Dold–Thom theorem

The most common version of its proof consists of showing that the composition of the homotopy group functors with the infinite symmetric product defines a reduced homology theory.

It is also very useful that there exists an isomorphism φ : πnSP(X) → H̃n(X) which is compatible with the Hurewicz homomorphism h: πn(X) → H̃n(X), meaning that one has a commutative diagram where i* is the map induced by the inclusion i: X = SP1(X) → SP(X).

One wants to show that the family of functors hn = πn ∘ SP defines a homology theory.

Dold and Thom chose in their initial proof a slight modification of the Eilenberg-Steenrod axioms, namely calling a family of functors (h̃n)n∈N0 from the category of basepointed, connected CW complexes to the category of abelian groups a reduced homology theory if they satisfy One can show that for a reduced homology theory (h̃n)n∈N0 there is a natural isomorphism h̃n(X) ≅ H̃n(X; G) with G = h̃1(S1).

In order to verify the compatibility with the Hurewicz homomorphism, it suffices to show that the statement holds for X = Sn.

However, by using the suspension isomorphisms for homotopy respectively homology groups, the task reduces to showing the assertion for S1.

One direct consequence of the Dold-Thom theorem is a new way to derive the Mayer-Vietoris sequence.

One gets the result by first forming the homotopy pushout square of the inclusions of the intersection A ∩ B of two subspaces A, B ⊂ X into A and B themselves.

It basically predicates the following: Note that SP(Y) has this property for every connected CW complex Y and that it therefore has the weak homotopy type of a generalised Eilenberg-MacLane space.

The theorem amounts to saying that all k-invariants of a path-connected, commutative and associative H-space with strict unit vanish.

Then the special H-space structure of X yields a map given by summing up the images of the coordinates.

But as there are natural homeomorphisms with Π denoting the weak product, f induces isomorphisms on πn for n ≥ 2.