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Tesi etd-06282016-121015

Thesis type
Tesi di laurea magistrale
Trisections of 4-manifolds
Corso di studi
relatore Lisca, Paolo
Parole chiave
  • low dimensional topology
  • trisections
  • 4-manifolds
Data inizio appello
Riassunto analitico
A common theme in topology is to reduce hard problems by decomposing<br>a manifold into simpler and standard pieces. Classical examples<br>in this direction include handle decompositions, Heegaard splittings of<br>3-manifolds, surgery theory and so on.<br>This thesis will present the concept of trisections for smooth, closed<br>and oriented 4-manifolds. Trisections were recently defined by David<br>T. Gay and Robion Kirby. A trisection is a decomposition of a<br>4-manifold X into three pieces X1, X2 and X3, each diffeomorphic to<br>a 4-dimensional handlebody; the triple intersection X1∩X2∩X3 is a closed<br>surface Σ, and each pair of double intersections (Xi∩Xj, Xk∩Xj ) provides<br>a Heegaard splitting for the boundary ∂Xj with Heegaard surface Σ.<br>Trisections can be thought of as a 4-dimensional analogue of Heegaard<br>splittings. As in the case of 3-manifolds, we are decomposing<br>into standard pieces, and we can recover the underlying manifold from<br>combinatorial data associated to the decomposition. In the original<br>definition the three 4-dimensional handlebodies are meant to be of the<br>same genus. However, in this thesis we are going to work in the slightly<br>general context of unbalanced trisections.<br>The authors proved that every smooth, closed and oriented<br>4-manifolds admits a trisection. Trisections are far from being unique;<br>however, two trisections representing the same 4-manifold X are related<br>by simple stabilization moves. We will show that uniqueness of a<br>trisection is guaranteed up to stabilizations.<br>The proofs of existence and uniqueness rely on results concerning<br>Morse 2-functions. These arise as a generalization of homotopies between<br>regular Morse functions. In other words, a Morse 2-function is<br>a generic map from a smooth n-manifold X to a 2-manifold, satisfying<br>some additional constraints. All the relevant tools, mostly hinging<br>upon Cerf theory, were developed by Gay and Kirby.<br><br>Between 2015 and 2016, trisections have been a very active topic<br>of research. Several authors managed to prove partial classification<br>results, to extend to the non-empty boundary case and<br>a group-theoretic approach to trisections was obtained in a recent paper. We will<br>give a complete classification of trisected 4-manifolds up to genus 2<br>(Theorem 4.4).<br>We will also present some developments concerning the compact<br>case, that is currently work in progress by Juanita Pinzon Caicedo and<br>Nick Castro. Lastly we will talk about the group-theoretic reformulation<br>due to Gay and Kirby.