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Infinite Symmetric Products

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 in 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 where the basepoint yields 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. That way, one can give a homotopical definition of homology.

Definition

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

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

Here, the symmetric group S_n acts on Xⁿ by permuting the factors. Hence the elements of SPⁿ(X) are the unordered n-tuples. 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 direct limits exist. Namely, one then has isomorphisms φ : X × Y → Y × X for any objects X and Y and can define the action of the transposition $(k\ k+1)\in S_{n}$ on Xⁿ as $\operatorname {Id} ^{k-1}\times \phi \times \operatorname {Id} ^{n-k-1}$, 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

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

If (X, e) is a based space, it is common to set SP⁰(X) = {e}. Further, Xⁿ can be embedded in 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

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

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 iff all its intersections with the SPⁿ(X) are open. We define the base point of SP(X) as [e]. Hence SP(X) is again a based space.

Examples

• SPⁿ(I) equals Δⁿ, the n-dimensional standard simplex, where I is the unit interval.
• SP(S²) is homeomorphic to the infinite-dimensional complex projective space CP^∞ as follows: The space CPⁿ can be identified with the nonzero polynomials over the Riemann sphere C ∪ {∞} of degree at most n up to scalar multiplication by sending a₀ + ... + a_nzⁿ to the line passing through (a₀, ..., a_n). Interpreting S² as C ∪ {∞} yields a map
  ${\begin{aligned}\qquad f\colon (S^{2})^{n}&\to \mathbb {C} \mathbb {P} ^{n},\\(a_{1},\dots ,a_{n})&\mapsto (z+a_{1})\cdots (z+a_{n}),\end{aligned}}$
  where the possible factors z + ∞ are omitted. One can check that this map indeed is continuous. As f(a₁, ..., a_n) stays unchanged under permutation of the a_i's, f gives 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.
• SP(S¹) is homotopy-equivalent to S¹: As in the previous example, one sees that SP(C - {0}) can be identified with the polynomials of the form (z + a₁) ⋅⋅⋅ (z + a_n) where all a_i ∈ C are nonzero. But this means that both the leading coefficient and the constant term are nonzero as well. So SPⁿ(S¹) is homotopy equivalent to CPⁿ minus the hyperplanes a₀ ≠ 0 and a_n ≠ 0 as SP is a homotopy functor. But this is again homotopic to S¹.

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 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.

Properties

H-Space strucutre

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) 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 product of Eilenberg-MacLane spaces if X is path-connected.

Functioriality

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 based 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.

Simplicial and CW structure

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 e ∈ X is a vertex, meaning that in this case SP(X) inherits a simplicial structure. However, one should note that Xⁿ and SPⁿ(X) do not need to have the weak topology if X has uncountably many simplices. 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, with the desired topology if X is an arbitrary simplicial complex.

Homotopy

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 applied to the infinite symmetric product affect the homotopy groups. For example, it has been shown that the map

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

is a weak homotopy equivalence, where ΣX = X ∧ S¹ denotes the reduced suspension and Ω stands for the loop the space.

Homology

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.

For a connected space X, one has furthermore

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

Induction then yields

$H_{*}(\operatorname {SP} (X))\cong \bigoplus _{n=1}^{\infty }H_{*}(\operatorname {SP} ^{n}(X),\operatorname {SP} ^{n-1}(X)).$

Related Constructions and Generalisations

S. Liao was one of the first to work with finite symmetric products. He introduced them in a slightly more general version as Γ-spaces for a subgroup Γ of the symmetric group S_n. 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 topological group over X with the base point as the zero element. In order to equip this group with a topology, Dold and Thom initially introduced it as the following quotient of SP(X ∨ X): Let τ : X ∨ X → X ∨ X be interchanging the summands. Furthermore, let ~ be the equivalence relation on SP(X ∨ X) generated by

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

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

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

As ~ 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] and it can be seen easily that this construction has similar properties as the one 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 based set. Define

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

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 functione e at no more than n points. B_n(G, X) gets equipped with the final topology of the map

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

Now, B_n(G, X) is a closed subset of B_n+1(G, X). 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 based spaces. 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

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

for based spaces X and Y having the homotopy type of a CW complex. Here, H̃_n denotes reduced homology. 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} → X is a cofibration.

