Trigenus

In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple [math]\displaystyle{ (g_1,g_2,g_3) }[/math]. 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 [math]\displaystyle{ M=V_1\cup V_2\cup V_3 }[/math] with [math]\displaystyle{ {\rm int} V_i\cap {\rm int} V_j=\varnothing }[/math] for [math]\displaystyle{ i,j=1,2,3 }[/math] and being [math]\displaystyle{ g_i }[/math] the genus of [math]\displaystyle{ V_i }[/math].

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

For non-orientable spaces the [math]\displaystyle{ {\rm trig} }[/math] has the form [math]\displaystyle{ {\rm trig}(M)=(0,g_2,g_3)\quad \mbox{or}\quad (1,g_2,g_3) }[/math] depending on the image of the first Stiefel–Whitney characteristic class [math]\displaystyle{ w_1 }[/math] under a Bockstein homomorphism, respectively for [math]\displaystyle{ \beta(w_1)=0\quad \mbox{or}\quad \neq 0. }[/math]

It has been proved that the number [math]\displaystyle{ g_2 }[/math] has a relation with the concept of Stiefel–Whitney surface, that is, an orientable surface [math]\displaystyle{ G }[/math] which is embedded in [math]\displaystyle{ M }[/math], has minimal genus and represents the first Stiefel–Whitney class under the duality map [math]\displaystyle{ D\colon H^1(M;{\mathbb{Z}}_2)\to H_2(M;{\mathbb{Z}}_2), }[/math], that is, [math]\displaystyle{ Dw_1(M)=[G] }[/math]. If [math]\displaystyle{ \beta(w_1)=0 \, }[/math] then [math]\displaystyle{ {\rm trig}(M)=(0,2g,g_3) \, }[/math], and if [math]\displaystyle{ \beta(w_1)\neq 0. \, }[/math] then [math]\displaystyle{ {\rm trig}(M)=(1,2g-1,g_3) \, }[/math].

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 [math]\displaystyle{ S^1 }[/math]", 2005, Soc. Mat. Mex. | pdf