FILTERING THE HUREWICZ MAP

FUMIKO KATO AND DAI TAMAKI

1. Introduction

The infinite symmetric product SP^{∞}(X) of a pointed space X was introduced by Dold and Thom in 1958 [DT58]. One of the most significant properties of SP^{∞}(X) is the existence of a natural isomorphism

π_{k}(SP^{∞}(X)) ≅ H̃_{k}(X)

for a CW-complex X. Under this isomorphism, the Hurewicz homomorphism

Hur : π_{k}(X) ⟶ H_{k}(X)

can be identified with the map

i_{∗} : π_{k}(X) ⟶ π_{k}(SP^{∞}(X))

induced by the canonical inclusion

i : X ↪ SP^{∞}(X).

Furthermore the Hurewicz map factors through the stable homotopy groups, namely the following diagram is commutative

[figure: π_{k}(X); H_{k}(X); π_{k+n}(Σ^{n}X); π_{k}^{S}(X).; Hur; E^{n}]

Let C_{n}(X) be the little cube model for Ω^{n}Σ^{n}X defined by C_{n}(j), the space of j little n-cubes of disjoint interiors [Ma72]. Using D_{n}(j), the space of j little n-cubes in I^{n} with any overlappings allowed, we have a little cube model D_{n}(X) for SP^{∞}(X) and the Hurewicz homomorphism

π_{k}(Ω^{n}Σ^{n}X) ≅ π_{k+n}(Σ^{n}X) ⟶ π_{k}(SP^{∞}(X)) ≅ H̃_{k}(X)

coincides with the map induced by the inclusion

C_{n}(X) ↪ D_{n}(X).

The purpose of this note is to study the filtration on D_{n}(X) defined by the multiplicities of overlappings of little cubes.

Let D_{n}^{i}(X) be the subspace of D_{n}(X) in which at most i cubes have a point in common in their interiors. (See §2 for the precise definition.) The main theorem of this note is the following.

Theorem 1. If (X, ∗) is a strong NDR pair and X is path-connected, then we have the followng natural weak homotopy equivalence

D_{n}^{i}(X) ≃_{w} Ω^{n} SP^{i} Σ^{n}X

where SP^{i}(Y) denotes the i-th symmetric product of a pointed space Y.

The proof of this theorem is exacly parallel to the proof of the weak homotopy equivalence

C_{n}(X) ≃_{w} Ω^{n}Σ^{n}X (1)

in [Ma72]. The idea is to construct a quasifibration model for the path-loop fibration

Ω^{n} SP^{i} Σ^{n}X ⟶ P Ω^{n−1} SP^{i} Σ^{n}X ⟶ Ω^{n−1} SP^{i} Σ^{n}X

In §2, we introduce a space E_{n}^{i}(X, A) ⊂ D_{n}^{i}(X) for a pointed pair (X, A) and define maps

π_{n}^{i} : E_{n}^{i}(X, A) ⟶ D_{n−1}^{i}(X/A)
α_{n}^{i} : D_{n}^{i}(X) ⟶ Ω^{n} SP^{i} Σ^{n}X
α̃_{n}^{i} : E_{n}^{i}(CX, X) ⟶ P Ω^{n−1} SP^{i} Σ^{n}X

which make the following diagram commutative

[figure: D_{n}^{i}(X); E_{n}^{i}(CX, X); D_{n−1}^{i}(ΣX); Ω^{n} SP^{i} Σ^{n}X; P Ω^{n−1} SP^{i} Σ^{n}X; Ω^{n−1} SP^{i} Σ^{n}X; α_{n}^{i}; π_{n}^{i}; α̃_{n}^{i}; α_{n−1}^{i}]

In §3, it is shown that

D_{n}^{i}(A) ⟶ E_{n}^{i}(X, A) ⟶^{π_{n}^{i}} D_{n−1}^{i}(X/A)

is a quasifibration and E_{n}^{i}(CX, X) is contractible. These two facts, together with the above diagram prove our Main Theorem.

In the final section §4, several remarks are given.

As is mentioned above, the idea of the proof is entirely due to May’s book [Ma72]. We claim no originality in the proof of Main Theorem. The only new idea in this note is the definition of the spaces D_{n}^{i}(X) and E_{n}^{i}(X, A).

