• Source: Pontryagin product

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
:

Δ

m


→
X


{\displaystyle f:\Delta ^{m}\to X}

and



g
:

Δ

n


→
Y


{\displaystyle g:\Delta ^{n}\to Y}

we can define the product map



f
×
g
:

Δ

m


×

Δ

n


→
X
×
Y


{\displaystyle 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
×
Y


{\displaystyle X\times Y}

. To do this one can subdivide




Δ

m


×

Δ

n




{\displaystyle \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
)
⊗

H

n


(
Y
;
R
)
→

H

m
+
n


(
X
×
Y
;
R
)


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


by proving that if



f


{\displaystyle f}

and



g


{\displaystyle g}

are cycles then so is



f
×
g


{\displaystyle f\times g}

and if either



f


{\displaystyle f}

or



g


{\displaystyle g}

is a boundary then so is the product.




Definition


Given an H-space



X


{\displaystyle X}

with multiplication



μ
:
X
×
X
→
X


{\displaystyle \mu :X\times X\to X}

, the Pontryagin product on homology is defined by the following composition of maps





H

∗


(
X
;
R
)
⊗

H

∗


(
X
;
R
)


→






×




H

∗


(
X
×
X
;
R
)


→







μ

∗






H

∗


(
X
;
R
)


{\displaystyle 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
×
X
→
X


{\displaystyle 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
)
=

⨁

n
=
0


∞



H

n


(
X
;
R
)


{\displaystyle 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.

Kata Kunci Pencarian:

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• Pontryagin duality
• H-space
• Indefinite inner product space
• Thom space
• Inductive dimension
• Classifying space for U(n)
• List of algebraic topology topics
• Duality (mathematics)