Trigenus

In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple ${\displaystyle (g_{1},g_{2},g_{3})}$. 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 ${\displaystyle M=V_{1}\cup V_{2}\cup V_{3}}$ with ${\displaystyle {\rm {int}}V_{i}\cap {\rm {int}}V_{j}=\varnothing }$ for ${\displaystyle i,j=1,2,3}$ and being ${\displaystyle g_{i}}$ the genus of ${\displaystyle V_{i}}$.

For orientable spaces, ${\displaystyle {\rm {trig}}(M)=(0,0,h)}$, where ${\displaystyle h}$ is ${\displaystyle M}$'s Heegaard genus.

For non-orientable spaces the ${\displaystyle {\rm {trig}}}$ has the form ${\displaystyle {\rm {trig}}(M)=(0,g_{2},g_{3})\quad {\mbox{or}}\quad (1,g_{2},g_{3})}$ depending on the image of the first Stiefel–Whitney characteristic class ${\displaystyle w_{1}}$ under a Bockstein homomorphism, respectively for ${\displaystyle \beta (w_{1})=0\quad {\mbox{or}}\quad \neq 0.}$

It has been proved that the number ${\displaystyle g_{2}}$ has a relation with the concept of Stiefel–Whitney surface, that is, an orientable surface ${\displaystyle G}$ which is embedded in ${\displaystyle M}$, has minimal genus and represents the first Stiefel–Whitney class under the duality map ${\displaystyle D\colon H^{1}(M;{\mathbb {Z} }_{2})\to H_{2}(M;{\mathbb {Z} }_{2}),}$, that is, ${\displaystyle Dw_{1}(M)=[G]}$. If ${\displaystyle \beta (w_{1})=0\,}$ then ${\displaystyle {\rm {trig}}(M)=(0,2g,g_{3})\,}$, and if ${\displaystyle \beta (w_{1})\neq 0.\,}$ then ${\displaystyle {\rm {trig}}(M)=(1,2g-1,g_{3})\,}$.

Theorem

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

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.
• "On the trigenus of surface bundles over ${\displaystyle S^{1}}$", 2005, Soc. Mat. Mex. | pdf