The spaces D_{n}^{i}(X) defines a filtration on D_{n}(X) ≃ SP^{∞}(X)

C_{n}(X) = D_{n}^{1}(X) ⊂ D_{n}^{2}(X) ⊂ ⋯ ⊂ D_{n}^{i}(X) ⊂ D_{n}^{i+1}(X) ⊂ ⋯ ⊂ D_{n}(X),

which gives a factorization of the Hurewicz map

π_{k}(Ω^{n}Σ^{n}X) = π_{k}(C_{n}(X)) = π_{k}(D_{n}^{1}(X)) ⟶ π_{k}(D_{n}^{2}(X)) ⟶

⋯ ⟶ π_{k}(D_{n}^{∞}(X)) = π_{k}(D_{n}(X)) ≅ π_{k}(SP^{∞}(X)) ≅ H̃_{k}(X),

hence the title. But nothing is known about this factorization besides the above theorem.

Acknowledgements: The result of this paper is the master’s thesis of the first author at Shinshu University in 1995. We would like to thank algebraic topologists at Shinshu University, T. Matsuda, H. Kachi, J. Mukai for their helpful comments and encouragement.

After the first author completed her thesis, K. Iwama, a graduate student at Shinshu University, carefully read the thesis and pointed out several gaps in the proof. His effort was very helpful in writing this note.

The original idea of this note is due to Fred Cohen. He suggested to filter the infinite symmetric product via multiplicities in configuration spaces. We are grateful to him for providing the idea.

2. Definitions

Let us begin by recalling the definition of little cube.

Definition 2.1. A little n-cube is an affine embedding

c : I^{n} ⟶ I^{n}

which maps each i-th face parallelly to the i-th face of I^{n} with orientation preserved. In other words, c can be written as

c = f_{1} × ⋯ × f_{n}

where each f_{i} is an orientation preserving affine embedding

f_{i} : I ⟶ I.

The space of little n-cubes is denoted by

D_{n}(1) = {c : I^{n} ⟶ I^{n}|little n-cube}

The topology is, of course, the compact-open topology.

Define

D_{n}(j) = \underbrace{D_{n}(1) × ⋯ × D_{n}(1)}_{jtimes}

This is the configuration space of j little n-cubes (with arbitrary overlappings allowed).

D_{n}(j) is equipped with an action of the symmetric group of j letters Σ_{j} by permuting the ordering of cubes.

An important fact is the compact-open topology on D_{n}(j) coincides with more familiar one.

Lemma 2.2. Define a map

ξ_{j} : D_{n}(j) ⟶ I^{2nj}

by

ξ_{j}(c_{1}, ⋯, c_{j}) = (c_{1}(a), c_{1}(b), ⋯, c_{j}(a), c_{j}(b))

where

a = (\frac{1}{4}, ⋯, \frac{1}{4})
b = (\frac{3}{4}, ⋯, \frac{3}{4})

Then ξ is a homeomorphism onto its image.

Proof. See [Ma72].

This lemma reduces the proofs of continuity of various maps between the spaces of little cubes to elementary calculations in the coordinates in I^{2nj}, making the proofs much easier.

Definition 2.3. For i ≤ j, define a subspace D_{n}^{i}(j) of D_{n}(j) by

(c_{1}, ⋯, c_{j}) ∈ D_{n}^{i}(j) ⟺ At most i of c_{1}(IntI^{n}), ⋯, c_{j}(IntI^{n}) overlap

at the same point.

D_{n}^{1}(j) is the space C_{n}(j) studied in [Ma72] and each D_{n}^{i}(j) is closed under the action of Σ_{j}. Thus we have a Σ_{j}-equivariant filtration on D_{n}(j) beginning with C_{n}(j)

C_{n}(j) = D_{n}^{1}(j) ⊂ ⋯ D_{n}^{i}(j) ⊂ D_{n}^{i+1}(j) ⊂ ⋯ ⊂ D_{n}^{j}(j) = D_{n}(j)

For a pointed space X, [Ma72] defined a space C_{n}(X) out of C_{n}(j)’s. The same construction works with D_{n}^{i}(j)’s for each i.

