In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple . 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 with
for and being the genus of .
For orientable spaces, ,
where is 's Heegaard genus.
For non-orientable spaces the has the form
depending on the
image of the first Stiefel–Whitney characteristic class under a Bockstein homomorphism, respectively for
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 , that is, . If then , and if