Quasifibrations

In algebraic topology, a quasifibration is a generalisation of fibrations and fiber bundles introduced by Albrecht Dold and René Thom. Roughly speaking, it is a continuous map p : E → B between topological spaces such that it has the same behaviour as a fibration regarding the (relative) homotopy groups of E, B and p^-1(x). Equivalently, one can define a quasifibration to be a map between topological spaces such that the inclusion of each fiber into its homotopy fiber is a weak equivalence. One of the main applications of quasifibrations lies in proving the Dold-Thom theorem.

Definition

A continuous surjective map of topological spaces p: E → B is called a quasifibration if it induces isomorphisms

$p_{*}\colon \pi _{i}(E,p^{-1}(x),y)\to \pi _{i}(B,x)$

for all x ∈ B, y ∈ p⁻¹(x) and i ≥ 0. For i = 0,1 one can only speak of bijections between the two sets.

By definition, a quasifibration p: E → B shares a key property of a fibration, namely inducing a long exact sequence of homotopy groups

${\begin{aligned}\dots \to \pi _{i+1}(B,x)\to \pi _{i}(p^{-1}(x),y)\to \pi _{i}(E,y)&\to \pi _{i}(B,x)\to \dots \\&\to \pi _{0}(B,x)\to 0\end{aligned}}$

as follows directly from the long exact sequence for the pair (E, p⁻¹(x)).

This long exact sequence is also functorial in the following sense: Any fibrewise map f : E → E′ induces a morphism between the exact sequences of the pairs (E, p⁻¹(x)) and (E′, p′^-1(x)) and therefore a morphism between the exact sequence for a quasifibration. Hence, the diagram

commutes with f₀ being the restriction of f to p^-1(x) and x′ being an element of the form p′(f(e)) for an e ∈ p^-1(x).

An equivalent definition is saying that a surjective map p : E → B is a quasifibration if the inclusion of the fiber p⁻¹(b) into the homotopy fiber F_b of p over b is a weak equivalence for all b ∈ B. To see this, recall that F_b is the fiber of q under b where q : E_p → B is the usual path fibration construction. Thus, one has

$E_{p}=\{(e,\gamma )\in E\times B^{I}:\gamma (0)=p(e)\}$

and q : E_p → B given by q(e, γ) = γ(1). Now consider the natural homotopy equivalence φ : E → E_p, given by φ(e) = (e, p(e)), where p(e) denotes the corresponding constant path. By definition, p factors through E_p such that one gets a commutative diagram

Applying π_n yields the alternative definition.

Examples

• Every Serre fibration is a quasifibration. This follows from the Homotopy lifting property.
• The projection of the letter L onto its base interval is a quasifibration, but not a fibration. More generally, the mapping M_f → I of the mapping cylinder of a map f: X → Y between connected CW-complexes is a quasifibration if and only if π_i(M_f, p⁻¹(b)) = 0 = π_i(I, b) holds for all i ∈ I and b ∈ B. But by the long exact sequence of the pair (M_f, p⁻¹(b)) and by Whitehead's theorem, this is equivalent to f being a homotopy equivalence. For topological spaces X and Y in general, it is equivalent to f being a weak homotopy equivalence. Furthermore, if f is not surjective, paths in I starting at 0 cannot be lifted to paths starting a point of Y outside the image of f in M_f. This means that the projection is not a fibration in this case.
• The map SP(p) : SP(X) → SP(X/A) induced by the projection p: X → X/A is a quasifibration for a CW-pair (X, A) consisting of two connected spaces. This is one of the main results used for the proof of the Dold-Thom theorm.

Properties

The following is a direct consequence from the alternative definition of a fibration using the homotopy fiber:

Theorem. Every quasifibration p : E → B factors through a fibration whose fibres are weakly homotopy equivalent to the ones of p.

