Infinite symmetric product
Encyclopedia
In algebraic topology
, 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,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−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 infinite symmetric product appears in the Dold–Thom theorem
.
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, the infinite symmetric product SP(X) of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn) and identifying any point with any other point given by permuting
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
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
James reduced product
In topology, a branch of mathematics, the James reduced product J 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 with...
.
The infinite symmetric product appears in the Dold–Thom theorem
Dold–Thom theorem
In algebraic topology, the Dold–Thom theorem, proved by , states that the homotopy group πi of the infinite symmetric product SP of X is the i-th singular reduced homology group of "X", usually denoted by Hi with an added ~ on the H....
.