Definition 2.4. For a pointed space X, define

D_{n}^{i}(X) = (∐_{j=0}^{∞} D_{n}^{i}(j) ×_{Σ_{j}} X^{j})/∼

where the relation ∼ is the equivalence relation generated by

(c_{1}, ⋯, c_{j}; x_{1}, ⋯, x_{j}) ∼ (c_{1}, ⋯, c_{k−1}, c_{k+1}, ⋯, c_{j}; x_{1}, ⋯, x_{k−1}, x_{k+1}, ⋯, x_{j})

if x_{k} is the base point.

For k ≥ 0, define

F_{k}D_{n}^{i}(X) = (∐_{j=0}^{k} D_{n}^{i}(j) ×_{Σ_{j}} X^{j})/∼

In order to prove (1), May introduced an auxiliary space E_{n}(X, A) for a pointed pair (X, A) which fits into a quasifibration

C_{n}(A) ⟶ E_{n}(X, A) ⟶ C_{n−1}(X/A).

We need an “overlapped” version of May’s construction.

Definition 2.5. For a pointed pair (X, A), define a subspace

E_{n}^{i}(j; X, A) ⊂ D_{n}^{i}(j) ×_{Σ_{j}} X^{j}

as follows: for (c, x) ∈ D_{n}^{i}(j) ×_{Σ_{j}} X^{j}

(c, x) ∈ E_{n}^{i}(j; X, A) ⟺ pr_{2}(c_{k}) ∈ D_{n−1}^{i}(j) if x_{k} ∈ X − A

where

pr_{2} : D_{n}(j) ⟶ D_{n−1}(j)

is the projection onto the last (n − 1) coordinates.

Define

E_{n}^{i}(X, A) = (∐_{j=0}^{∞} E_{n}^{i}(j; X, A))/∼

F_{k}E_{n}^{i}(X, A) = (∐_{j=0}^{k} E_{n}^{i}(j; X, A))/∼

Thus we have filtrations

(2)  ∗ = F_{0}D_{n}^{i}(X) ⊂ F_{1}D_{n}^{i}(X) ⊂ ⋯ ⊂ F_{∞}D_{n}^{i}(X) = D_{n}^{i}(X)
(3)  ∗ = F_{0}E_{n}^{i}(X, A) ⊂ F_{1}E_{n}^{i}(X, A) ⊂ ⋯ ⊂ F_{∞}E_{n}^{i}(X, A) = E_{n}^{i}(X, A).

Remark 2.6. For i ≥ j,

E_{n}^{i}(j; X, A) = D_{n}(j) ×_{Σ_{j}} X^{j}.

Therefore

E_{n}^{∞}(X, A) = D_{n}(X).

Definition 2.7. For a pointed pair (X, A), define

π_{n} : D_{n}(X) ⟶ D_{n−1}(X/A)

by the following composition

D_{n}(X) ⟶^{(pr_{2})_{∗}} D_{n−1}(X) ⟶^{D_{n−1}(π)} D_{n−1}(X/A)

where (pr_{2})_{∗} is the map induced by

pr_{2} : D_{n}(j) ⟶ D_{n−1}(j)

and D_{n−1}(π) is induced by the projection

π : X ⟶ X/A.

It is obvious that π_{n} is surjective. By the definition of E_{n}^{i}(X, A), the restriction of π_{n} to E_{n}^{i}(X, A) gives a surjective map

π_{n}^{i} : E_{n}^{i}(X, A) ⟶ D_{n−1}^{i}(X/A)

It is proved in §3 that π_{n}^{i} is a quasifibration with fiber D_{n}^{i}(A).

In the rest of this section we define maps

α_{n}^{i} : D_{n}^{i}(X) ⟶ Ω^{n} SP^{i} Σ^{n}X
α̃_{n}^{i} : E_{n}^{i}(CX, X) ⟶ P Ω^{n−1} SP^{i} Σ^{n}X

which make the following diagram commutative

[figure: D_{n}^{i}(X); E_{n}^{i}(CX, X); D_{n−1}^{i}(ΣX); Ω^{n} SP^{i} Σ^{n}X; P Ω^{n−1} SP^{i} Σ^{n}X; Ω^{n−1} SP^{i} Σ^{n}X; α_{n}^{i}; π_{n}^{i}; α̃_{n}^{i}; α_{n−1}^{i}]

(4)

The map α_{n}^{i} is defined by the following composition

D_{n}^{i}(X) ⟶^{D_{n}^{i}(E^{n})} D_{n}^{i}(Ω^{n}Σ^{n}X) ⟶^{ᾱ_{n}^{i}} Ω^{n} SP^{i} Σ^{n}X

where E^{n} : X ⟶ Ω^{n}Σ^{n}X is the Freudenthal suspension map. The map ᾱ_{n}^{i} is defined as follows.

