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(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.