Symmetric product (topology)

For a topological space X, the n^th symmetric product of X is the space

${\displaystyle \operatorname {SP} ^{n}(X):=X^{n}/S_{n},}$

that is, the orbit space given by the quotient of the n-fold product of X by the natural action of the symmetric group defined by

${\displaystyle {\begin{aligned}S_{n}\times X^{n}&\longrightarrow X^{n}\\(\sigma ,x)&\longmapsto \sigma (x).\end{aligned}}}$[1][2]

The infinite symmetric product SP(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X², X³, ... obtained by identifying points (x₁,...,x_n) with (x₁,...,x_n,e) and identifying any point with any other point given by permuting its coordinates. In other words its underlying set is the free commutative monoid generated by X (with unit e), and is the abelianization of the James reduced product.

• Dold, Albrecht; Thom, René (1956), "Une généralisation de la notion d'espace fibré. Application aux produits symétriques infinis", Les Comptes rendus de l'Académie des sciences, 242: 1680–1682, MR 0077121
• Dold, Albrecht; Thom, René (1958), "Quasifaserungen und unendliche symmetrische Produkte", Annals of Mathematics, Second Series, 67: 239–281, doi:10.2307/1970005, ISSN 0003-486X, JSTOR 1970005, MR 0097062

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