Definition 2.8. Define

β_{n} : D_{n}(1) ⟶ Ω^{n}S^{n}

by

β_{n}(c)(s) = \{ \begin{matrix}c^{−1}(s) & if s ∈ Im c \\ ∗ & otherwise \end{matrix}

for c ∈ D_{n}(1), s ∈ I^{n}.

Since D_{n}(j) = D_{n}(1)^{j}, β_{n} induces

β_{n}^{j} : D_{n}(j) ⟶ (Ω^{n}S^{n})^{j}.

Consider the following composition

I^{n} × (D_{n}(j) ×_{Σ_{j}} (Ω^{n}X)^{j}) ⟶^{id×β_{n}^{j}×id} I^{n} × (Ω^{n}S^{n})^{j} ×_{Σ_{j}} (Ω^{n}X)^{j}

⟶^{id×∘} I^{n} × (Ω^{n}X)^{j}/Σ_{j} ⟶^{Δ^{j}×id} Δ^{j}(I^{n}) × (Ω^{n}X)^{j}/Σ_{j} ⟶^{eval} X^{j}/Σ_{j} = SP^{j}(X)

where ∘ is the map given by composing elements in Ω^{n}S^{n} and Ω^{n}X) and eval is the evaluation map.

The above composition is compatible with the base-point relation and defines the following map.

I^{n} × (∐_{j=0}^{∞} D_{n}(j) ×_{Σ_{j}} (Ω^{n}X)^{j})/∼ ⟶ (∐_{j=0}^{∞} SP^{j}(X))/∼
∥  ∥
I^{n} × D_{n}(Ω^{n}X)  SP^{∞}(X)

Let

ᾱ_{n} : D_{n}(Ω^{n}X) ⟶ Ω^{n} SP^{∞} X

be the adjoint of the above map.

Lemma 2.9. β_{n}^{j} is continuous and for any t ∈ I^{n} and c ∈ D_{n}^{i}(j),

β_{n}^{j}(c)(t, ⋯, t) ∈ SP^{i}(S^{n}).

Thus ᾱ_{n} restricts to give us a well-defined continuous map

ᾱ_{n}^{i} : D_{n}^{i}(Ω^{n}Σ^{n}X) ⟶ Ω^{n} SP^{i} Σ^{n}X.

The definition of α̃_{n}^{i} is similar.

Definition 2.10. Define

β̃_{n} : D_{n}(1) ⟶ Map_{∗}((CS^{n−1}, S^{n−1}), (CS^{n−1}, S^{n−1}))

by

β̃_{n}(c)(t, s) = \{ \begin{matrix}c^{−1}(t, s) & if (t, s) ∈ Im c \\ (1, c′′^{−1}(s)) & if t ≥ c′(1) and s ∈ Im c′′ \\ ∗ & otherwise \end{matrix}

for c = (c′, c′′) ∈ D_{1}(1) × D_{n−1}(1).

8 FUMIKO KATO AND DAI TAMAKI

Composed with the evaluation map

Map(CS^{n−1}, CS^{n−1}) × P Ω^{n−1}S^{n} ⟶ P Ω^{n−1}S^{n},

we have a Σ_{j}-equivariant map

β̃_{n}^{j} : E_{n}^{∞}(j; P Ω^{n−1}X, Ω^{n}X) ⟶ (P Ω^{n−1}X)^{j}.