A corollary of this theorem is that all fibres of a quasifibration are weakly homotopy equivalent if the base space is path-connected as this is the case for fibrations.

Checking whether a given map is a quasifibration tends to be quite tedious. The following two theorems are designed to make this problem a bit easier. They will make use of the following notion. Let p : E → B be a continuous map. A subset U ⊂ p(E) is called distinguished (with respect to p) if p : p⁻¹(U) → U is a quasifibration.

Theorem. If the open subsets U,V and U ∩ V are distinguished with respect to the continuous map p : E → B, then so is U ∪ V.

Theorem. Let p : E → B be a continuous map where B is the inductive limit of a sequence B₁ ⊂ B₂ ⊂ ... All B_n are moreover assumed to satisfy the first separation axiom. If all the B_n are distinguished, then p is a quasifibration.

To see that the latter statement holds, one only needs to bear in mind that continuous images of compact sets in B already lie in some B_n. That way, one can reduce it to the case where the assertion is known. These two theorems mean that it suffices to show that a given map is a quasifibration on certain subsets. Then one can patch these together in order to see that it holds for bigger subsets and then using a limiting argument to see that the map is a quasifibration on the whole space. This procedure has e.g. been used in the main part of the proof of the Dold-Thom theorem.

The Dold-Thom Theorem

In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW-complex are the same as its reduced homology groups. 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. One of the main tools used in doing so are quasifibrations. It has been extended and generalised in various ways, the strongest one probably being the Almgren isomorphism theorem.

There are also several theorems constituting relations between homotopy and homology, for example the Hurewicz theorem. Another approach is given by stable homotopy theory. Thanks to the Freudenthal suspension theorem, one can see that this actually defines a homology theory. Nevertheless, none of these allow one to directly reduce homology to homotopy.

The Theorem

Dold-Thom theorem. For a connected CW complex X one has $\pi _{n}\operatorname {SP} (X)\cong {\tilde {H}}_{n}(X),$ where H̃_n denotes reduced homology and SP stands for the infite symmetric product.

It is also very useful that there exists an isomorphism φ : π_n(SP(X)) → H̃_n(X) which is compatible with the Hurewicz homomorphism h : π_n(X) → H̃_n(X), meaning that one has a commutative diagram

where i_* is the map induced by the inclusion i : X = SP¹(X) → SP(X).

The following example illustrates that the requirement of X being a CW complex cannot be dropped offhand: Let X = CH ∨ CH be the wedge sum of two copies of the cone of the Hawaiian earring. The common point of the two copies is supposed to be the point 0 ∈ H meeting every circle. On the one hand, H₁(X) is an infinite group while H₁(CH) is trivial. On the other hand, π₁(SP(X)) ≅ π₁(SP(CH)) × π₁(SP(CH)) holds since φ : SP(X) × SP(Y) → SP(X ∨ Y) defined by φ([x₁, ..., x_n], [y₁, ..., y_n]) = ([x₁, ..., x_n, y₁, ..., y_n]) is a homeomorphism for compact X and Y.

But this implies that either π₁(SP(CH)) ≅ H₁(CH) or π₁(SP(X)) ≅ H₁(X) does not hold.

Sketch of the Proof

One wants to show that the family of functors h_n = π_i ∘ SP defines a homology theory. However, Dold and Thom preferred in their initial proof a slight modification of the Eilenberg-Steenrod axioms, namely calling a family of functors (h_n)_{n ∈ N0} from the category of basepointed, connected CW complexes to the category of abelian groups a reduced homology theory if they satisfy

