Trigenus


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
then.

Theorem

A manifold S is a Stiefel–Whitney surface in M, if and only if S and M−int are orientable.