Symmetric product (topology)

In algebraic topology, the n^th symmetric product of a topological space consists of the unordered n-tuples of its elements. If one fixes a basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the colimit over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor.

From an algebraic point of view, the infinite symmetric product is the free commutative monoid generated by the space minus the basepoint, the basepoint yielding the identity element. That way, one can view it as the abelian version of the James reduced product.

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. That way, one can give a homotopical definition of homology.

Let X be a topological space and n ≥ 1 a natural number. Define the n^th symmetric product of X or the n-fold symmetric product of X as the space

${\displaystyle \operatorname {SP} ^{n}(X)=X^{n}/S_{n}.}$

Here, the symmetric group S_n acts on Xⁿ by permuting the factors. Hence, the elements of SPⁿ(X) are the unordered n-tuples of elements of X. Write [x₁, ..., x_n] for the point in SPⁿ(X) defined by (x₁, ..., x_n) ∈ Xⁿ.

Note that one can define the n^th symmetric product in any category where products and colimits exist. Namely, one then has canonical isomorphisms φ : X × Y → Y × X for any objects X and Y and can define the action of the transposition ${\displaystyle (k\ k+1)\in S_{n}}$ on Xⁿ as ${\displaystyle \operatorname {Id} ^{k-1}\times \phi \times \operatorname {Id} ^{n-k-1}}$, thereby inducing an action of the whole S_n on Xⁿ. This means that one can consider symmetric products of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive law X × (Y ∐ Z) ≅ X × Y ∐ X × Z holds and therefore one gets

${\displaystyle \operatorname {SP} ^{n}(X\amalg Y)=\coprod _{k=0}^{n}\operatorname {SP} ^{k}(X)\times \operatorname {SP} ^{n-k}(Y).}$

If (X, e) is a based space, it is common to set SP⁰(X) = {e}. Further, Xⁿ can then be embedded into Xⁿ⁺¹ by sending (x₁, ..., x_n) to (x₁, ..., x_n, e). This clearly induces an embedding of SPⁿ(X) into SPⁿ⁺¹(X). Therefore, the infinite symmetric product can be defined as

${\displaystyle \operatorname {SP} (X)=\operatorname {colim} \operatorname {SP} ^{n}(X).}$

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. This means that a subset of SP(X) is open if and only if all its intersections with the SPⁿ(X) are open. We define the basepoint of SP(X) as [e]. That way, SP(X) becomes a based space as well.

One can generalise this definition as well to pointed categories where products and colimits exist. Namely, in this case one has a canonical map Xⁿ → Xⁿ⁺¹, induced by the identity Xⁿ → Xⁿ and the zero map Xⁿ → X. So this results in a direct system of the symmetric products, too and one can therefore define its colimit as the infinite symmetric product.

• SPⁿ(I) is the same as the n-dimensional standard simplex Δⁿ, where I denotes the unit interval.
• SPⁿ(S¹) can be identified with the space of conjugacy classes of unitary n × n-matrices, where S¹ is supposed to be the circle. This is because such a class is uniquely determined by the eigenvalues of an element of the class, all lying in S¹. At first, one can easily see that this space is homotopy-equivalent to S¹: As SPⁿ is a homotopy functor (see Properties), the space in question is homotopy-equivalent to SPⁿ(C − {0}). Consider the map SPⁿ(C − {0}) → P_n into the space P_n of polynomials over C of degree at most n, mapping [w₁, ..., w_n] to (z - w₁) ⋅⋅⋅ (z - w_n). This way, one can identify SPⁿ(C − {0}) with the space of monic polynomials of degree n having constant term different from zero, i.e. C^{n − 1} × (C − {0}), which is homotopy-equivalent to S¹. This implies that the infinite symmetric product SP(S¹) is homotopy-equivalent to S¹ as well. However, one knows considerably more about the space SPⁿ(S¹). Namely, that the map
  ${\displaystyle {\begin{aligned}\qquad \operatorname {SP} ^{n}(S^{1})&\to S^{1},\\{\color {white}.}[w_{1},\dots ,w_{n}]&\mapsto w_{1}\cdots w_{n}\end{aligned}}}$
  is a fibre bundle with fibre being homeomorphic to the (n − 1)-dimensional standard simplex ∆ⁿ⁻¹. It is orientable if and only if n is odd.^[1][2]
• SP(S²) is homeomorphic to the infinite-dimensional complex projective space CP^∞ as follows: The space CPⁿ can be identified with the space of nonzero polynomials of degree at most n over C up to scalar multiplication by sending a₀ + ... + a_nzⁿ to the line passing through (a₀, ..., a_n). Interpreting S² as the Riemann sphere C ∪ {∞} yields a map
  ${\displaystyle {\begin{aligned}\qquad f\colon (S^{2})^{n}&\to \mathbf {CP} ^{n},\\(a_{1},\dots ,a_{n})&\mapsto (z+a_{1})\cdots (z+a_{n}),\end{aligned}}}$
  where the possible factors z + ∞ are omitted. One can check that this map indeed is continuous.^[3] As f(a₁, ..., a_n) remains unchanged under permutation of the a_i's, f induces a continuous bijection SPⁿ(S²) → CPⁿ. But as both are compact Hausdorff spaces, this map is a homeomorphism. Letting n go to infinity shows that the assertion holds.