We define

¯α̃_{n}^{∞} : E_{n}^{∞}(P Ω^{n−1}X, Ω^{n}X) ⟶ P Ω^{n−1} SP^{∞} X

to be the map induced by the adjoint of the compostion

I^{n} × E_{n}^{∞}(j; P Ω^{n−1}X, Ω^{n}X)/Σ_{j} ⟶^{Δ^{j}×β̃_{n}^{j}} Δ^{j}(I^{n}) × (P Ω^{n−1}X)^{j}/Σ_{j} ⟶^{eval} X^{j}/Σ_{j} = SP^{j}(X).

Let

Ẽ^{n} : CX ⟶ P Ω^{n−1}Σ^{n}X

be the map defined in [Ma72] making the following diagram commutative

[figure: X; CX; ΣX; Ω^{n}Σ^{n}X; P Ω^{n−1}Σ^{n}X; Ω^{n−1}Σ^{n}X.; E^{n}; Ẽ^{n}; E^{n−1}]

Define

α̃_{n}^{∞} : E_{n}^{∞}(CX, X) ⟶ P Ω^{n−1} SP^{∞} Σ^{n}X

by the composition

E_{n}^{∞}(CX, X) ⟶^{E_{n}^{∞}(Ẽ^{n})} E_{n}^{∞}(P Ω^{n−1}X, Ω^{n}X) ⟶^{¯α̃_{n}^{∞}} P Ω^{n−1} SP^{∞} Σ^{n}X.

Restricting to E_{n}^{i}(CX, X) we have the following map

α̃_{n}^{i} : E_{n}^{i}(CX, X) ⟶ P Ω^{n−1} SP^{i} Σ^{n}X.

It is elementary to check the following. Lemma 2.11. The diagram (4) is commutative.

3. Proof of the Main Theorem

The proof of the main theorem follows from the following two propositions.

Proposition 3.1. If (X, ∗) is an NDR pair, then E_{n}^{i}(CX, X) is contractible for any n, i ≥ 1, where CX denotes the (reduced) cone on X.

Proposition 3.2. Let (X, A) be a strong NDR pair of pointed spaces and suppose that (A, ∗) is a strong NDR pair and A is path-connected. Then the following sequence is a quasifibration.

D_{n}^{i}(X) ⟶ E_{n}^{i}(X, A) ⟶^{π_{n}^{i}} D_{n−1}^{i}(X/A)

for i, n ≥ 1.

With these two propositions, Main Theorem is proved as follows:

The case n = 0 is immediate.

Lemma 3.3. For a pointed space X, we have

D_{0}^{i}(X) = SP^{i}(X).

Proof. By definition

D_{0}^{i}(j) = \{ \begin{matrix}\{∗\} & if j ≤ i \\ ∅ & if j > i. \end{matrix}

Thus the result follows from the definition of SP^{i}(X).

By using the commutative diagram (4), the long exact sequence of homotopy groups and the above two propositions, the theorem follows by induction on n.

In order to prove Proposition 3.1, we make use of the filtration (3). For proving the contractibility of a filtered space, the following fact is useful.

Lemma 3.4. For a filtered space

X_{0} ⊂ X_{1} ⊂ ⋯ X_{k} ⊂ X_{k+1} ⊂ ⋯ ⊂ X,

suppose X = colim_{k} X_{k} with weak topology.

If X_{k} is contractible in X_{k+1} and (X_{k+1}, X_{k}) is an NDR pair for each k, then X is contractible.

Proof. See [St67].

Let (X, A) be an NDR pair with nondegenerate base point. Using NDR representations of (X, A) and (A, ∗), May defines an NDR representation (h̃_{k+1}, ũ_{k+1}) of (F_{k+1}C_{n}(X), F_{k}C_{n}(X)) which restricts to give an NDR representation of (F_{k+1}E_{n}(X, A), F_{k}E_{n}(X, A)). Since he only uses the NDR representation of (X, ∗), these maps are actually defined on F_{k+1}D_{n}(X)× I and F_{k+1}D_{n}(X). Thus we have the following.

Lemma 3.5. If (X, ∗) is an NDR pair, (F_{k+1}D_{n}^{i}(X), F_{k}D_{n}^{i}(X)) is an NDR pair. If (X, A) and (A, ∗) are NDR pairs then (F_{k+1}E_{n}^{i}(X, A), F_{k}E_{n}^{i}(X, A)) is an NDR pair.

