Trigenus
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In low dimensional topology, the trigenus is an invariant consisting of a triplet
- (g1,g2,g3)
assigned to closed 3-manifolds. The definition is 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 into
with
for i,j = 1,2,3 and being gi the genus of Vi.
For orientable spaces
- trig(M) = (0,0,h)
where h is M's Heegaard genus.
For non-orientable spaces the trig has the form as
depending on the image of the first Stiefel-Whitney characteristic class w1 under a Bockstein homomorphism, respectively for
It has been proved that the number g2 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
i.e.
-
- Dw1(M) = [G]
so,
,
if 
or
,
if 
[edit] References
- 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.
