Classifying combinatorial 4-manifolds up to complexity
An \((n + 1)\)-colored graph \((G, c)\) is a finite graph \(G\) along with an edge coloring map \(c : E(G) \to \Delta_n = \{i \in \mathbb Z \;|\; 0 \leq i \leq n\}\) by \(n+1\) colors, where \(E(G)\) is the set of edges of \(G\). To \((G, c)\) we can associate a cellular complex \(K(G)\) of dimension \(n\), such that each \(n\)-simplex of \(K(G)\) (with vertices arbitrarily colored by \(\Delta_n\)) corresponds to a vertex of \(G\). If two vertices of \(G\) are joined by an edge colored by \(i \in \Delta_n\) then the corresponding \(n\)-simplexes are attached along the faces opposite to their vertices colored by \(i\) by a color-preserving identification. \(K(G)\) is defined as the quotient space. If \(M\) is a PL \(n\)-manifold and \(K(G)\) is PL-homeomorphic to \(M\) for some \((G, c)\), we say that \((G, c)\) represents \(M\). If \(K(G)\) has exactly \(n+1\) vertices we say that \((G, c)\) is a crystallization of \(M\). By a theorem of \textit{M. Pezzana} [Atti Sem. Mat. Fis. Univ. Modena 23, 269--277 (1974; Zbl 0314.57005)] and [Boll. Unione Mat. Ital. 12, 98--105 (1975; Zbl 0327.57003)] any closed PL \(n\)-manifold admits a crystallization. Then we can define the complexity \(c(M)\) as the minimum number of vertices that occur in the crystallizations of \(M\). Put \(\widetilde c(M) = c(M) - 2\), the reduced complexity of \(M\). This is a finite-to-one PL invariant with values in the non-negative even integers. In the paper under review the authors give a classification of the closed connected PL \(4\)-manifolds of reduced complexity at most 14. Roughly speaking, the main theorem states that \(\widetilde c(S^4) = 0\), \(\widetilde c(\mathbb CP^2) = 6\), \(\widetilde c(S^1 \times S^3) = \widetilde c(S^1 \, \widetilde\times\, S^3) = 8\), \(\widetilde c(S^2 \times S^2) = 12\), \(\widetilde c(\mathbb RP^4) = 14\), and there are no other prime closed 4-manifolds of reduced complexity \(\leq 14\) up to PL homeomorphisms. In order to prove this theorem, the authors consider the various cases that occur for a crystallization with at most 16 vertices. Another key ingredient is a lower bound for the reduced complexity, namely \(\widetilde c(M) \geq 6 \chi(M) + 20\, rk(M) - 12\), where \(rk(M)\) denotes the rank of the fundamental group of \(M\) (the authors give also a proof of this inequality). The paper ends with some conjectures related to the PL (\(=\) smooth) Poincaré conjecture in dimension four. For instance, it is conjectured that the reduced complexity is additive with respect to connected sum (this implies the PL Poincaré conjecture in dimension four). Moreover, the authors give some lower bounds for the reduced complexity of certain nonsingular algebraic surfaces. For instance, in case of the smooth hypersurface \(S_d \subset \mathbb CP^3\) of degree \(d\) they obtain \(\widetilde c(S_d) \geq 6(d^3 -4d^2 +6d-2)\).
scientific article; zbMATH DE number 5794610