Symmetric product (topology)

If one fixes a basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones.

One of its essential applications is the Dold-Thom theorem, stating that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as the reduced homology groups of that complex.

Moreover, if the category is cartesian closed, the distributive law X × (Y ∐ Z) ≅ X × Y ∐ X × Z holds and therefore one gets If (X, e) is a based space, it is common to set SP0(X) = {e}.

Therefore, the infinite symmetric product can be defined as A definition avoiding category theoretic notions can be given by taking SP(X) to be the union of the increasing sequence of spaces SPn(X) equipped with the direct limit topology.

The identifications on the other concentric copies of the boundary yield the quotient space Z of Dn × ∂Dn, obtained by identifying (x, y) with (y, x) whenever both coordinates lie in ∂Dn.

If x is the midpoint of the segment zz′, there is no way to distinguish between z and w, but this is not a problem since f takes values in the quotient space Z.

This means that SP(X) is the quotient of J(X) obtained by identifying points that differ only by a permutation of coordinates.

As such, the Dold-Thom theorem implies that all its k-invariants vanish, meaning that it has the weak homotopy type of a generalised Eilenberg-MacLane space if X is path-connected.

Furthermore, SPn(X) is also a subsimplex of SPn+1(X) if the basepoint e ∈ X is a vertex, meaning that SP(X) inherits a simplicial structure in this case as well.

[9] So the distinction between the two topologies will not cause any differences for purposes of homotopy, e.g. One of the main uses of infinite symmetric products is the Dold-Thom theorem.

It has also been studied how other constructions combined with the infinite symmetric product affect the homotopy groups.

[11] For a simplicial set K, one has furthermore Passing to geometric realisations, one sees that this statement holds for connected CW complexes as well.

In order to equip this group with a topology, Dold and Thom initially introduced it as the following quotient over the infinite symmetric product of the wedge sum of X with a copy of itself: Let τ : X ∨ X → X ∨ X be interchanging the summands.

McCord gave a construction generalising both SP(X) and Z[X]: Let G be a monoid with identity element 1 and let (X, e) be a pointed set.

Define Bn(G, X) to be the subset of B(G, X) consisting of all maps that differ from the constant function 1 at no more than n points.

[16] Then B(G, X) can be equipped with the direct limit topology, making it again a compactly generated space.

As in the preceding cases, one sees that a based homotopy ft : X → Y induces a homotopy B(Id, ft) : B(G, X) → B(G, Y) for an abelian topological monoid G. Using this construction, the Dold-Thom theorem can be generalised.

Namely, for a discrete module M over a commutative ring with unit one has for based spaces X and Y having the homotopy type of a CW complex.

[18] Here, H̃n denotes reduced homology and [X, Z] stands for the set of all based homotopy classes of basepoint-preserving maps X → Z.

It is noteworthy as well that B(G, S1) is a classifying space for G if G is a topological group such that the inclusion {1} → G is a cofibration.