Dold–Thom theorem
Encyclopedia
In algebraic topology
, the Dold–Thom theorem, proved by , states that the homotopy group
πi(SP(X)) of the infinite symmetric product
SP(X) of X is the i-th singular reduced homology
group of "X", usually denoted by Hi(X,Z) with an added ~ (tilde) on the H.
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 Dold–Thom theorem, proved by , states that the homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
πi(SP(X)) of the infinite symmetric product
Infinite symmetric product
In algebraic topology, the infinite symmetric product SP of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, .....
SP(X) of X is the i-th singular reduced homology
Reduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero...
group of "X", usually denoted by Hi(X,Z) with an added ~ (tilde) on the H.