Covering-monopole map and higher degree in non-commutative geometry
Tsuyoshi Kato
Kyoto University, Japan
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Abstract
We analyze the monopole map over the universal covering space of a compact four-manifold. We induce the property of local properness of the covering-monopole map under the condition of closedness of the Atiyah–Hitchin–Singer (AHS) complex. In particular, we construct a higher degree of the covering-monopole map when the linearized equation is isomorphic. This induces a homomorphism between the -groups of the group -algebra. We apply a non-linear analysis on the covering space, which is related to cohomology. We also obtain various Sobolev estimates on the covering spaces.
By applying the Singer conjecture on cohomology, we propose a conjecture of an aspherical version of the -inequality. This is satisfied for a large class of four-manifolds, including some complex surfaces of general type.
Cite this article
Tsuyoshi Kato, Covering-monopole map and higher degree in non-commutative geometry. J. Noncommut. Geom. 15 (2021), no. 3, pp. 995–1071
DOI 10.4171/JNCG/430