Although calculating SP(Sⁿ) for n ≥ 3 turns out to be quite difficult, one can still describe SP²(Sⁿ) quite well as the mapping cone of a map ΣⁿRP^n-1 → Sⁿ, where Σⁿ stands for applying the reduced suspension n times and RPⁿ⁻¹ is the (n − 1)-dimensional real projective space: One can view SP²(Sⁿ) as a certain quotient of Dⁿ × Dⁿ by identifying Sⁿ with Dⁿ/∂Dⁿ. Interpreting Dⁿ × Dⁿ as the cone on its boundary Dⁿ × ∂Dⁿ ∪ ∂Dⁿ × Dⁿ, the identifications for SP² respect the concentric copies of the boundary. Hence, it suffices to only consider these. The identifications on the boundary ∂Dⁿ × Dⁿ ∪ Dⁿ × ∂Dⁿ of Dⁿ × Dⁿ itself yield Sⁿ. This is clear as this is a quotient of Dⁿ × ∂Dⁿ and as ∂Dⁿ is collapsed to one point in Sⁿ. 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. Define a map f: Dⁿ × RPⁿ⁻¹ → Z by sending a pair (x, L) to (w, z). Here, z ∈ ∂Dⁿ and w ∈ Dⁿ are chosen on the line through x parallel to L such that x is their midpoint. 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. Therefore, f is well-defined. As f(x, L) = f(x, L′) holds for every x ∈ ∂Dⁿ, f factors through ΣⁿRPⁿ⁻¹ and is easily seen to be a homeomorphism on this domain.

As SP(X) is the free commutative monoid generated by X − {e} with identity element e, it can be thought of as a commutative analogue of the James reduced product J(X). This means that SP(X) is the quotient of J(X) obtained by identifying points that differ only by a permutation of coordinates. Therefore, the H-space structure on J(X) induces one on SP(X) if X is a CW complex, making it a commutative and associative H-space with strict identity. 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.^[4] However, if X is an arbitrary space, the multiplication on SP(X) may fail to be continuous.^[5]

SPⁿ is a homotopy functor: A map f: X → Y clearly induces a map SPⁿ(f) : SPⁿ(X) → SPⁿ(Y) given by SPⁿ(f)[x₁, ..., x_n] = [f(x₁), ..., f(x_n)]. A homotopy between two maps f, g: X → Y yields one between SPⁿ(f) and SPⁿ(g). Also, one can easily see that the diagram

commutes, meaning that SP is a functor as well. Similarly, SP is even a homotopy functor on the category of pointed spaces and basepoint-preserving homotopy classes of maps. In particular, X ≃ Y implies SPⁿ(X) ≃ SPⁿ(Y), but in general not SP(X) ≃ SP(Y) as homotopy equivalence may be affected by requiring maps and homotopies to be basepoint-preserving. However, this is not the case if one requires X and Y to be connected CW complexes.^[6]

SP(X) inherits certain structures of X: For a simplicial complex X, one can also install a simplicial structure on Xⁿ such that each n-permutation is either the identity on a simplex or a homeomorphism from one simplex to another. This means that one gets a simplicial structure on SPⁿ(X). 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.^[7] However, one should note that Xⁿ and SPⁿ(X) do not need to have the weak topology if X has uncountably many simplices.^[8] An analogous statement can be made if X is a CW complex. Nevertheless, it is still possible to equip SP(X) with the structure of a CW complex such that both topologies have the same compact sets if X is an arbitrary simplicial complex.^[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 states that the reduced homology groups coincide with the homotopy groups of the infinite symmetric product of a connected CW complex. This allows one to reformulate homology only using homotopy which can be very helpful in algebraic geometry. It also means that the functor SP maps Moore spaces M(G, n) to Eilenberg-MacLane spaces K(G, n). Therefore, it yields a natural way to construct the latter spaces given the proper Moore spaces.

