High Quality Content by WIKIPEDIA articles! In low-dimensional topology, the trigenus is an invariant consisting of a triplet (g1,g2,g3) assigned to closed 3-manifolds. The definition is by minimizing the genera of three orientable handle bodies — with no intersection between their interiors— which decompose the manifold as far as the Heegaard genus need only two. That is, a decomposition into scriptstyle M=V_1cup V_2cup V_3 with scriptstyle {rm int} V_icap {rm int} V_j=varnothing for i,j = 1,2,3 and being gi the genus of Vi. For orientable spaces scriptstyle {rm trig}(M)=(0,0,h) where h is M's Heegaard genus. For non-orientable spaces the trig has the form as scriptstyle {rm trig}(M)=(0,g_2,g_3)quad mbox{or}quad (1,g_2,g_3) depending on the image of the first Stiefel-Whitney characteristic class w1 under a Bockstein homomorphism, respectively for scriptstyle beta(w_1)=0quad mbox{or}quad neq 0.

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