Trigenus

In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple $$(g_1,g_2,g_3)$$. It is obtained by minimizing the genera of three orientable handle bodies &mdash; with no intersection between their interiors&mdash; which decompose the manifold as far as the Heegaard genus need only two.

That is, a decomposition $$ M=V_1\cup V_2\cup V_3$$ with $$ {\rm int} V_i\cap {\rm int} V_j=\varnothing$$ for $$i,j=1,2,3$$ and being $$g_i$$ the genus of $$V_i$$.

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 $$w_1$$ under a Bockstein homomorphism, respectively for $$\beta(w_1)=0\quad \mbox{or}\quad \neq 0.$$

It has been proved that the number $$g_2$$ 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 $$D\colon H^1(M;{\mathbb{Z}}_2)\to H_2(M;{\mathbb{Z}}_2), $$, that is, $$Dw_1(M)=[G]$$. If $$ \beta(w_1)=0 \,$$ then $$ {\rm trig}(M)=(0,2g,g_3) \,$$, and if $$ \beta(w_1)\neq 0. \,$$ then $$ {\rm trig}(M)=(1,2g-1,g_3) \,$$.

Theorem
A manifold S is a Stiefel–Whitney surface in M, if and only if S and M&minus;int(N(S)) are orientable.