Ternary relation: Difference between revisions

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In [[low-dimensional topology]], the '''trigenus''' of a [[closed manifold|closed]] [[3-manifold]] is an invariant consisting of an ordered triple <math>(g_1,g_2,g_3)</math>.  It is obtained by minimizing the genera of three ''[[orientable]]'' [[handlebody|handle bodies]] &mdash; with no intersection between their interiors&mdash; which decompose the manifold as far as the [[Heegaard splitting|Heegaard]] genus need only two.
 
That is, a decomposition <math> M=V_1\cup V_2\cup V_3</math> with
<math> {\rm int} V_i\cap {\rm int} V_j=\varnothing</math>
for <math>i,j=1,2,3</math> and being <math>g_i</math> the genus of <math>V_i</math>.
 
For orientable spaces, <math>{\rm trig}(M)=(0,0,h)</math>,
where <math>h</math> is <math>M</math>'s [[Heegaard genus]].
 
For non-orientable spaces the <math>{\rm trig}</math> has the form  <math>{\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 class|Stiefel–Whitney characteristic class]] <math>w_1</math> under a [[Bockstein homomorphism]], respectively for
<math>\beta(w_1)=0\quad \mbox{or}\quad \neq 0.</math>
 
It has been proved that the number <math>g_2</math> has a relation with the concept of [[Stiefel–Whitney surface]], that is, an orientable surface <math>G</math> which is embedded in <math>M</math>, has minimal genus and represents the first Stiefel–Whitney class under the duality map <math>D\colon H^1(M;{\mathbb{Z}}_2)\to H_2(M;{\mathbb{Z}}_2), </math>, that is, <math>Dw_1(M)=[G]</math>. If <math> \beta(w_1)=0 \,</math> then <math> {\rm trig}(M)=(0,2g,g_3) \,</math>, and if <math> \beta(w_1)\neq 0. \,</math>
then <math> {\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&minus;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>S^1</math>", 2005, Soc. Mat. Mex. [http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf | pdf]
 
[[Category:Geometric topology]]
[[Category:3-manifolds]]

Revision as of 09:08, 23 November 2013

In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple (g1,g2,g3). 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=V1V2V3 with intViintVj= 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 trig(M)=(0,g2,g3)or(1,g2,g3) depending on the image of the first Stiefel–Whitney characteristic class w1 under a Bockstein homomorphism, respectively for β(w1)=0or0.

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:H1(M;2)H2(M;2),, that is, Dw1(M)=[G]. If β(w1)=0 then trig(M)=(0,2g,g3), and if β(w1)0. then trig(M)=(1,2g1,g3).

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 S1", 2005, Soc. Mat. Mex. | pdf