- 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 :eta(w_1)=0quad mbox{or}quad eq 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: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 eta(w_1)=0 , or:m trig}(M)=(1,2g-1,g_3) ,, if eta(w_1) eq 0. ,
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.
Wikimedia Foundation. 2010.