Trigenus

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

(g1,g2,g3)

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_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 gi the genus of Vi.

For orientable spaces

trig(M) = (0,0,h)

where h is M's Heegaard genus.

For non-orientable spaces the trig has the form as

{\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 w1 under a Bockstein homomorphism, respectively for

\beta(w_1)=0\quad \mbox{or}\quad \neq 0

It has been proved that the number g2 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),

i.e.

Dw1(M) = [G]

so,

{\rm trig}(M)=(0,2g,g_3) \,,

if \beta(w_1)=0 \, or

{\rm trig}(M)=(1,2g-1,g_3) \,,

if \beta(w_1)\neq 0 \,

[edit] 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.
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