What is "trigenus"

In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple (g, g, g). It is obtained 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 M = V ∪ V ∪ V with ${\rm int} V_i\cap {\rm int} V_j=\varnothing$ for i, j = 1, 2, 3 and being g the genus of V.

For orientable spaces, ${\rm trig}(M)=(0,0,h)$, where h is M's Heegaard genus.

For non-orientable spaces the ${\rm trig}$ has the form ${\rm trig}(M)=(0,g_2,g_3)\quad \mbox{or}\quad (1,g_2,g_3)$ depending on the image of the first Stiefel–Whitney characteristic class w under a Bockstein homomorphism, respectively for β(w) = 0 or  ≠ 0.

It has been proved that the number g has a relation with the concept of Stiefel–Whitney surface, that is, an orientable surface G which is embedded in M, has minimal genus and represents the first Stiefel–Whitney class under the duality map D: H(M; Z) → H(M; Z), , that is, Dw(M) = [G]. If β(w) = 0  then ${\rm trig}(M)=(0,2g,g_3) \,$, and if β(w) ≠ 0.  then ${\rm trig}(M)=(1,2g-1,g_3) \,$.