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In mathematics, the Pontryagin product, introduced by Lev Pontryagin (1939), is a product on the homology of a topological space induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an abelian group, the Pontryagin product on an H-space, and the Pontryagin product on a loop space.

Cross product

In order to define the Pontryagin product we first need a map which sends the direct product of the m-th and n-th homology group to the (m+n)-th homology group of a space. We therefore define the cross product, starting on the level of singular chains. Given two topological spaces X and Y and two singular simplices $f:\Delta ^{m}\to X$ and $g:\Delta ^{n}\to Y$ we can define the product map $f\times g:\Delta ^{m}\times \Delta ^{n}\to X\times Y$ , the only difficulty is showing that this defines a singular (m+n)-simplex in $X\times Y$ . To do this one can subdivide $\Delta ^{m}\times \Delta ^{n}$ into (m+n)-simplices. It is then easy to show that this map induces a map on homology of the form

H_{m}(X;R)\otimes H_{n}(Y;R)\to H_{m+n}(X\times Y;R)

by proving that if $f$ and $g$ are cycles then so is $f\times g$ and if either $f$ or $g$ is a boundary then so is the product.

Definition

Given an H-space $X$ with multiplication $\mu :X\times X\to X$ , the Pontryagin product on homology is defined by the following composition of maps

H_{*}(X;R)\otimes H_{*}(X;R){\xrightarrow[{}]{\times }}H_{*}(X\times X;R){\xrightarrow[{}]{\mu _{*}}}H_{*}(X;R)

where the first map is the cross product defined above and the second map is given by the multiplication $X\times X\to X$ of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology. Then $H_{*}(X;R)=\bigoplus _{n=0}^{\infty }H_{n}(X;R)$ .

References

Brown, Kenneth S. (1982). Cohomology of groups. Graduate Texts in Mathematics. Vol. 87. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90688-1. MR 0672956.
Pontryagin, Lev (1939). "Homologies in compact Lie groups". Recueil Mathématique (Matematicheskii Sbornik). New Series. 6 (48): 389–422. MR 0001563.
Hatcher, Hatcher (2001). Algebraic topology. Cambridge: Cambridge University Press. ISBN 978-0-521-79160-1.

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-In [[mathematics]], the '''Pontryagin product''', introduced by {{harvs|txt|last=Pontryagin|first=Lev|authorlink=Lev Pontryagin|year=1939}}, is a product on the homology of a [[topological space]] induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an [[abelian group]], the Pontryagin product on an [[H-space]], and the Pontryagin product on a [[loop space]].
+{{Short description|Product on the homology of a topological space induced by a product on the topological space}}
+In [[mathematics]], the '''Pontryagin product''', introduced by {{harvs|txt|last=Pontryagin|first=Lev|authorlink=Lev Pontryagin|year=1939}}, is a product on the [[Homology (mathematics)|homology]] of a [[topological space]] induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an [[abelian group]], the Pontryagin product on an [[H-space]], and the Pontryagin product on a [[loop space]].
-==Cross Poduct==
+==Cross product==
-In order to define the Pontryagin product we first need a map which sends the direct product of the m-th and n-th homology group to the (m+n)-th homology group of a space. We therefore define the cross product, starting on the level of singular chains. Given two topological spaces X and Y and two singular simplices <math>f:\Delta^m\to X</math> and <math>g:\Delta^n\to Y</math> we can define the product map <math>f\times g:\Delta^m\times\Delta^n\to X\times Y</math>, the only difficulty is showing that this defines a singular (m+n)-simplex in <math> X\times Y</math>. To do this one can subdivide <math>\Delta^m\times\Delta^n</math> into (m+n)-simplices. It is then easy to show that this map induces a map on homology of the form
+In order to define the Pontryagin product we first need a map which sends the direct product of the m-th and n-th homology group to the (m+n)-th homology group of a space. We therefore define the cross product, starting on the level of [[singular chain]]s. Given two topological spaces X and Y and two [[singular simplex|singular simplices]] <math>f:\Delta^m\to X</math> and <math>g:\Delta^n\to Y</math> we can define the product map <math>f\times g:\Delta^m\times\Delta^n\to X\times Y</math>, the only difficulty is showing that this defines a singular (m+n)-simplex in <math> X\times Y</math>. To do this one can subdivide <math>\Delta^m\times\Delta^n</math> into (m+n)-simplices. It is then easy to show that this map induces a map on homology of the form
-:<math> H_m(X;R)\otimes H_n(X;R)\to H_{m+k}(X\times Y;R)</math>
+:<math> H_m(X;R)\otimes H_n(Y;R)\to H_{m+n}(X\times Y;R)</math>
 by proving that if <math>f</math> and <math>g</math> are cycles then so is <math>f\times g</math> and if either <math>f</math> or <math>g</math> is a boundary then so is the product.
 ==Definition==
-Given an [[H-space]] <math>X</math> with multiplication <math>\mu:X\times X\to X</math> we define the '''Pontryagin product''' on Homology by the following composition of maps
+Given an [[H-space]] <math>X</math> with multiplication <math>\mu:X\times X\to X</math>, the '''Pontryagin product''' on homology is defined by the following composition of maps
 :<math> H_*(X;R)\otimes H_*(X;R)\xrightarrow[]{\times} H_*(X\times X;R) \xrightarrow[]{\mu_*} H_*(X;R) </math>
-where the first map is the cross product defined above and the second map is given by the multiplication <math> X\times X\to X</math> of the [[H-space]], and <math> H_*(X;R) = \bigoplus_{n=0}^\infty H_n(X;R)</math>.
+where the first map is the cross product defined above and the second map is given by the multiplication <math> X\times X\to X</math> of the [[H-space]] followed by application of the homology functor to obtain a homomorphism on the level of homology. Then <math> H_*(X;R) = \bigoplus_{n=0}^\infty H_n(X;R)</math>.
 ==References==
+{{More footnotes needed|date=December 2020}}
-*{{Citation | last1=Brown | first1=Kenneth S. | authorlink = Kenneth Brown (mathematician) | title=Cohomology of groups | url=https://books.google.com/books?id=PMqb2DppvCsC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-90688-1 |mr=672956 | year=1982 | volume=87}}
+* {{Cite book| last1=Brown | first1=Kenneth S. | authorlink = Kenneth Brown (mathematician) | title=Cohomology of groups | url=https://books.google.com/books?id=PMqb2DppvCsC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-90688-1 |mr=672956 | year=1982 | volume=87}}
-*{{Citation | last1=Pontrjagin | first1=Lev | authorlink = Lev Pontryagin | title=Homologies in compact Lie groups |mr=0001563 | year=1939 | journal=Recueil Mathématique (Matematicheskii Sbornik) N.S. | volume=6 | issue = 48  | pages=389–422}}
+* {{Cite journal | last1=Pontryagin | first1=Lev | authorlink = Lev Pontryagin | title=Homologies in compact Lie groups |mr=0001563 | year=1939 | journal=Recueil Mathématique (Matematicheskii Sbornik) |series=New Series | volume=6 | issue = 48  | pages=389–422}}
-*[[Allen Hatcher]], [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] Cambridge University Presses, Cambridge, 2001. xii+544 pp. {{isbn|978-0-521-79160-1}} and {{isbn|0-521-79540-0}}
+* {{Cite book | last1=Hatcher | first1= Hatcher | authorlink = Allen Hatcher |url = http://pi.math.cornell.edu/~hatcher/AT/ATpage.html | title = Algebraic topology | publisher = [[Cambridge University Press]] | location = Cambridge | year = 2001 | isbn = 978-0-521-79160-1}}
 [[Category:Homology theory]]