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 C2p (4) (resp. ~C2p (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(M4)=p-1 iff 2p is the minimum order of a crystallization of M4.
THEOREM
Let M4 be a closed connected PL 4-manifold. Then:
- k (M4)=0 iff M4 is PL-homeomorphic to S4;
- k (M4)=3 iff M4 is PL-homeomorphic to CP2;
- k (M4)=4 iff M4 is PL-homeomorphic to either S1 x S3 or S1 ~ x S3;
- k (M4)=6 iff M4 is PL-homeomorphic to either S2 x S2 or CP2 #CP2 or CP2 # (-CP2);
- k (M4)=7 iff M4 is PL-homeomorphic to either RP4 or CP2 # (S1 x S3) or CP2 # (S1 ~ x S3);
- k (M4)=8 iff M4 is PL-homeomorphic to either #2(S1 x S3) or #2 (S1 ~ x S3).
Moreover :
- no PL 4-manifold M4 exists with k(M4) =1,2,5;
- no exotic PL 4-manifold exists, with k(M4) <= 8;
- any PL 4-manifold M4 with k(M4)=9 is simply connected (with second Betti number β2= 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 th2012), 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.