Rylands Library Papyrus P52 - Revision history

en>RSeuhs: /* Brent Nongbri */ fix typo

2014-12-08T16:16:03Z

Brent Nongbri: fix typo

← Older revision		Revision as of 18:16, 8 December 2014
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en>FrescoBot: Bot: link syntax and minor changes

2014-02-09T01:02:57Z

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en>TomHennell: /* Date */ edit duplicate mattedr

2014-02-03T16:47:29Z

Date: edit duplicate mattedr

← Older revision		Revision as of 18:47, 3 February 2014
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	~~She~~ is ~~known~~ by the ~~name~~ of ~~Myrtle Shryock~~. ~~Years~~ in the ~~past he moved~~ to ~~North Dakota~~ and ~~his family loves it~~. ~~Supervising~~ is ~~my profession~~. ~~To do aerobics~~ is a ~~factor that I~~'~~m totally addicted~~ to.<br><br>~~Look at my web~~-~~site~~ - [http://~~Fhort~~.~~com~~/~~diettogo30380 fhort~~.~~com~~]		In [[mathematics]], specifically in [[algebraic topology]], the '''cup product''' is a method of adjoining two [[cocycle]]s of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space ''X'' into a graded ring, ''H''<sup>∗</sup>(''X''), called the [[cohomology ring]]. The cup product was introduced in work of [[James Waddell Alexander II\|J. W. Alexander]], [[Eduard Čech]] and [[Hassler Whitney]] from 1935–1938, and, in full generality, by [[Samuel Eilenberg]] in 1944.

			==Definition==
			In [[singular cohomology]], the '''cup product''' is a construction giving a product on the [[graded ring\|graded]] [[cohomology ring]] ''H''<sup>∗</sup>(''X'') of a [[topological space]] ''X''.

			The construction starts with a product of [[Cochain (algebraic topology)\|cochain]]s: if ''c''<sup>''p''</sup> is a ''p''-cochain and
			''d''<sup>''q''</sup> is a ''q''-cochain, then
			:<math>(c^p \smile d^q)(\sigma) = c^p(\sigma \circ \iota_{0,1, ... p}) \cdot d^q(\sigma \circ \iota_{p, p+1 ,..., p + q})</math>
			where σ is a [[Singular homology\|singular]] (''p'' + ''q'') -[[simplex]] and <math>\iota_S , S \subset \{0,1,...,p+q \} </math>
			is the canonical [[embedding]] of the simplex spanned by S into the <math>(p+q)</math>-simplex whose vertices are indexed by <math>\{0,...,p+q \}</math>.

			Informally, <math> \sigma \circ \iota_{0,1, ..., p}</math> is the ''p''-th '''front face''' and <math>\sigma \circ \iota_{p, p+1, ..., p + q}</math> is the ''q''-th '''back face''' of σ, respectively.

			The [[coboundary]] of the cup product of cocycles c<sup>''p''</sup> and d<sup>''q''</sup> is given by
			:<math>\delta(c^p \smile d^q) = \delta{c^p} \smile d^q + (-1)^p(c^p \smile \delta{d^q}).</math>
			The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. Thus, the cup product operation passes to cohomology, defining a bilinear operation
			: <math> H^p(X) \times H^q(X) \to H^{p+q}(X). </math>

			==Properties==
			The cup product operation in cohomology satisfies the identity
			:<math>\alpha^p \smile \beta^q = (-1)^{pq}(\beta^q \smile \alpha^p)</math>
			so that the corresponding multiplication is [[supercommutative\|graded-commutative]].

			The cup product is [[functor]]ial, in the following sense: if
			:<math>f\colon X\to Y</math>
			is a continuous function, and
			:<math>f^\colon H^(Y)\to H^*(X)</math>
			is the induced [[homomorphism]] in cohomology, then
			:<math>f^(\alpha \smile \beta) =f^(\alpha) \smile f^*(\beta),</math>
			for all classes α, β in ''H'' <sup></sup>(''Y''). In other words, ''f'' <sup></sup> is a (graded) [[ring homomorphism]].

