Symmetric product (topology)

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 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]

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,...,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.

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

1. "symmetric product of circles in nLab". ncatlab.org. Retrieved 2017-08-23.
2. 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.
3. "Dold-Thom theorem in nLab". ncatlab.org. Retrieved 2017-08-23.