The contraction of F_{k}E_{n}(X, A) in F_{k+1}E_{n}(X, A) defined in [Ma72] can be also used in our case without modification, since it is defined in the way it moves only points in X, not in C_{n}(j). Thus we have

Lemma 3.6. If (X, ∗) is an NDR pair, then F_{k}E_{n}^{i}(CX, X) is contractible in F_{k+1}E_{n}^{i}(CX, X).

And Propostion 3.1 follows.

For the proof of Proposition 3.2, we use the Dold-Thom criterion following May.

Theorem 3.7 (Dold-Thom Criterion). Let {F_{j}B} be a filtration on a space B. A continuous map

p : E ⟶ B

is a quasifibration if the following conditions are satisfied.

1. F_{0}B = ∗

2. Any open subset V ⊂ F_{j}B − F_{j−1}B is distinguished, i.e.

p|_{p^{−1}(V)} : p^{−1}(V) ⟶ V

is a quasifibration.

3. For each j, there exist an open subset U ⊂ F_{j}B with F_{j−1}B ⊂ U and continuous maps

φ : I × U ⟶ U
Φ : I × p^{−1}(U) ⟶ p^{−1}(U)

satisfying the following properites.

(a) φ(0, x) = x

(b) φ(I × F_{j−1}B) ⊂ F_{j−1}B

(c) φ({1} × U) ⊂ F_{j−1}B

(d) Φ(0, x) = x

(e) p ∘ Φ = φ ∘ (1 × p)

(f) For any point x ∈ U

Φ : {1} × p^{−1}(x) ⟶ p^{−1}(φ(1, x))

is a weak homotopy equivalence.

May proves that

C_{n}(X) ⟶ E_{n}(X, A) ⟶ C_{n−1}(X/A)

is a quasifibration by showing that the filtration {F_{j}C_{n−1}(X/A)} satisfies the conditions of the Dold-Thom criterion. Let us check if his proof works for the map

π_{n} : E_{n}^{i}(X, A) ⟶ D_{n}^{i}(X/A)

and the filtration {F_{j}D_{n}^{i}(X/A)}.

Condition 1 in Theorem 3.7 is obvious.

In order to prove Condition 2 in Theorem 3.7 is satisfied, May proves that there exists a fiber homotopy equivalence

π_{n}^{−1}(V) ≃ V × C_{n}(A)

for any open V ⊂ F_{j}C_{n−1}(X/A) − F_{j−1}C_{n−1}(X/A). His homotopy equivalence

π_{n} × q : π_{n}^{−1}(V) ⟶ V × C_{n}(A)
w : V × C_{n}(A) ⟶ π_{n}^{−1}(V)

are defined based on the following maps.

Definition 3.8. Define

σ^{+} : D_{n−1}(1) ⟶ D_{n}(1)
τ^{−} : D_{n}(1) ⟶ D_{n}(1)

by

σ^{+}(c′′) = g^{+} × c′′
τ^{−}(c) = (g^{−} × 1^{n−1}) ∘ c

where

g^{+}(s) = \frac{1}{2}(1 + s)
g^{−}(s) = \frac{1}{2}s.

Since these maps extend to

σ^{+} : D_{n−1}^{i}(j) ⟶ D_{n}^{i}(j)
τ^{−} : D_{n}^{i}(j) ⟶ D_{n}^{i}(j)

we have extensions of May’s maps

π_{n}^{−1}(V) ⟶ V × D_{n}^{i}(A)
V × D_{n}^{i}(A) ⟶ π_{n}^{−1}(V)

for any open V ⊂ F_{j}D_{n−1}^{i}(X/A) − F_{j−1}D_{n−1}^{i}(X/A).

Homotopies are defined by using the homotopy τ^{−} ≃ 1. But this homotopy can be easily seen to extend to

D_{n}^{i}(j) × I ⟶ D_{n}^{i}(j).

Thus the Condition 2 in Theorem 3.7 is satisfied.

For the Condition 3 in Theorem 3.7, let

K = {[c; π(x_{1}), ⋯, π(x_{j})] ∈ F_{j}D_{n−1}^{i}(X/A)|There exists an r with x_{r} ∈ w^{−1}([0, 1)).}

where (k, w) is a strong NDR representation of (X, A). Then

U = F_{j−1}D_{n−1}^{i}(X/A) ∪ K

is an open subspace of F_{j}D_{n−1}^{i}(X/A) containing F_{j−1}D_{n−1}^{i}(X/A).

