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| 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.
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| ==Definition==
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| 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''.
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| The construction starts with a product of [[Cochain (algebraic topology)|cochain]]s: if ''c''<sup>''p''</sup> is a ''p''-cochain and
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| ''d''<sup>''q''</sup> is a ''q''-cochain, then
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| :<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>
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| where σ is a [[Singular homology|singular]] (''p'' + ''q'') -[[simplex]] and <math>\iota_S , S \subset \{0,1,...,p+q \} </math>
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| 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>. | |
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| 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.
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| The [[coboundary]] of the cup product of cocycles c<sup>''p''</sup> and d<sup>''q''</sup> is given by
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| :<math>\delta(c^p \smile d^q) = \delta{c^p} \smile d^q + (-1)^p(c^p \smile \delta{d^q}).</math>
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| 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
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| : <math> H^p(X) \times H^q(X) \to H^{p+q}(X). </math>
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| ==Properties==
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| The cup product operation in cohomology satisfies the identity
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| :<math>\alpha^p \smile \beta^q = (-1)^{pq}(\beta^q \smile \alpha^p)</math>
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| so that the corresponding multiplication is [[supercommutative|graded-commutative]].
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| The cup product is [[functor]]ial, in the following sense: if
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| :<math>f\colon X\to Y</math>
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| is a continuous function, and
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| :<math>f^*\colon H^*(Y)\to H^*(X)</math>
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| is the induced [[homomorphism]] in cohomology, then
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| :<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]]. | |
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| ==Interpretation==
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| 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:
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| <math> \displaystyle C^\bullet(X) \times C^\bullet(X) \to C^\bullet(X \times X) \overset{\Delta^*}{\to} C^\bullet(X) </math>
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| 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>.
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| 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]].
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| 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>
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| ==Examples==
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| 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).
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| ==Other definitions==
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| ===Cup product and differential forms===
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| In [[de Rham cohomology]], the cup product of differential forms is induced by the [[wedge product]]. In other words, the wedge product of
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| two closed differential forms belongs to the de Rham class of the cup product of the two original de Rham classes.
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| ===Cup product and geometric intersections===
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| [[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.]]
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| 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. | |
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| 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.
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| ==Massey products==
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| [[File:BorromeanRings.svg|thumb|[[Massey product]]s generalize cup product, allowing one to define "higher order linking numbers", the [[Milnor invariants]].]]
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| {{main|Massey product}}
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| 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).
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| ==See also==
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| *[[singular homology]]
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| *[[homology theory]]
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| *[[cap product]]
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| *[[Massey product]]
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| ==References==
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| * James R. Munkres, "Elements of Algebraic Topology", Perseus Publishing, Cambridge Massachusetts (1984) ISBN 0-201-04586-9 (hardcover) ISBN 0-201-62728-0 (paperback)
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| * [[Glen E. Bredon]], "Topology and Geometry", Springer-Verlag, New York (1993) ISBN 0-387-97926-3
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| * Allen Hatcher, "[http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology]", Cambridge Publishing Company (2002) ISBN 0-521-79540-0
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| [[Category:Homology theory]]
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| [[Category:Algebraic topology]]
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| [[Category:Binary operations]]
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"After 7 years of trying other methods, I finally got pregnant, 4 weeks after applying these holistic methods at the age of 44, thank you". Having a deep posture provides the male sperm a quicker and easier access to an egg. s body approaches the time when she is most fertile to conceive naturally, the amount of cervical mucous discharge by the vagina increases. This will assist you time your fertile period far better to ensure that it is possible to boost your chances of conceiving naturally. If you've searched for them in i - Tunes then you know what I'm talking about.
Incorporate nutritious foods into your meal plan and begin exercising to get your body in shape. Some women will run out of 'good eggs' sooner than others will, and there is not really a remedy for that. Although, there is no basis to this theory, and little is known what causes the issue to begin with. Most of these the application is limited and therefore are of very few consciousness, they certainly usually do not teach the teens generation concerning the risk-free really like process, risk-free habits and what's more the employing of the contraceptives. helps you track your period and find out when it is due and so forth.
Becoming in tune with the body's discreet signs that it transmits to you every month throughout the monthly period and fertile periods is one good way of boosting your possibilities of victory when trying to get pregnant. But, I can tell you that many probably do struggle for money, and many will tell you themselves that they wish they would have waited to have a baby. Â Choose the best vitamins, Fertil - Aid for Women: Female Fertility Supplement">herbs and supplements that work for you in order to achieve your goal. Getting pregnant is possible and many women have successfully brought a baby into this world. Alcohol can reduce fertility by as much as 50% and decrease sperm counts.
The very same variables that control the birth of a male also help to encourage the birth of a female, only the positions and conditions are altered. Sexually transmitted diseases causes of infertility. Experts suggest that instead of focusing all of your sexual energy during one week of the month, spread the wealth and have sex several times every week. Stay away from sodas, marijuana, cocaine and so on. A healthy diet to improve fertility is basically to consuming a well balanced diet of carbohydrates, protein and fats.
The good news is that by reducing your exposure you'll actually get pregnant faster and easier. Folic acid supplements are some of the best supplements that you can take when trying to conceive. When you are bearing in mind how you can best get pregnant rapidly, try to bear in mind that pregnancy is awfully tough to control or force. Merely performing intercourse isn't necessarily all there is to getting pregnant. Unlike conventional methods, the answer to the question of how to get pregnant can be painstakingly obtained through correct guidance and holistic therapeutic solutions.
In case you loved this short article and you want to receive much more information regarding quick ways to get pregnant kindly visit the website.