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.
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 .
A manifold S is a Stiefel–Whitney surface in M, if and only if S and M−int(N(S)) are orientable .
- J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel–Whitney surfaces and decompositions of 3-manifolds into handlebodies, Topology Appl. 60 (1994), 267–280.
- J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel–Whitney surfaces and the trigenus of non-orientable 3-manifolds, Manuscripta Math. 100 (1999), 405–422.
- "On the trigenus of surface bundles over ", 2005, Soc. Mat. Mex. | pdf