It has also been studied how other constructions combined with the infinite symmetric product affect the homotopy groups. For example, it has been shown that the map

${\displaystyle \rho \colon \operatorname {SP} (X)\to \Omega \operatorname {SP} (\Sigma X),\quad \rho [x_{1},\dots ,x_{n}](t)=[(x_{1},t),\dots ,(x_{n},t)]}$

is a weak homotopy equivalence, where ΣX = X ∧ S¹ denotes the reduced suspension and ΩY stands for the loop space of the pointed space Y.^[10]

Unsurprisingly, the homology groups of the symmetric product cannot be described as easily as the homotopy groups. Nevertheless, it is known that the homology groups of the symmetric product of a CW complex are determined by the homology groups of the complex. More precisely, if X and Y are CW complexes and R is a principal ideal domain such that H_i(X, R) ≅ H_i(Y, R) for all i ≤ k, then H_i(SPⁿ(X), R) ≅ H_i(SPⁿ(Y), R) holds as well for all i ≤ k. This can be generalised to Γ-products, defined in the next section.^[11]

For a simplicial set K, one has furthermore

${\displaystyle H_{*}(\operatorname {SP} ^{n+1}(K))\cong H_{*}(\operatorname {SP} ^{n+1}(K),\operatorname {SP} ^{n}(K))\oplus H_{*}(\operatorname {SP} ^{n}(K)).}$

Passing to geometric realisations, one sees that this statement holds for connected CW complexes as well.^[12] Induction yields furthermore

${\displaystyle H_{*}(\operatorname {SP} (K))\cong \bigoplus _{n=1}^{\infty }H_{*}(\operatorname {SP} ^{n}(K),\operatorname {SP} ^{n-1}(K)).}$^[13]

S. Liao introduced a slightly more general version of symmetric products, called Γ-products for a subgroup Γ of the symmetric group S_n.^[14] The operation was the same and hence he defined X^Γ = Xⁿ/Γ as the Γ-product of X. That allowed him to study cyclic products, the special case for Γ being the cyclic group, as well.

When establishing the Dold-Thom theorem, they also considered the "quotient group" Z[X] of SP(X). This is the free abelian group over X with the basepoint as the zero element. If X is a CW complex, it is even a topological group. 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. Furthermore, let ~ be the equivalence relation on SP(X ∨ X) generated by

${\displaystyle x\sim x+y+\operatorname {SP} (\tau )(y)}$

for x, y ∈ SP(X ∨ X). Then one can define Z[X] as

${\displaystyle \mathbb {Z} [X]=\operatorname {SP} (X\vee X)/\sim .}$

Since ~ is compatible with the addition in SP(X ∨ X), one gets an associative and commutative addition on Z[X]. One also has the topological inclusions X ⊂ SP(X) ⊂ Z[X]^[15] and it can easily be seen that this construction has properties similar to the ones of SP, like being a functor.

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

${\displaystyle B(G,X)=\{u\colon X\to G:u(e)=1{\text{ and }}u(x)=1{\text{ for all but finitely many }}x\in X\}.}$

Then B(G, X) is again a monoid under pointwise multiplication which will be denoted by ⋅. Let gx denote the element of B(G, X) taking the value g at x and being 1 elsewhere for g ∈ G, x ∈ X − {e}. Moreover, ge shall denote the function being 1 everywhere, the unit of B(G, X).

In order to install a topology on B(G, X), one needs to demand that X be compactly generated and that G be an abelian topological monoid. Define B_n(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. B_n(G, X) gets equipped with the final topology of the map

${\displaystyle {\begin{aligned}\mu _{n}\colon (G\times X)^{n}&\to B_{n}(G,X),\\((g_{1},x_{1}),\dots ,(g_{n},x_{n}))&\mapsto g_{1}x_{1}\cdots g_{n}x_{n}.\end{aligned}}}$

Now, B_n(G, X) is a closed subset of B_n+1(G, X).^[16] Then B(G, X) can be equipped with the direct limit topology, making it again a compactly generated space. One can then identify SP(X) respectively Z[X] with B(N, X) respectively B(Z, X).

Moreover, B(⋅,⋅) is functorial in the sense that B: C × D → C is a bifunctor for C being the category of abelian topological monoids and D being the category of pointed CW complexes.^[17] Here, the map B(φ, f) : B(G, X) → B(H, Y) for a morphism φ: G → H of abelian topological monoids and a continuous map f: X → Y is defined as

${\displaystyle B(\varphi ,f)(g_{1}x_{1}\cdots g_{n}x_{n})=(\varphi g_{1})(fx_{1})\cdots (\varphi g_{n})(fx_{n})}$

for all g_i ∈ G and x_i ∈ X. As in the preceding cases, one sees that a based homotopy f_t : X → Y induces a homotopy B(Id, f_t) : 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

${\displaystyle [X,B(M,Y)]\cong \prod _{n=0}^{\infty }{\tilde {H}}^{n}(X,{\tilde {H}}_{n}(Y,M))}$

for based spaces X and Y having the homotopy type of a CW complex.^[18] Here, H̃_n denotes reduced homology and [X, Z] stands for the set of all based homotopy classes of basepoint-preserving maps X → Z. As M is a module, [X, B(M, Y)] has an obvious group structure. Inserting X = Sⁿ and M = Z yields the Dold-Thom theorem for Z[X].

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.^[19]

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• Symmetric product in arbitrary categories? on MathOverflow