Covering monopole map and higher degree in non commutative geometry

Abstract

In this paper we develop analysis of the monopole maps over the universal covering space of a compact four manifold. We induce a property on local properness of the covering monopole map under the condition of closeness of the AHS complex. In particular we construct a higher degree of the covering monopole map when the linearized equation is isomorphic, which induces a homomorphism between K group of the group C^* algebras. It involves non linear analysis on the covering space, which is related to L^p cohomology. We also obtain various Sobolev estimates on the covering spaces. As a possible application, we propose an aspherical version of 10/8 inequality, combining with Singer conjecture on L^2 cohomology. It is satisfied for a large class of four manifolds which includes some complex surfaces of general type.

replacement from the previous version by the technical reason to remove irrelevant characters

References in corpus (1)

• Lp-cohomology of symmetric spaces

Cited by in corpus (4)

• An infinite-dimensional index theorem and the Higson-Kasparov-Trout algebra
• $L^2$ harmonic theory, Seiberg-Witten theory and asymptotics of differential forms
• Higher Nahm transform in non commutative geometry
• Induced map on K theory for certain Γ-equivariant maps between Hilbert spaces