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

Abstract

Let $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ denote the locally finite infinite ordered and unordered configuration spaces of the complex plane. We prove that both $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ are aspherical. We further obtain a locally finite analogue of the braid exact sequence, \[ 1\longrightarrow H^{lf}(\infty)\longrightarrow B^{lf}(\infty)\longrightarrow \Aut(\N)\longrightarrow 1, \] where $H^{lf}(\infty)=\pi_1(Conf^{lf}_{\infty}(\C))$ and $B^{lf}(\infty)=\pi_1(Conf^{lf}_{\infty}(\C)//\Aut(\N))$, the fundamental group of the homotopy quotient of $Conf^{lf}_{\infty}(\C)$ by $\Aut(\N)$. 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 $F:X\times\C\to\C$. In particular, if $\pi_1(X)$ is free and $H^2(X;\Z)=0$, then every countably infinite--sheeted covering space over $X$ is realizable.