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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 | |
| ''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 | |
| :<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>
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| 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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There is however a trick that you must stay with if you want success. Studies conducted over the years have helped women overcome infertility. If probable, wait to have sexual intercourse until finally you are entirely performed taking your medication. You could be the next and is definitely worth your try. The somewhat controversial study used uterine cells from both groups of women.
There is always the chance of getting pregnant during unprotected sex, no matter what time of the month it is. Sims are able to have sex with members of the opposite sex as well as members of the same sex. Even though it sounds crazy, the best option would be to give teens some form of birth control and actually allow them to have sex in their house. If you think you can't have a baby, it is possible that your body will act that way. If not every day; even every other day will help because sperm can survive inside you for up to five days in readiness for when that egg decides to appear.
In fact, it can be used as a remedy for curing quite a few other ailments and it is a superior resource of nutrition. For successful attempts at conception, your natural flow of energy must be in balance through full function of your kidneys, liver and spleen. Studies show that people who do these things have easier pregnancies and healthier babies. s sad that I have to mention this but some people need to hear it. This app is great for expecting mothers as well as husbands, mother in laws, aunts, and any other family member or friends who are with you during those labor contractions.
In IVF, sperm and eggs are combined in a petri dish and then the resulting embryos are either transferred into the uterus or frozen for future use. I hope you found the above information on how to get pregnant with twins naturally to be helpful. Learn about how your body works, about what works in your favor and how to read your fertile signs. I hope that you have found these tips on getting pregnant:now. Herbs can help detoxify the liver, which is important to insulin regulation, and stimulate ovary functions.
Howewer there is an amazing , fast , in a period of 2 months, please visit this link to know it :. In the average 28 day cycle (the first day of menstruation being day one) ovulation takes place around day 13 to Day 15, its the best time to get pregnant. In this position, the gravity will aid the sperm in making its way to the fallopian tube to meet the ovary. That is why it is vital that couples do things to mitigate stress. Since every female is different from one another, this includes her hormone balance which in turn affects her period cycles, there is no exact science saying when a woman is fertile and when one is not.
If you have any kind of concerns pertaining to where and how to make use of what are the best positions to get pregnant, you could call us at our website.