User:Marsupilamov/Dold-Thom theorem

In topology, a branch of mathematics,

Let (X,e) be a pointed topological space.

Reduced product of a space
The James reduced product J(X) is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk&minus;1,e,xk+1,...,xn) with (x1,...,xk&minus;1, xk+1,...,xn). In other words, its underlying set is the free monoid generated by X (with unit e). It was introduced by.

For a connected CW complex X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X.

Infinite symmetric product
The commutative analogue of the James reduced product is called the infinite symmetric product.

Definition
The infinite symmetric product SP(X) is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk&minus;1,e,xk+1,...,xn) with (x1,...,xk&minus;1, xk+1,...,xn) 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.

The Dold-Thom theorem
The infinite symmetric product appears in the Dold–Thom theorem, proved by. It states that the homotopy group πi(SP(X)) of the infinite symmetric product SP(X) of X is the homology Hi(X,Z) of the singular complex of the suspension of X.