Ternary relation

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 $M=V_1\cup V_2\cup V_3$ with $\mathrm{i}\mathrm{n}\mathrm{t}{V}_i\cap \mathrm{i}\mathrm{n}\mathrm{t}{V}_j=\varnothing$ for $i,j=1,2,3$ and being $g_i$ the genus of $V_i$.

For orientable spaces, $\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{g}(M)=(0,0,h)$, where $h$ is $M$'s Heegaard genus.

For non-orientable spaces the $\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{g}$ has the form $\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{g}(M)=(0,g_2,g_3)\text{or}(1,g_2,g_3)$ depending on the image of the first Stiefel–Whitney characteristic class $w_1$ under a Bockstein homomorphism, respectively for $\beta (w_1)=0\text{or}\ne 0.$

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 $D:H^1(M;{\mathbb{\mathbb{Z}}}_2)\to H_2(M;{\mathbb{\mathbb{Z}}}_2),$, that is, $D{w}_1(M)=[G]$. If $\beta (w_1)=0$ then $\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{g}(M)=(0,2g,g_3)$, and if $\beta (w_1)\ne 0.$ then $\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{g}(M)=(1,2g-1,g_3)$.

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 $S^1$", 2005, Soc. Mat. Mex. | pdf