Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of "standard" type, that is, all connected sums of $\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of $\mathbb{CP}^2$ is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of $\mathbb{CP}^2$ by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear in Advances in Geometr
Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely gem-complexity and regular genus. In the present paper we prove that, for any closed connected PL 4-manifold M , its gem-complexity k (M ) and its regular genus G(M ) satisfy:where rk(π 1 (M )) = m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of "standard type", involved in existing crystallization catalogues, and their connected sums.MSC 2010 : Primary 57Q15. Secondary 57Q05, 57N13, 05C15.
Let M be a 3-manifold with boundary and let b 1 (∂M ) be the first Betti number of the boundary ∂M with Z/2Z coefficients. We have shown that if (Γ, γ) is a crystallization of M where ∂M is connected, then |V (Γ)| ≥ 2 + 3 b 1 (∂M ), and the bound is sharp.We have also shown that if (Γ, γ) is a crystallization of M and |V (Γ)| < 8 + 3 b 1 (∂M ) then M is a handlebody. Let (Γ, γ) be a crystallization of M where ∂M has exactly two components. Then we have proved that |V (Γ)| ≥ 8 + 3 b 1 (∂M ), and this bound is sharp when the both boundary components are spheres or handles.MSC 2010 : Primary 57Q15. Secondary 05C15; 57N10; 57N15; 57Q05.
Let ∆ be a d-dimensional normal pseudomanifold, d ≥ 3. A relative lower bound for the number of edges in ∆ is that g 2 of ∆ is at least g 2 of the link of any vertex. When this inequality is sharp ∆ has relatively minimal g 2 . For example, whenever the one-skeleton of ∆ equals the one-skeleton of the star of a vertex, then ∆ has relatively minimal g 2 . Subdividing a facet in such an example also gives a complex with relatively minimal g 2 . We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of 3-dimensional ∆ with relatively minimal g 2 whenever ∆ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body.Complete combinatorial descriptions of ∆ with g 2 (∆) ≤ 2 are due to Kalai [12] (g 2 = 0), Nevo and Novinsky [13] (g 2 = 1) and Zheng [21] (g 2 = 2). In all three cases ∆ is the boundary of a simplicial polytope. Zheng observed that for all d ≥ 0 there are triangulations of S d * RP 2 with g 2 = 3. She asked if this is the only nonspherical topology possible for g 2 (∆) = 3. As another application of relatively minimal g 2 we give an affirmative answer when ∆ is 3-dimensional.
For d ≥ 2, the regular genus of a closed connected PL d-manifold M is the least genus (resp., half of genus) of an orientable (resp., a non-orientable) surface into which a crystallization of M imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL 4-manifold M , it is known that its regular genus G(M ) is at least 2χ(M ) + 5m − 4, where m is the rank of the fundamental group of M . In this article, we introduce the concept of "weak semi-simple crystallization" for every closed connected PL 4-manifold M , and prove that G(M ) = 2χ(M ) + 5m − 4 if and only if M admits a weak semi-simple crystallization. We then show that the PL invariant regular genus is additive under the connected sum within the class of all PL 4-manifolds admitting a weak semi-simple crystallization. Also, we note that this property is related to the 4-dimensional Smooth Poincaré Conjecture.MSC 2010 : Primary 57Q15. Secondary 57Q05; 57N13; 05C15.
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