jones (eds.)

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