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On the Classification of Irregular Dihedral Branched Covers of Four-Manifolds University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2015 On the Classification of Irregular Dihedral Branched Covers of Four-Manifolds Alexandra Kjuchukova University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons Recommended Citation Kjuchukova, Alexandra, "On the Classification of Irregular Dihedral Branched Covers of Four-Manifolds" (2015). Publicly Accessible Penn Dissertations. 1817. https://repository.upenn.edu/edissertations/1817 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1817 For more information, please contact [email protected]. On the Classification of Irregular Dihedral Branched Covers of Four-Manifolds Abstract We 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. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Julius Shaneson Keywords branched covers, classification, four-manifolds, knot theory, singularity, topology Subject Categories Mathematics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/1817 ON THE CLASSIFICATION OF IRREGULAR DIHEDRAL BRANCHED COVERS OF FOUR-MANIFOLDS Alexandra Kjuchukova A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2015 Julius Shaneson, Class of 1939 Professor of Mathematics Supervisor of Dissertation David Harbater, Professor of Mathematics Graduate Group Chairperson Dissertation Committee: Julius Shaneson, Class of 1939 Professor of Mathematics Tony Pantev, Professor of Mathematics Antonella Grassi, Professor of Mathematics | deep in whose reeds great elephants decay, I, an island, sail, and my shoes toss | Ted Berrigan ii for Julius Shaneson iii ABSTRACT ON THE CLASSIFICATION OF IRREGULAR DIHEDRAL BRANCHED COVERS OF FOUR-MANIFOLDS Alexandra Kjuchukova Julius Shaneson I 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 embed- ded 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, I 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, I 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 iv same type. In the second construction, there is one singularity, whose type is the connected sum of a knot with itself. v Contents 1 Introduction 1 2 Background 5 2.1 Basic definitions . .5 2.2 Knot Theory and preliminary results . .8 2.3 Brief note on notation . 16 3 Irregular Dihedral Branched Covers of Four-Manifolds 17 3.1 Necessary condition for the existence of a dihedral cover . 17 3.2 Sufficient condition in the case of two-bridge slice singularities . 38 3.3 Construction for other singularity types . 47 3.4 A family of knot invariants . 56 4 Further Questions 62 Appendix 64 Bibliography 67 vi Chapter 1 Introduction The study of branched covers of Sn dates back to the 1920s when Alexander dis- covered their astonishing generality: he proved that every closed orientable PL n-manifold is a PL branched cover of Sn [1]. Since this seminal work, the classifica- tion of branched covers has been an active area of research { see, for example, [2], [12], [22]. A typical question of interest is to find the minimum number of sheets, or the least complex, according to some criterion, branching set needed to realize all man- ifolds in a given dimension as covers of the sphere. So far, the answers are known only in dimension three: by a well-known result ([10], [15]), every closed oriented three-manifold is a three-fold irregular cover of S3 branched over a knot. Intuitively, a knot in S3 is the \simplest possible" branching set over a three- dimensional base (by opposition to, say, a link or a self-intersecting curve). There- 1 fore, we can restate the result about covers of S3 in the following way: allowing a complicated branching set does not enrich the family of branched covers of the three-sphere. The situation in dimension four is considerably more subtle. A four- manifold Y which can be realized as a branched cover of S4 with branching set a locally flat oriented submanifold must have signature equal to zero (see [23]). On the other hand, by a result of Montesions [16], every closed oriented PL four- manifold is a four-fold cover of S4 branched over an immersed PL surface. The middle ground between these two results remains poorly understood. For example, what manifolds can be realized as covers of S4 branched over an embedded, but not necessarily locally flat, oriented surface of a given genus? More specifically, what singularities does the branching set need to have in order to realize a given manifold Y as a cover of S4? These are among the motivating questions of this thesis. However, the work presented here is not restricted to covers of the sphere. Given any simply-connected closed oriented topological four-manifold X, we ask: which closed oriented topological four-manifolds Y are homeomorphic to branched covers of X, and with what singularities on the branching set? In Chapter 3 we prove a necessary condition for a four-manifold Y to be homeomorphic to a p-fold irregular dihedral branched cover of the pair (X; B), where X is a closed oriented topolog- ical four-manifold, and the branching set B ⊂ X is an oriented surface embedded locally flatly in X except at finitely many points with specified singularity types. In a subsequent section, I show that, under some additional assumptions on the 2 singularity type, the necessary condition is sufficient as well. The sufficient condition just described is obtained via a procedure for construct- ing an irregular branched cover with a specified slice knot singularity. Namely, fix an odd square-free integer p, a closed oriented four-manifold X and a surface B embedded locally flatly in X. The first step is to embed in X a surface B1, homeomorphic to B, so that the embedding B1 ⊂ X admits a slice singularity. The second step is to construct an irregular p-fold cover of X branched over B1. Third, I give a procedure for obtaining an infinite family of covers by modifying the branching set away from the singularity. If, in addition, X is simply-connected and π1(X − B; x0) = Z=2Z, the covers obtained by this method are simply-connected manifolds. A consequence of this construction is that any slice knot which itself admits an irregular p-fold dihedral cover can be realized as the unique singularity of a branched cover between four-manifolds. Whether the same is true for all knots remains an interesting open question. In Section 3.3, I describe two more procedures for constructing infinite families of irregular p-fold branched covers over singular branching sets. These procedures admit a more general type of singularities but work over a more restricted set of four-manifolds X. The first procedure yields a manifold cover; the branching set has two components. One of the components is locally flat, and the other has two singularities of the same type, α, which is not necessarily slice. The second procedure produces a stratified space as a cover, and the branching set has two 3 components: one is again locally flat, and the other has one singularity of composite type, α#α.
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