Cataloguing PL 4-Manifolds by Gem-Complexity

Abstract

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices).Possible interactions with the (not completely known) relationship among the different classifications in the TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.

Topology in colored tensor models via crystallization theory

“…Again, by using known simple and semi-simple crystallizations of the involved manifolds and performing graph connected sums, we obtain the required example. [12] or, equivalently, Theorem 4.6 of the survey paper [14]. Finally, in order to prove the last statement, note that D G (M 4 ) ≤ 59 implies k(M 4 )+β 2 (M 4 ) ≤ 9.…”

“…non-orientable) d-pseudomanifold: the vertices of the graph represent the d-simplices of P and the colored edges of the graph describe the pairwise gluing in P of the (d−1)-faces of its maximal simplices (the graph thus becomes the dual 1-skeleton of P ). In this framework, many results have been achieved during the last 40 years; noteworthy are the classification results obtained in dimensions 3 and 4 with respect to the PL-manifold invariants regular genus and gem-complexity, specifically introduced and investigated in GEM theory with geometric topology aims (see for example [10] for the 3-dimensional case, [12] and [14] for the 4-dimensional one). In the present paper we show that the G-degree, which arises with physics motivations, can be linked with both these invariants: thanks to known results about them, new ideas are obtained about the meaning of the G-degree.…”

“…The thesis now easily follows by making use of the up-to-date results about topological classification of simply connected PL 4-manifolds (see [23] and [29]), exactly as in the proof of [12,Proposition 23] With regard to the PL classification of PL 4-manifolds, the following statements collect some classifying results involving the G-degree: in particular, Proposition 33 (resp. Proposition 35) provides the complete list of all orientable (resp.…”

The Tensor Track: An Update

“…The emphasis in the topological as well as in the quantum gravity community has been on dimension/rank 3 and 4. But while the topological community, chiefly interested in classifying and encoding such manifolds [29][30][31], has focused on reduction moves allowing to find their simplest colored triangulation, the quantum gravity community, chiefly interested in summing triangulations pondered by the Einstein-Hilbert action, has focused on an almost inverse process, namely finding infinite families of leading triangulations for this action. It happens that the most important family of this type, the melonic parallel/series family [17], in fact reduces through simple moves to the unique bipartite d-regular graph with two vertices corresponding to the simplest triangulation of the trivial spheric topology.…”

“…Nevertheless classifying and summing are subtly related issues, and we can expect progress from dialogue between the two communities, even if the typical integers of interest to topologists (regular genus, gem complexity [31]), are different from those of interest to the quantum gravity community, such as the Gurau degree which governs the standard tensor 1/N expansion [32][33][34]. The latter indeed include metric properties of the underlying triangulation in addition to its topological properties.…”