  Covering spaces of graphs have long been useful for studying expanders (as
"graph lifts") and unique games (as the "label-extended graph"). In this paper
we advocate for the thesis that there is a much deeper relationship between
computational topology and the Unique Games Conjecture. Our starting point is
Linial's 2005 observation that the only known problems whose inapproximability
is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are
instances of Maximum Section of a Covering Space on graphs. We then observe
that the reduction between these two problems (Khot-Kindler-Mossel-O'Donnell,
FOCS 2004; SICOMP, 2007) gives a well-defined map of covering spaces. We
further prove that inapproximability for Maximum Section of a Covering Space on
(cell decompositions of) closed 2-manifolds is also equivalent to the Unique
Games Conjecture. This gives the first new "Unique Games-complete" problem in
over a decade.
  Our results partially settle an open question of Chen and Freedman (SODA
2010; Disc. Comput. Geom., 2011) from computational topology, by showing that
their question is almost equivalent to the Unique Games Conjecture. (The main
difference is that they ask for inapproximability over
$\mathbb{Z}/2\mathbb{Z}$, and we show Unique Games-completeness over
$\mathbb{Z}/k\mathbb{Z}$ for large $k$.) This equivalence comes from the fact
that when the structure group $G$ of the covering space is Abelian - or more
generally for principal $G$-bundles - Maximum Section of a $G$-Covering Space
is the same as the well-studied problem of 1-Homology Localization.
  Although our most technically demanding result is an application of Unique
Games to computational topology, we hope that our observations on the
topological nature of the Unique Games Conjecture will lead to applications of
algebraic topology to the Unique Games Conjecture in the future.
