Models of simply-connected trivalent 2-dimensional stratifolds
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ORIGINAL ARTICLE
Models of simply-connected trivalent 2-dimensional stratifolds J. C. Go´mez-Larran˜aga1 • F. Gonza´lez-Acun˜a2 • Wolfgang Heil3 Received: 7 March 2019 / Accepted: 9 March 2020 Ó Sociedad Matemática Mexicana 2020
Abstract Trivalent 2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where three sheets meet. We develop operations on their associated labeled graphs that will effectively construct from a single vertex all graphs that represent simply connected trivalent 2-stratifolds. Keywords Stratifold Simply connected Trivalent graph
Mathematics Subject Classification 57M20 57M05 57M15
1 Introduction In topological data analysis one studies high dimensional data sets by extracting shapes. Many of these shapes are 2-dimensional simplicial complexes where it is computationally possible to calculate topological invariants such as homology groups (see for example [14]). 2-complexes that are amenable for more detailed analysis are 2-dimensional stratified spaces, which occur in the study of persistent homology of high-dimensional data [2, 3]. Special classes of these stratified spaces are foams, which occur as special spines of 3-dimensional manifolds [15, 17]. Khovanov [13] used trivalent foams to construct a bigraded homology theory whose & Wolfgang Heil [email protected] J. C. Go´mez-Larran˜aga [email protected] F. Gonza´lez-Acun˜a [email protected] 1
Centro de Investigacio´n en Matema´ticas, A.P. 402, 36000 Guanajuato, GTO, Mexico
2
Instituto de Matema´ticas, UNAM, 62210 Cuernavaca, Morelos, Mexico
3
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
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Euler characteristic is a quantum sl(3) link invariant and Carter [4] presented an analog of the Reidemeister-type moves for knotted foams in 4-space. One would like to obtain a classification of 2-dimensional stratified spaces in terms of algebraic invariants, but this class is too large, even when restricted to foams. However, there is a smaller class of 2-dimensional stratified spaces, namely those with empty 0stratum, called 2-stratifolds, for which such a classification is feasible. A closed 2stratifold is a 2-dimensional cell complex X that contains a collection of finitely many simple closed curves, the components of the 1-skeleton X ð1Þ of X, such that X X ð1Þ is a 2-manifold and a neighborhood of each component C of X ð1Þ consists of n 3 sheets (a precise definition is given in Sect. 2). In particular, if the number of sheets is three, then X is called trivalent. Such 2-stratifolds also arise in the study of categorical invariants of 3-manifolds. For example if G is a non-empty family of groups that is closed under subgroups, one would like to determine which (closed) 3-manifolds have G-category equal to 3. In [5] it is shown that such manifolds have a decomposition into three compact 3-submanifolds H1 ; H2 ; H3 , where the intersection of Hi \ Hj (for i 6¼ j) is a compact 2-manifold, and each Hi is Gcontractible (i.e.
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