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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Classifying spaces and Dirac operators coupled to instantons


Author: Marc Sanders
Journal: Trans. Amer. Math. Soc. 347 (1995), 4037-4072
MSC: Primary 58D27; Secondary 55P99, 55R45, 57R57, 58G03
MathSciNet review: 1311915
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Abstract: Let $ M(k,SU(l))$ denote the moduli space of based gauge equivalence classes of $ SU(l)$ instantons on principal bundles over $ {S^4}$ with second Chern class equal to $ k$. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces as $ l$ increases relative to $ k$. This "coupling" procedure produces maps $ {\partial _u}:M(k,SU(l)) \to BU(k)$, and we prove that in the limit over $ l$ such maps recover Kirwan's $ [$K$ ]$ homotopy equivalence $ M(k,SU) \simeq BU(k)$. We also compute, for any $ k$ and $ l$, the image of the homology map $ {({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z)$. Finally, we prove all the analogous results for $ Sp(l)$ instantons.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1311915-7
PII: S 0002-9947(1995)1311915-7
Keywords: Instantons, Dirac operators, classifying space
Article copyright: © Copyright 1995 American Mathematical Society