Trigenus

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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[edit]

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

References[edit]

  • 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