Rayleigh–Ritz method - Revision history

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Description of method: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10480)

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	In [~~[mathematics]~~], ''~~'cellular homology''' in~~ [~~[algebraic topology]~~] is a [~~[homology theory]]~~ for ~~[[CW-complex]]es. It agrees with [[singular homology]~~], and ~~can provide an effective means of computing homology modules~~.		I woke up another day and realised - Today I've been single for some time and after [http://lukebryantickets.citizenswebcasting.com Luke bryan my kind Of Night tour] much intimidation from pals I today locate myself signed up for internet dating. They promised me that there are lots of standard, sweet and enjoyable folks to meet, so here goes the toss!<br>I try and maintain as physically fit as potential being at the gym many times weekly. I love my sports and attempt to perform or watch while many a possible. I shall frequently at Hawthorn fits being winter. Note: If you contemplated purchasing an athletics I don't mind, I've experienced the carnage of fumbling suits at stocktake sales.<br>My [http://Www.Dict.cc/englisch-deutsch/friends.html friends] and fam are magnificent and hanging out with them at bar gigs or dinners is definitely imperative. I have never been in to cabarets as I realize that you can not have a good dialogue using the sound. Additionally, [http://lukebryantickets.hamedanshahr.com tickets for luke bryan concert] I have 2 very adorable and unquestionably cheeky canines that are almost always keen to meet fresh folks.<br><br>

	~~== Definition ==~~		Review my web site :: is the luke bryan concert sold out - [http://www.ladyhawkshockey.org ladyhawkshockey.org],

	~~If ''X'' is a CW-complex with [[n-skeleton]] ''X<sub>n</sub>'', the cellular homology modules are defined as the [[homology group]]s of the cellular [[chain complex]]~~

	~~:<math> \cdots \to H_{n+1}( X_{n+1}, X_n ) \to H_n( X_n, X_{n-1} ) \to H_{n-1}( X_{n-1}, X_{n-2} ) \to \cdots . </math>~~

	~~[<math>X_{-1}</math> is the empty set]~~

	~~The group~~

	~~:<math>H_n( X_n, X_{n-1} ) \,</math>~~

	is [[Free module\|free]], with generators which can be identified with the ''n''-cells of ''X''. Let <math>e_n^{\alpha}</math> be an ''n''-cell of ''X'', let <math>\chi_n^{\alpha} : \partial e_n^{\alpha}\cong S^{n-1} \to X_{n-1}</math> be the attaching map, and consider the composite maps

	~~:<math>\chi_n^{\alpha\beta}:S^{n-1} \to X_{n-1} \to X_{n-1}/(X_{n-1}-e_{n-1}^{\beta})\cong S^{n-1}</math>~~

	~~where <math>e_{n-1}^{\beta}</math> is an <math>(n-1)</math>-cell of ''X'' and the second map is the quotient map identifying <math>(X_{n-1}-e_{n-1}^{\beta})</math> to a point.~~

	~~The [[boundary map]]~~

	:~~<math>d_n~~:~~H_n(X_n,X_{n-1}) \to H_{n-1}(X_{n-1},X_{n-2}) \,</math>~~

	~~is then given by the formula~~

	~~:<math>d_n(e_n^{\alpha})=\sum_{\beta}\deg(\chi_n^{\alpha\beta})e_{n-1}^{\beta}\, </math>~~

	~~where <math>deg(\chi_n^{\alpha\beta})</math> is the [[Degree of a continuous mapping\|degree]] of <math>\chi_n^{\alpha\beta}</math> and the sum~~ is ~~taken over all <math>(n-1)</math>-cells of ''X'', considered as generators of <math>H_{n-1}(X_{n-1},X_{n-2})\,</math>.~~

	~~== Other properties ==~~

	~~One sees from~~ the ~~cellular chain complex that the ''n''~~-~~skeleton determines all lower-dimensional homology:~~

	~~:<math>H_k(X) \cong H_k(X_n) </math>~~

	~~for ''k'' < ''n''.~~

	An important consequence of the cellular perspective is that if a CW-complex has no cells in consecutive dimensions, all its homology modules are free. For example, [[complex projective space]] '''CP'''<sup>''n''</sup> has a cell structure with one cell in each even dimension; it follows that for 0 ≤ ''k'' ≤ ''n'',

	~~:<math> H_{2k}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} </math>~~

	~~and~~

	~~:<math> H_{2k+1}(\mathbb{CP}^n) = 0 .</math>~~

	~~== Generalization ==~~

	The [[Atiyah–Hirzebruch spectral sequence\|Atiyah-Hirzebruch spectral sequence]] is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary [[Extraordinary homology theory\|extraordinary (co)homology theory]].

	~~== Euler characteristic ==~~

	For a cellular complex ''X'', let ''X<sub>j</sub>'' be its ''j''-th skeleton, and ''c<sub>j</sub>'' be the number of ''j''-cells, i.e. the rank of the free module ''H<sub>j</sub>''(''X<sub>j</sub>'', ''X''<sub>''j''-1</sub>). The [[Euler characteristic]] of ''X'' is defined by

	~~:<math>\chi (X) = \sum _0 ^n (-1)^j c_j.</math>~~

	~~The Euler characteristic is a homotopy invariant. In fact, in terms of the [[Betti number]]s of ''X'',~~

	~~:<math>\chi (X) = \sum _0 ^n (-1)^j \; \mbox{rank} \; H_j (X). </math>~~

	~~This can be justified as follows. Consider the long exact sequence of [[relative homology]] for the triple (''X<sub>n</sub>'', ''X''<sub>''n'' - 1 </sub>, ∅):~~

	~~:<math> \cdots \to H_i( X_{n-1}, \empty) \to H_i( X_n, \empty) \to H_i( X_{n}, X_{n-1} ) \to \cdots . </math>~~

	~~Chasing exactness through the sequence gives~~

	~~:<math>~~
	~~\sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, \empty)~~

	~~= \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, X_{n-1}) \; + \; \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_{n-1}, \empty).</math>~~

	~~The same calculation applies to the triple (''X''<sub>''n'' - 1</sub>, ''X''<sub>''n'' - 2</sub>, ∅), etc. By induction,~~

	~~:<math>~~

	~~\sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, \empty)~~

	~~= \sum_{j = 0} ^n \; \sum_{i = 0} ^j (-1)^i \; \mbox{rank} \; H_i (X_j, X_{j-1})~~

	~~= \sum_{j = 0} ^n (-1)^j c_j.</math>~~

	~~==References==~~
	* A. Dold: ''Lectures on Algebraic Topology'', Springer ISBN 3-540-58660-1.

	* A. Hatcher: ''Algebraic Topology'', Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the [http://www.~~math~~.~~cornell~~.~~edu/~hatcher/ author's homepage].~~

	~~[[Category:Homology theory]~~]

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