1. If f ≃ g : X → Y, then f_* = g_* : h_n(X) → h_n(Y), where ≃ denotes homotopy equivalence.
2. There are natural boundary homomorphisms ∂ : h_n(X/A) → h_n−1(A) for each CW-pair (X, A), yielding an exact sequence
   $\qquad \dots \xrightarrow {\partial } {\tilde {h}}_{n}(A)\xrightarrow {i_{*}} {\tilde {h}}_{n}(X)\xrightarrow {q_{*}} {\tilde {h}}_{n}(X/A)\xrightarrow {\partial } {\tilde {h}}_{n-1}(A)\xrightarrow {i_{*}} \dots$
   where i is the inclusion and q is the quotient map.
3. h_n(S¹) = 0 for n ≠ 1.
4. Let (X_λ) be the system of compact subspaces of a pointed space X containing the base point. Then (X_λ) is a direct system together with the inclusions. Denote by $i_{\lambda }\colon X_{\lambda }\to X$ respectively $i_{\lambda }^{\mu }\colon X_{\lambda }\to X_{\mu }$ the inclusion if X_λ ⊂ X_μ. h_n(X_λ) is a direct system as well with the morphisms ${i_{\lambda }^{\mu }}_{*}$. Then the homomorphism
   $\qquad i_{*}\colon \varinjlim H_{n}(X_{\lambda })\to H_{n}(X),$
   induced by the $i_{\lambda *}$ is required to be an isomorphism.

One can show that for a reduced homology theory (h_n) there is a natural isomorphism h_n(X) ≅ H̃_n(X;G) with G = h₁(S¹).

Clearly, h_n is a functor fulfilling property 1 as SP is a homotopy functor. Moreover, the third property is clear since one has SP(S¹) ≃ S¹. So it only remains to verify the axioms 2 and 4. The crux of this undertaking will be the first point. This is where quasifibrations come into play:

The goal is to prove that the map p_* : SP(X) → SP(X/A) induced by the quotient map p : X → X/A is a quasifibration for a CW-pair (X, A) consisting of connected complexes. First of all, X will be replaced by the mapping cylinder of the inclusion A → X. This will not change anything as SP is a homotopy functor. It suffices to prove by induction that p_*: E_n → B_n is a quasifibration with B_n = SPⁿ(X/A) and E_n = p_*⁻¹(B_n). For n = 0 this is trivially fulfilled. In the induction step, one decomposes B_n into an open neighbourhood of B_n−1 and B_n - B_n−1 and shows that these two sets are, together with their intersection, distinguished, i.e. that p restricted to the preimages of these two sets each is a quasifibration. It can be shown that B_n is then already distinguished itself. Therefore, p_* is indeed a quasifibration and the long exact sequence of such a one implies that axiom 2 is satisfied as p_*⁻¹([e]) ≅ SP(A) holds.

Verifying the fourth axiom can be done quite elementary, in contrast to the previous one.

Compatibility with the Hurewicz Homomorphism

In order to verify the compatibility with the Hurewicz homomorphism, it suffices to show that the statement holds for X = Sⁿ since then, one gets a prism

where all sides except the one at the bottom commute. However, by using the suspension isomorphisms for homotopy respectively homology groups, the task reduces to showing the assertion for S¹. But in this case the inclusion SP¹(S¹) → SP(S¹) is a homotopy equivalence.

One should bear in mind that there is a variety of different proofs although this one is seemingly the most popular. For example, proofs have been established via factorisation homology or simplicial sets. One can also proof the theorem making use of other notions of a homology theory (the Eilenberg-Steenrod axioms e.g.).

Applications

Mayer-Vietoris sequence

One direct consequence of the Dold-Thom theorem is a new proof of 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. Then one applies SP to that square and finally π_* to the resulting pullback square.

A Theorem of Moore

Another application is a new proof of a theorem first stated by Moore. It basically predicates the following:

Theorem. A path-connected, commutative and associative H-space X with a strict identity element has the weak homotopy type of a generalised Eilenberg-MacLane space.

Note that SP(Y) has this property for every path-connected space Y and that it therefore has the weak homotopy type of a generalised Eilenberg-MacLane space. This is equivalent to saying that all k-invariants of a path-connected, commutative and associative H-space with strict unity vanish.

