On branched covering representation of 4-manifolds

TL;DR

This paper introduces new branched covering representations for bounded and non-compact 4-manifolds, extending known results for closed 4-manifolds, with specific conditions on degree and branching sets.

Contribution

It provides novel branched covering constructions for non-compact 4-manifolds and extends the concept to topological manifolds, including a universal 4-fold cover of any closed 4-manifold.

Findings

Existence of simple branched covers for bounded 4-manifolds with controlled degree and branching set

Existence of simple branched covers for open 4-manifolds with controlled degree and branching set

Any closed oriented topological 4-manifold is a 4-fold branched cover of S^4

Abstract

We provide new branched covering representations for bounded and/or non-compact 4-manifolds, which extend the known ones for closed 4-manifolds. Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1) if $M$ is compact with (possibly empty) boundary, there exists a simple branched cover $p:M \to S^4 - \mathop{\mathrm{Int}}(B^4_1 \cup \dots \cup B^4_n)$, where the $B^4_i$'s are disjoint PL 4-balls, $n \geq 0$ is the number of boundary components of $M$; (2) if $M$ is open, there exists a simple branched cover $p : M \to S^4 - \mathop{\mathrm{End}} M$, where $\mathop{\mathrm{End}} M$ is the end space of $M$ tamely embedded in $S^4$. In both cases, the degree $d(p)$ and the branching set $B_p$ of $p$ can be assumed to satisfy one of these conditions: (1) $d(p)=4$ and $B_p$ is a properly self-transversally immersed locally flat PL surface; (2) $d(p)=5$ and $B_p$ is a properly embedded locally flat PL surface. In the compact (resp. open) case, by relaxing the assumption on the degree we can have $B^4$ (resp. $R^4$) as the base of the covering. We also define the notion of branched covering between topological manifolds, which extends the usual one in the PL category. In this setting, as an interesting consequence of the above results, we prove that any closed oriented topological 4-manifold is a 4-fold branched covering of $S^4$. According to almost-smoothability of 4-manifolds, this branched cover could be wild at a single point.