In algebraic topology, the symmetric product of a topological space X consists of unordered n-tuples of distinct points in X. The infinite symmetric product is the colimit of this process, and appears in the Dold–Thom theorem.
Definition
For a topological space X, the nth 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]
Infinite symmetric product
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,...,xn) with (x1,...,xn,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.
Category-theoretic definition
The infinite symmetric product is also defined as the colimit
\( {\displaystyle \operatorname {SP} ^{\infty }(X):=\operatorname {colim} \operatorname {SP} ^{n}(X).} \) [3]
References
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
Specific
"symmetric product of circles in nLab". ncatlab.org. Retrieved 2017-08-23.
Blagojevic, Pavle; Grujic, Vladimir; Zivaljevic, Rade (2004-08-30). B. Dragovic; B. Sazdovic (eds.). Symmetric products of surfaces; a unifying theme for topology and physics. Proceedings of Summer School in Modern Mathematical Physics. SFIN XV (A3). 3. Institute of Physics, Belgrade. arXiv:math/0408417. Bibcode:2004math......8417B.
"Dold-Thom theorem in nLab". ncatlab.org. Retrieved 2017-08-23.
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