In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple
(
g
2
3
.
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
{\displaystyle {\rm {int}}V_{i}\cap {\rm {int}}V_{j}=\varnothing }
the genus of
For orientable spaces,
{\displaystyle {\rm {trig}}(M)=(0,0,h)}
's Heegaard genus.
For non-orientable spaces the
{\displaystyle {\rm {trig}}}
has the form
{\displaystyle {\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
under a Bockstein homomorphism, respectively for
β (
It has been proved that the number
has a relation with the concept of Stiefel–Whitney surface, that is, an orientable surface
which is embedded in
, has minimal genus and represents the first Stiefel–Whitney class under the duality map
β (
{\displaystyle {\rm {trig}}(M)=(0,2g,g_{3})\,}
β (
{\displaystyle {\rm {trig}}(M)=(1,2g-1,g_{3})\,}
A manifold S is a Stiefel–Whitney surface in M, if and only if S and M−int(N(S)) are orientable.