It remains to find homotopies

φ : I × U ⟶ U
Φ : I × π_{n}^{−1}(U) ⟶ π_{n}^{−1}(U)

and prove that

Φ : {1} × π_{n}^{−1}([c, x]) ⟶ π_{n}^{−1}([c, x])

is a weak homotopy equivalence for any [c, x] ∈ U.

However corresponding maps and homotopies in [Ma72] are defined in terms of the paths in A and NDR representations of (X, A)^{j} and (F_{j}D_{n−1}^{i}(X/A), F_{j−1}D_{n−1}^{i}(X/A)), which is defined by an NDR representation (X, A)^{j}. Thus May’s proof works without modification and Proposition 3.2 follows.

4. Concluding Remarks

We finish this note by making several remarks.

Since D_{n}(j) is Σ_{j}-equivariantly contractible, we have a natural homotopy equivalence

D_{n}(X) ≃ SP^{∞}(X).

We have

E_{n}^{∞}(X, A) = D_{n}(X)

as stated in Remark 2.6.

Therefore the quasifibration in Proposition 3.2 for i = ∞ is equivalent to the quasifibration in [DT58]

SP^{∞}(A) ⟶ SP^{∞}(X) ⟶ SP^{∞}(X/A).

Thus Proposition 3.2 unifies May’s result and the result of Dold and Thom.

Nakaoka [Naka] proved a natural homology equivalence

H_{∗}(SP^{∞}(X)) ≅ ⨁_{j=1}^{∞} H_{∗}(SP^{j}(X)/SP^{j−1}(X))

namely a weak homotopy equivalence

SP^{∞}(SP^{∞}(X)) ≃_{w} SP^{∞}(∨_{j=1}^{∞} SP^{j}(X)/SP^{j−1}(X))

or

D_{∞}(D_{n}(X)) ≃_{w} D_{∞}(∨_{j=1}^{∞}F_{j}D_{n}^{i}(X)/F_{j−1}D_{n}^{i}(X)).

On the other hand, Snaith [Sn74] proved a stable weak homotopy equivalence

Ω^{n}Σ^{n}X ≃_{S} ⋁_{j=1}^{∞} F_{j}C_{n}(X)/F_{j−1}C_{n}(X)

or a weak homotopy equivalence

C_{∞}(C_{n}(X)) ≃_{w} C_{∞}(∨_{j=1}^{∞}F_{j}C_{n}(X)/F_{j−1}C_{n}(X)).

Following the proof of [CMT], it is easy to see that we have a homology splitting

H_{∗}(Ω^{n} SP^{i} Σ^{n}X) ≅ ⨁_{j=1}^{∞} H_{∗}(F_{j}D_{n}^{i}(X)/F_{j−1}D_{n}^{i}(X)).

Note that this homology equivalence cannot be improved to a stable homotopy equivalence when i ≥ 2, since D_{n}^{i}(X) contains SP^{i}(X) as filtered spaces.

References

[CMT] F.R. Cohen, J.P. May and L.R. Taylor. Splitting of certain spaces CX. Math. Proc. Camb. Phil. Soc., 84:465–496, 1978.

[DT58] A. Dold and R. Thom. Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math., 67:239–281, 1958.

[Ma72] J.P. May. The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York, 1972.

[Naka] M. Nakaoka. Cohomology of symmetric products. Journal of the Institute of Polytechnics, Osaka City University, 8:121–145.

[Sn74] V.P. Snaith. A stable decomposition of Ω^{n}S^{n}X. J. London Math. Soc. (2), 7:577–583, 1974.

[St67] N.E. Steenrod. A convinient category of topological spaces. Mich. Math. J., 14:133–152, 1967.

Department of Mathematical Sciences,, Shinshu University,, Matsumoto, 390-8621, Japan

  E-mail address: fumichan@sonet.or.jp

Department of Mathematical Sciences,, Shinshu University,, Matsumoto, 390-8621, Japan

  E-mail address: rivulus@math.shinshu-u.ac.jp