			==Interpretation==
			It is possible to view the cup product <math> \smile \colon H^p(X) \times H^q(X) \to H^{p+q}(X)</math> as induced from the following composition:

			<math> \displaystyle C^\bullet(X) \times C^\bullet(X) \to C^\bullet(X \times X) \overset{\Delta^*}{\to} C^\bullet(X) </math>

			in terms of the [[chain complex]]es of <math>X</math> and <math>X \times X</math>, where the first map is the [[Künneth formula\|Künneth map]] and the second is the map induced by the [[diagonal functor\|diagonal]] <math> \Delta \colon X \to X \times X</math>.

			This composition passes to the quotient to give a well-defined map in terms of cohomology, this is the cup product. This approach explains the existence of a cup product for cohomology but not for homology: <math> \Delta \colon X \to X \times X</math> induces a map <math>\Delta^* \colon H^\bullet(X \times X) \to H^\bullet(X)</math> but would also induce a map <math>\Delta_* \colon H_\bullet(X) \to H_\bullet(X \times X)</math>, which goes the wrong way round to allow us to define a product. This is however of use in defining the [[cap product]].

			Bilinearity follows from this presentation of cup product, i.e. <math> (u_1 + u_2) \smile v = u_1 \smile v + u_2 \smile v </math> and <math> u \smile (v_1 + v_2) = u \smile v_1 + u \smile v_2. </math>

			==Examples==
			Cup products may be used to distinguish manifolds from wedges of spaces with identical cohomology groups. The space <math>X:= S^2\vee S^1\vee S^1</math> has the same cohomology groups as the torus ''T'', but with a different cup product. In the case of ''X'' the multiplication of the [[cochain]]s associated to the copies of <math>S^1</math> is degenerate, whereas in ''T'' multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to '''Z''' (more generally ''M'' where this is the base module).

			==Other definitions==

			===Cup product and differential forms===
			In [[de Rham cohomology]], the cup product of differential forms is induced by the [[wedge product]]. In other words, the wedge product of
			two closed differential forms belongs to the de Rham class of the cup product of the two original de Rham classes.

			===Cup product and geometric intersections===
			[[File:Linking Number 1.svg\|thumb\|The [[linking number]] can be defined in terms of a non-vanishing cup product on the complement of a link. The complement of these two linked circles deformation retracts to a torus, which has a non-vanishing cup product.]]
			When two submanifolds of a [[smooth manifold]] intersect [[Transversality (mathematics)\|transversely]], their intersection is again a submanifold. By taking the fundamental homology class of these manifolds, this yields a bilinear product on homology. This product is dual to the cup product, i.e. the homology class of the intersection of two submanifolds is the Poincaré dual of the cup product of their Poincaré duals.

			Similarly, the [[linking number]] can be defined in terms of intersections, shifting dimensions by 1, or alternatively in terms of a non-vanishing cup product on the complement of a link.

			==Massey products==
			[[File:BorromeanRings.svg\|thumb\|[[Massey product]]s generalize cup product, allowing one to define "higher order linking numbers", the [[Milnor invariants]].]]
			{{main\|Massey product}}
			The cup product is a binary (2-ary) operation; one can define a ternary (3-ary) and higher order operation called the [[Massey product]], which generalizes the cup product. This is a higher order [[cohomology operation]], which is only partly defined (only defined for some triples).

			==See also==
			*[[singular homology]]
			*[[homology theory]]
			*[[cap product]]
			*[[Massey product]]

			==References==
			* James R. Munkres, "Elements of Algebraic Topology", Perseus Publishing, Cambridge Massachusetts (1984) ISBN 0-201-04586-9 (hardcover) ISBN 0-201-62728-0 (paperback)
			* [[Glen E. Bredon]], "Topology and Geometry", Springer-Verlag, New York (1993) ISBN 0-387-97926-3
			* Allen Hatcher, "[http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology]", Cambridge Publishing Company (2002) ISBN 0-521-79540-0

			[[Category:Homology theory]]
			[[Category:Algebraic topology]]
			[[Category:Binary operations]]

en>TomHennell: /* Greek text */ tense

2012-08-02T15:38:01Z

Greek text: tense

New page

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