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 — with no intersection between their interiors— which decompose the manifold as far as the Heegaard genus need only two.
That is, a decomposition M = V 1 ∪ V 2 ∪ V 3 with i n t V i ∩ i n t V j = ∅ for i , j = 1 , 2 , 3 and being g i the genus of V i .
For orientable spaces, t r i g ( M ) = ( 0 , 0 , h ) , where h is M 's Heegaard genus.
For non-orientable spaces the t r i g has the form t r i g ( M ) = ( 0 , g 2 , g 3 ) or ( 1 , g 2 , g 3 ) depending on the image of the first Stiefel–Whitney characteristic class w 1 under a Bockstein homomorphism, respectively for β ( w 1 ) = 0 or ≠ 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 : H 1 ( M ; Z 2 ) → H 2 ( M ; Z 2 ) , , that is, D w 1 ( M ) = [ G ] . If β ( w 1 ) = 0 then t r i g ( M ) = ( 0 , 2 g , g 3 ) , and if β ( w 1 ) ≠ 0. then t r i g ( M ) = ( 1 , 2 g − 1 , g 3 ) .
