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 $C^*$-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.

我们分析了紧凑四流形的通用覆盖空间上的单极子映射。我们在 Atiyah-Hitchin-Singer (AHS) 复合体的封闭条件下引入了覆盖单极子映射的局部性质。特别是，当线性化方程是同构时，我们构造了更高阶的覆盖单极子映射。这导致了群 $C^*$-algebra 的 $K$-群之间的同态。我们对与 $L^p$ 上同调相关的覆盖空间应用非线性分析。我们还获得了对覆盖空间的各种 Sobolev 估计。

通过将 Singer 猜想应用于 $L^2$ 上同调，我们提出了 $\frac{10}{8}$-不等式的非球面版本的猜想。这对于一大类四流形来说是满足的，包括一些一般类型的复杂表面。