Proof

Let G_n = π_n(X). Then there exists a map M(G_n,n) → X inducing an isomorphism on π_n for n ≥ 2 and an isomorphism on H₁ when n = 1 for a Moore space M(G_n,n). These give a map

$\bigvee _{n}M(G_{n},n)\to X$

if one takes the maps to be basepoint-preserving. Then the special H-space structure of X yields a map

$f\colon \operatorname {SP} \left(\bigvee _{n}M(G_{n},n)\right)\to X$

given by summing up the images of the coordinates. As there is a natural homeomorphism

$\operatorname {SP} \left(\bigvee _{\alpha }X_{\alpha }\right)\cong \prod _{\alpha }\operatorname {SP} (X_{\alpha }),$

where ∏ denotes the weak product, f induces isomorphisms on π_n for n ≥ 2. But as ... is the Hurewicz homomorphism, it also induces isomorphisms on π₁. Thanks to the Dold-Thom theorem, each SP(M(G_n, n)) is now an Eilenberg-MacLane space K(G_n, n). This also implies that the natural inclusion of the weak product ∏_n SP(M(G_n, n)) into the cartesian product is a weak homotopy equivalence. Therefore, X has the weak homotopy type of a generalised Eilenberg-MacLane space.

Eilenberg-MacLane Spaces as Abelian Topological Groups

Another application of the Dold-Thom theorem is the construction of Eilenberg-MacLane spaces with the structure of an abelian topological group. For this purpose, let G be an abelian group. For a connected CW-complex X let Z[X] be the free abelian group generated by X. Then the kernel of the map Z[X] → Z is homotopy equivalent to the infinite symmetric product SP(X) of X. If one takes X now to be a Moore space M(G, n), the Dold-Thom theorem yields a K(G, n) having the strucutre of an abelian group. Note that this construction made use of another consequence of the Dold-Thom theorem, namely that the functor SP maps Moore spaces to Eilenberg-MacLane spaces, yielding another way to construct Eilenberg-MacLane spaces. One does not even need to consider Moore spaces if one takes the generalised construction of the infinite symmetric product introduced by McCord, written as B(G, X). Namely, it is known that the space B(G, Sⁿ) is a K(G, n) having an abelian group structure when one considers G as a topological group equipped with the discrete topology.

Algebraic Geometry

What distinguishes the Dold-Thom theorem from other alternative foundations of homology like Cech or Alexander-Spanier cohomology is that it is of particular interest for algebraic geometry since it allows to reformulate homology only using homotopy. Since applying methods from algebraic topology can be quite insightful in this field, one tries to transfer these to algebraic geometry. This could be achieved for homotopy theory, but for homology theory only in a rather limited way using a formulation via sheaves. So the Dold-Thom theorem allows to give a foundation of homology having an algebraic analogue.

References

• Aguilar, Marcelo; Gitler, Samuel; Prieto, Carlos (2008). Algebraic Topology from a Homotopical Viewpoint. Springer Science & Business Media. ISBN 978-0-387-22489-3.
• Bandklayder, Lauren (2019), "The Dold-Thom Theorem via Factoriation Homology", Journal of Homotopy and Related Sources, 14 (2): 579–593
• Dold, Albrecht (1958), "Homology of Symmetric Products and other Functor of Complexes", Annals of Mathematics: 54–80
• Dold, Albrecht; Lashof, Richard (1959), "Principal Quasifibrations and Fibre Homotopy Equivalence of Bundles", Illinois Journal of Mathematics, 2 (2): 285–305
• Dold, Albrecht; Thom, René (1958), "Quasifaserungen und unendliche symmetrische Produkte", Annals of Mathematics, Second Series, 67 (2): 239–281, doi:10.2307/1970005, ISSN 0003-486X, JSTOR 1970005, MR 0097062
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External Links

• Why the Dold-Thom theorem? on MathOverflow
• The Dold-Thom theorem for infinity categories? on MathOverflow
• Quasifibrations and Homotopy Pullbacks on MathOverflow
• Symmetric product in arbitrary categories? on MathOverflow
• Group structure on Eilenberg-MacLane spaces on StackExchange
• Quasifibrations from the Lehigh University