Kjuchukova, Alexandra
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Publication On the Classification of Irregular Dihedral Branched Covers of Four-Manifolds(2015-01-01) Kjuchukova, AlexandraWe prove a necessary condition for a four-manifold $Y$ to be homeomorphic to a $p$-fold irregular dihedral branched cover of a given four-manifold $X$, with a fixed branching set $B$. The branching sets considered are closed oriented surfaces embedded locally flatly in $X$ except at one point with a specified cone singularity. The necessary condition obtained is on the rank and signature of the intersection form of $Y$ and is given in terms of the rank and signature of the intersection form of $X$, the self-intersection number of $B$ in $X$ and classical-type invariants of the singularity. Secondly, we show that, for an infinite class of singularities, the necessary condition is sharp. That is, if the singularity is a two-bridge slice knot, every pair of values of the rank and signature of the intersection form which the necessary condition allows is in fact realized by a manifold dihedral cover. In a slightly more general take on this problem, for an infinite class of simply-connected four-manifolds $X$ and any odd square-free integer $p>1$, we give two constructions of infinite families of $p$-fold irregular branched covers of $X$. The first construction produces simply-connected manifolds as the covering spaces, while the second produces simply-connected stratified spaces with one singular stratum. The branching sets in the first of these constructions have two singularities of the same type. In the second construction, there is one singularity, whose type is the connected sum of a knot with itself.