Mielke, Thomas Martin (1995) Mapping spaces, configuration spaces and gauge theory. PhD thesis, University of Warwick.

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Abstract

The present thesis considers the space of connections modulo based gauge equivalence on a principal SU(2) bundle over a closed simply-connected smooth four-dimensional manifold M. Up to homotopy equivalence, this is the space of basepoint-preserving maps from M to BSU('2), the classifying space of SU(2). It depends only on the homotopy type of M which is characterized by the intersection form.

The Z/pZ-homology of the mapping space for p a prime not equal to 3 is computed and given in terms of the data associated to the intersection form. For the prime 3, partial results are obtained. The main method is to consider a fibration associated to a CW decomposition of M and to show that the corresponding Eilenberg- Moore spectral sequence collapses. These results generalize from manifolds to spaces homotopy equivalent to a bouquet of 2-spheres with a single 4-cell attached.

For the possible homotopy types the space of connections modulo gauge equivalence ran attain, a classification is obtained in the following sense. The homotopy type of this space is uniquely determined by the rank, type and signature modulo eight of the intersection form. On the other hand, the homotopy type determines the rank, type and signature modulo four of the intersection form. Both results together give a complete classification for the case of spin manifolds. The homotopy types of the spaces of connections modulo gauge equivalence over two spin manifolds agree if and only if the intersection forms are of the same rank. These results use a classification of unimodular bilinear forms over the ring Z/4Z.

In a final part, a map is constructed from the labelled configuration spaces of points in the manifold to the mapping space. This map is shown to be asymptotically surjective in homology with Z/2Z-coefficients. For homology with general coefficients, classes are constructed which are not approximated by this map.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Gauge fields (Physics), Homotopy theory, Configuration space
Official Date:	March 1995
Dates:	Date Event March 1995 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Jones, John
Sponsors:	Studienstiftung des Deutschen Volkes ; Commission of the European Communities ; University of Warwick
Format of File:	pdf
Extent:	viii, 116 leaves
Language:	eng

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