Parallelizable manifold - Revision history

en>David Eppstein: /* References */ Richard L. Bishop

2014-06-17T06:27:35Z

← Older revision		Revision as of 08:27, 17 June 2014
Line 1:		Line 1:
			Greetings. Allow me start by telling you the writer's title - Phebe. Minnesota has usually been his house but his spouse wants them to transfer. My day job is a meter reader. One of the very very best issues in the world for him is to gather badges but he is struggling to find time for it.<br><br>Take a look at my web-site ... [http://kitakeluarga.com/index.php?do=/profile-12707/info/ http://kitakeluarga.com/]
	~~In [[mathematics]], '''reduced homology'~~'~~' is a minor modification made to [[homology theory]] in [[algebraic topology]], designed to make a point have all its [[homology group]]~~s ~~zero~~. ~~This change is required~~ to ~~make statements without some number of exceptional cases ([[Alexander duality]] being an example)~~.

	~~If ''P''~~ is a ~~single-point space, then with~~ the ~~usual definitions~~ the ~~integral homology group~~

	~~:''H''<sub>0</sub>(''P'')~~

	is ~~an [[infinite cyclic group]], while~~ for ~~''i'' ≥ 1 we have~~

	~~:''H''~~<~~sub~~>~~''i''~~<~~/sub~~>~~(''P'') = {0}.~~

	~~More generally if ''X'' is~~ a ~~[[simplicial complex]] or finite [[CW complex]], then the group ''H''<sub>0</sub>(''X'') is the [[free abelian group]] with the [[connected space\|connected component]]s of ''X'' as generators~~. ~~The reduced homology should replace this group, of rank ''r'' say, by one of rank ''r'' − 1~~. ~~Otherwise the homology groups should remain unchanged~~. ~~An ''ad hoc'' way to do this is to think of a 0-th homology class not as a~~ [~~[formal sum]] of connected components, but as such a formal sum where the coefficients add up to zero.~~

	~~In the usual definition of [[homology (mathematics)\|homology]] of a space ''X'', we consider the chain complex~~
	:~~<math>\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n~~
	~~\overset{\partial_n}{\longrightarrow\,}C_{n-1}~~
	~~\overset{\partial_{n-1}}{\longrightarrow\,}~~
	~~\dotsb~~
	~~\overset{\partial_2}{\longrightarrow\,}~~
	~~C_1~~
	~~\overset{\partial_1}{\longrightarrow\,}~~
	~~C_0\overset{\partial_0}{\longrightarrow\,} 0</math>~~

	~~and define the homology groups by <math>H_n(X) = \ker \partial_n~~ / ~~\mathrm{im}(\partial_{n+1})<~~/~~math>~~.

	~~To define reduced homology, we start with the ''augmented'' chain complex~~

	~~<math>\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n~~
	~~\overset{\partial_n}{\longrightarrow\,}C_{n-1}~~
	~~\overset{\partial_{n-1}}{\longrightarrow\,}~~
	~~\dotsb~~
	~~\overset{\partial_2}{\longrightarrow\,}~~
	~~C_1~~
	~~\overset{\partial_1}{\longrightarrow\,}~~
	~~C_0\overset{\epsilon}{\longrightarrow\,} \mathbb{Z} \to 0~~
	~~</math>~~

	~~where <math>\epsilon \left( \sum_i n_i \sigma_i \right) = \sum_i n_i <~~/~~math>~~. ~~Now we define the ''reduced'' homology groups by~~

	~~:<math> \tilde{H_n}(X)~~ = ~~\ker(\partial_n)~~ / ~~\mathrm{im}(\partial_{n+1})</math> for positive ''n'' and <math>\tilde{H}_0(X) = \ker(\epsilon) / \mathrm{im}(\partial_1)</math>.~~

	~~One can show that <math> H_0(X) = \tilde{H}_0(X) \oplus \mathbb{Z}<~~/~~math>; evidently <math>H_n(X) = \tilde{H}_n(X)<~~/~~math> for all positive ''n''.~~

	Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the [[tensor product]], or ''reduced'' [[cohomology group]]s from the [[cochain complex]] made by using a [[Hom functor]], can be applied.


	~~==References==~~
	* Hatcher, A., (2002) ''[http://~~www~~.~~math.cornell.edu~~/~hatcher/AT/ATchapters.html Algebraic Topology]'' Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.


