Cataloguing PL 4-manifolds by gem-complexity
The authors describe an algorithm which can determine whether two crystallizations give the same (homeomorphic/PL-homeomorphic) PL-manifold or not. Using their algorithm they prove, amongst others, the following. Let \((\Gamma, \gamma)\) be a crystallization of a closed, connected, PL 4-manifold \(M\) and let \(\nu(\Gamma)\) be the order of \((\Gamma, \gamma)\). (i) If \(\nu(\Gamma) \leq 18\) and \(M\) is handle-free then \(M\) is PL-homeomorphic to \(S^4\), \(\mathbb{CP}^2\), \(S^2\times S^2\), \(\mathbb{CP}^2\#\mathbb{CP}^2\), \(\mathbb{CP}^2\#(-\mathbb{CP}^2)\) or \(\mathbb{RP}^4\). (ii) If \(\nu(\Gamma) =20\) and \(M\) is handle-free then either \(M\) is simply connected or \(M\) is PL-homeomorphic to \(\mathbb{RP}^4\). (iii) If \(\nu(\Gamma)\leq 132\) and \(M\) is simply connected then the second Betti number \(\beta_2(M)\) of \(M\) is at most \((\nu(\Gamma) -2)/6\) and \(M\) is homeomorphic to \(S^4\#(\#_q\mathbb{CP}^2)\#(\#_r(-\mathbb{CP}^2))\) or \(S^4\#(\#_{s}(S^2\times S^2))\) for some \(q, r, s \geq 0\) (with \(q+r = \beta_2(M)=2s\)). The algorithm may be useful to decide whether different known triangulations of a 4-manifold \(M\) give the same PL-structure on \(M\) or not.
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