The asphericity of locally finite infinite configuration spaces and Weierstrass entire coverings

Let $Con{f}_{\mathrm{\infty }}^{lf}(\text{C})$ and $C_{\mathrm{\infty }}^{lf}(\text{C})$ denote the locally finite infinite ordered and unordered configuration spaces of the complex plane. We prove that both $Con{f}_{\mathrm{\infty }}^{lf}(\text{C})$ and $C_{\mathrm{\infty }}^{lf}(\text{C})$ are aspherical. We further obtain a locally finite analogue of the braid exact sequence, $$1⟶H^{lf}(\mathrm{\infty })⟶B^{lf}(\mathrm{\infty })⟶\text{Aut}(\mathbb{N})⟶1,$$ where $H^{lf}(\mathrm{\infty })={\pi }_1(Con{f}_{\mathrm{\infty }}^{lf}(\text{C}))$ and $B^{lf}(\mathrm{\infty })={\pi }_1(Con{f}_{\mathrm{\infty }}^{lf}(\text{C})\mathrm{/}\mathrm{/}\text{Aut}(\mathbb{N}))$, the fundamental group of the homotopy quotient of $Con{f}_{\mathrm{\infty }}^{lf}(\text{C})$ by $\text{Aut}(\mathbb{N})$. Building on this, we classify connected countably infinite–sheeted covering spaces and give a criterion for when such a covering can be realized from the zero set of a family of entire functions $F:X\times \text{C}\to \text{C}$. In particular, if ${\pi }_1(X)$ is free and $H^2(X;\mathbb{Z})=0$, then every countably infinite–sheeted covering space over $X$ is realizable.