	~~{{DEFAULTSORT:Reduced Homology}}~~
	~~[[Category:Homology theory]~~]

en>Mgnbar: /* Examples */ clarified "in 3 dimensions; manifold dimension, not ambient space dimension

2014-01-20T15:09:51Z

Examples: clarified "in 3 dimensions; manifold dimension, not ambient space dimension

← Older revision		Revision as of 17:09, 20 January 2014
Line 1:		Line 1:
	~~Oscar~~ is ~~what my spouse enjoys~~ to ~~call me and I totally dig that name~~. ~~One~~ of the ~~things she enjoys most~~ is ~~to study comics and she~~'~~ll be beginning some thing else along~~ with it. ~~Since she was eighteen she~~'~~s been operating~~ as a ~~meter reader~~ but ~~she~~'~~s usually wanted her own business~~. ~~Puerto Rico is exactly~~ where he and ~~his wife live~~.<br><br>~~Feel free~~ to ~~surf to my web page ::~~ [http://~~immbooks~~.~~com~~/~~blogs~~/~~post~~/~~44599 at home std testing~~]
			In [[mathematics]], '''reduced homology''' is a minor modification made to [[homology theory]] in [[algebraic topology]], designed to make a point have all its [[homology group]]s zero. This change is required to make statements without some number of exceptional cases ([[Alexander duality]] being an example).

			If ''P'' is a single-point space, then with the usual definitions the integral homology group

			:''H''<sub>0</sub>(''P'')

			is an [[infinite cyclic group]], while for ''i'' ≥ 1 we have

			:''H''<sub>''i''</sub>(''P'') = {0}.

			More generally if ''X'' is a [[simplicial complex]] or finite [[CW complex]], then the group ''H''<sub>0</sub>(''X'') is the [[free abelian group]] with the [[connected space\|connected component]]s of ''X'' as generators. The reduced homology should replace this group, of rank ''r'' say, by one of rank ''r'' − 1. Otherwise the homology groups should remain unchanged. An ''ad hoc'' way to do this is to think of a 0-th homology class not as a [[formal sum]] of connected components, but as such a formal sum where the coefficients add up to zero.

			In the usual definition of [[homology (mathematics)\|homology]] of a space ''X'', we consider the chain complex
			:<math>\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n
			\overset{\partial_n}{\longrightarrow\,}C_{n-1}
			\overset{\partial_{n-1}}{\longrightarrow\,}
			\dotsb
			\overset{\partial_2}{\longrightarrow\,}
			C_1
			\overset{\partial_1}{\longrightarrow\,}
			C_0\overset{\partial_0}{\longrightarrow\,} 0</math>

			and define the homology groups by <math>H_n(X) = \ker \partial_n / \mathrm{im}(\partial_{n+1})</math>.

			To define reduced homology, we start with the ''augmented'' chain complex

			<math>\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n
			\overset{\partial_n}{\longrightarrow\,}C_{n-1}
			\overset{\partial_{n-1}}{\longrightarrow\,}
			\dotsb
			\overset{\partial_2}{\longrightarrow\,}
			C_1
			\overset{\partial_1}{\longrightarrow\,}
			C_0\overset{\epsilon}{\longrightarrow\,} \mathbb{Z} \to 0
			</math>

			where <math>\epsilon \left( \sum_i n_i \sigma_i \right) = \sum_i n_i </math>. Now we define the ''reduced'' homology groups by

			:<math> \tilde{H_n}(X) = \ker(\partial_n) / \mathrm{im}(\partial_{n+1})</math> for positive ''n'' and <math>\tilde{H}_0(X) = \ker(\epsilon) / \mathrm{im}(\partial_1)</math>.

			One can show that <math> H_0(X) = \tilde{H}_0(X) \oplus \mathbb{Z}</math>; evidently <math>H_n(X) = \tilde{H}_n(X)</math> for all positive ''n''.

			Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the [[tensor product]], or ''reduced'' [[cohomology group]]s from the [[cochain complex]] made by using a [[Hom functor]], can be applied.


			==References==
			* Hatcher, A., (2002) ''[http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology]'' Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.


			{{DEFAULTSORT:Reduced Homology}}
			[[Category:Homology theory]]

en>David Eppstein: /* References */ link stasheff

2012-03-25T06:24:45Z

References: link stasheff

New page

Oscar is what my spouse enjoys to call me and I totally dig that name. One of the things she enjoys most is to study comics and she'll be beginning some thing else along with it. Since she was eighteen she's been operating as a meter reader but she's usually wanted her own business. Puerto Rico is exactly where he and his wife live.<br><br>Feel free to surf to my web page :: [http://immbooks.com/blogs/post/44599 at home std testing]