Ternary relation - Revision history

en>BiT: /* Typing relation */

2014-04-14T04:43:39Z

Typing relation

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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.
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-That is, a decomposition <math> M=V_1\cup V_2\cup V_3</math> with
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-For orientable spaces, <math>{\rm trig}(M)=(0,0,h)</math>,
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en>Ancheta Wis: /* Congruence relation */ fix dab

2013-11-23T08:08:23Z

Congruence relation: fix dab

← Older revision		Revision as of 10:08, 23 November 2013
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	~~20 year old Architectural~~, ~~Building~~ and ~~Surveying Specialists~~ ~~Walton from Saint-Quentin~~, has ~~several hobbies~~ that ~~include bonsai trees~~, ~~ganhando dinheiro na internet~~ and ~~calligraphy~~. ~~Gets encouragement by going to La Grand~~-~~Place~~.<br><br>~~Also visit my webpage~~ ... [http://~~comoganhardinheironainternet~~.~~comoganhardinheiro101~~.~~com ganhar dinheiro~~]		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]] — with no intersection between their interiors— 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−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]]

en>JoergenB: /* Congruence relation */ lkfix

2012-04-23T07:45:59Z

Congruence relation: lkfix

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