CATALOGUES OF PL 4-MANIFOLDS
In dimension 4, all orientable (resp. non-orientable) closed connected PL 4-manifolds admitting a coloured triangulation with at most 2p 4-simplices are obtained by means of catalogue C^{2p} _{(4) } (resp. ^{~}C^{2p} _{(4) } ) of rigid dipole-free bipartite (resp. non-bipartite) 4-dimensional crystallizations of up to 2p vertices.
The generating 4 -dimensional algorithm is described in [M.R. Casali - P. Cristofori, Cataloguing PL 4-manifolds by gem-complexity, The Electronic Journal of Combinatorics 22(4) (2015), #P4.25]. It has been implemented in a C++ program, whose output data (up to 2p=20) are presented in the following table, according to the number of vertices.
In [M.R. Casali - P. Cristofori, Cataloguing PL 4-manifolds by gem-complexity, The Electronic Journal of Combinatorics 22(4) (2015), #P4.25], an n-dimensional classifying algorithm is also described: it automatically subdivides a given set of PL n-manifolds (via coloured triangulationsor, equivalently, via crystallizations) in classes whose elements are PL-homeomorphic.
The algorithm, implemented in the case n=4 (program Γ 4-class), succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds which admit a coloured triangulation with at most 18 4-simplices.
The obtained 4-dimensional classification results are collected in the following theorem, by making use of a suitable graph-defined PL- invariant, called gem-complexity:
a PL 4-manifold is said to have gem-complexity k(M^{4})=p-1 iff 2p is the minimum order of a crystallization of M^{4}.
THEOREM
Let M^{4 }be a closed connected PL 4-manifold. Then:
- k (M^{4})=0 iff M^{4} is PL-homeomorphic to S^{4};
- k (M^{4})=3 iff M^{4 }is PL-homeomorphic to CP^{2};
- k (M^{4})=4 iff M^{4} is PL-homeomorphic to either S^{1 }x S^{3} or S^{1 } ~ x S^{3};
- k (M^{4})=6 iff M^{4 }is PL-homeomorphic to either S^{2 }x S^{2} or CP^{2 }#CP^{2 }or CP^{2 }# (-CP^{2});
- k (M^{4})=7 iff M^{4 }is PL-homeomorphic to either RP^{4} or CP^{2 }# (S^{1 }x S^{3}) or CP^{2 }# (S^{1 } ~ x S^{3});
- k (M^{4})=8 iff M^{4 }is PL-homeomorphic to either #_{2}(S^{1 }x S^{3}) or #_{2 }(S^{1 } ~ x S^{3}).
Moreover :
- no PL 4-manifold M^{4 }exists with k(M^{4}) =1,2,5;
- no exotic PL 4-manifold exists, with k(M^{4}) <= 8;
- any PL 4-manifold M^{4 }with k(M^{4})=9 is simply connected (with second Betti number β_{2}= 3).
For details and related results, see:
- M.R. Casali, Catalogues of PL-manifolds and complexity estimations via crystallization theory, Oberwolfach Reports, Report No. 24/2012 - Workshop “TRIANGULATIONS” (April 29 ^{th }- May 04 ^{th}2012), 58-61. DOI: http://dx.doi.org/10.4171/OWR/2012/24
- M.R. Casali - P. Cristofori, Coloured graphs representing PL 4-manifolds, Electronic Notes in Discrete Mathematics 40(2013), 83-87. DOI: https://doi.org/10.1016/j.endm.2013.05.016
- M.R. Casali - P. Cristofori, Cataloguing PL 4-manifolds by gem-complexity, The Electronic Journal of Combinatorics 22(4) (2015), #P4.25. https://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p25
- M.R. Casali – P. Cristofori – C. Gagliardi, Classifying PL 4-manifolds via crystallizations: results and open problems, in: " A Mathematical Tribute to Professor José María Montesinos Amilibia”, Universidad Complutense Madrid (2016). [ISBN: 978-84-608-1684-3]. See here.