Trigenus

In low-dimensional topology, the trigenus is an invariant consisting of a triplet

:$(g\_1,g\_2,g\_3)$

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

:$M=V\_1cup\; V\_2cup\; V\_3$

with

:$\{\; m\; int\}\; V\_icap\; \{\; m\; int\}\; V\_j=varnothing$

for $i,j=1,2,3$ and being $g\_i$ the genus of $V\_i$.

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

For non-orientable spaces the $\{\; m\; trig\}$ has the form as:$\{\; m\; trig\}(M)=(0,g\_2,g\_3)quad\; mbox\{or\}quad\; (1,g\_2,g\_3)$ depending on theimage of the first Stiefel-Whitney characteristic class $w\_1$ under a Bockstein homomorphism, respectively for :

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

:$Dcolon\; H^1(M;\{mathbb\{Z\_2)\; o\; H\_2(M;\{mathbb\{Z\_2),$ i.e.::$Dw\_1(M)=\; [G]$so, :$\{\; m\; trig\}(M)=(0,2g,g\_3)\; ,$, if or:$\{\; m\; trig\}(M)=(1,2g-1,g\_3)\; ,$, if

Theorem

It is true that: "S" is a Stiefel-Whitney surface in "M", iff "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.