Covering-monopole map and higher degree in non-commutative geometry

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 $K$-groups of the group $C^{\ast }$-algebra. We apply a non-linear analysis on the covering space, which is related to $L^p$ cohomology. We also obtain various Sobolev estimates on the covering spaces.

By applying the Singer conjecture on $L^2$ cohomology, we propose a conjecture of an aspherical version of the $\frac{10}{8}$-inequality. This is satisfied for a large class of four-manifolds, including some complex surfaces of general type.