Stiefel-Whitney surfaces and the tri-genus of non-orientable 3-manifolds
Every non-orientable closed connected 3-manifold \(M\) can be expressed as a union of three orientable handlebodies \(V_1,V_2,V_3\) whose interiors are pairwise disjoint. If \(g_i\) is the genus of \(\partial V_i\), and if \(g_1\leq g_2\leq g_3\), then \(M\) is said to have a splitting of type \((g_1,g_2, g_3)\). Then the tri-genus of \(M\) is defined as the minimum triple \((g_1,g_2, g_3)\) with respect to the lexicographic ordering. Let \(\beta:H^1 (M;\mathbb{Z}_2)\to H^2(M;\mathbb{Z})\) be the Bockstein connecting homomorphism induced by the exact coefficient sequence \(0\to\mathbb{Z} \to\mathbb{Z} \to\mathbb{Z}_2\to 0\). If the image of the first Stiefel-Whitney class \(w_1(M)\in H^1(M;\mathbb{Z}_2)\) under \(\beta\) vanishes, then \(M\) has tri-genus \((0,2g,g_3)\), where \(g\) is the minimal genus of a 2-sided Stiefel-Whitney surface of \(M\) (a Stiefel-Whiney surface is a closed connected surface \(F\) in \(M\) that represents the nontrivial element of the torsion subgroup of \(H_2(M;\mathbb{Z})\). The homology class \([F]\in H_2(M;\mathbb{Z}_2)\) is the Poincaré dual of \(w_1(M)\) and \(M-F\) is an orientable 3-manifold). The authors prove that, if \(\beta w_1(M)\neq 0\), then \(M\) has tri-genus \((1,2g-1,g_3)\), where \(g\) is the minimal genus of an 1-sided Stiefel-Whitney surface of \(M\). Computations of the tri-genus for certain graph manifolds complete the paper.
scientific article; zbMATH DE number 1399920