Trigenus

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
$V_i\cap V_j=\varnothing$
for $i,j=1,2,3$ and being $g_i$ the genus of $V_i$.

For orientable spaces, $(M)=(0,0,h)$,
where $h$ is $M$'s Heegaard genus.

For non-orientable spaces the $$ has the form $(M)=(0,g_2,g_3)\quad \mbox\quad (1,g_2,g_3)$
depending on the
image of the first Stiefel–Whitney characteristic class $w_1$ under a Bockstein homomorphism, respectively for
$\beta(w_1)=0\quad \mbox\quad \neq 0.$

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 $D\colon H^1(M;_2)\to H_2(M;_2),$, that is, Dw_1(M)= /math>. If $\beta(w_1)=0 \,$ then $(M)=(0,2g,g_3) \,$, and if $\beta(w_1)\neq 0. \,$
then ${\rm trig}(M)=(1,2g-1,g_3) \,$.

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

Geometric topology
3-manifolds