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 {\rm int} V_i\cap {\rm int} V_j=\varnothing for i,j=1,2,3 and being g_i the genus of V_i . For orientable spaces, {\rm trig}(M)=(0,0,h) , where h is M 's Heegaard genus. For non-orientable spaces the {\rm trig} has the form {\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 w_1 under a Bockstein homomorphism, respectively for \beta(w_1)=0\quad \mbox{or}\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;{\mathbb{Z}}_2)\to H_2(M;{\mathbb{Z}}_2), , that is, Dw_1(M)=[G] . If \beta(w_1)=0 \, then {\rm trig}(M)=(0,2g,g_3) \, , and if \beta(w_1)\neq 0. \, then {\rm trig}(M)=(1,2g-1,g_3) \, .

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 {\rm int} V_i\cap {\rm int} V_j=\varnothing for i,j=1,2,3 and being g_i the genus of V_i . For orientable spaces, {\rm trig}(M)=(0,0,h) , where h is M 's Heegaard genus. For non-orientable spaces the {\rm trig} has the form {\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 w_1 under a Bockstein homomorphism, respectively for \beta(w_1)=0\quad \mbox{or}\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;{\mathbb{Z}}_2)\to H_2(M;{\mathbb{Z}}_2), , that is, Dw_1(M)=[G] . If \beta(w_1)=0 \, then {\rm trig}(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

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