ON THE HOMOLOGY OF CONFIGURATION SPACES C((M, M_{o}) × R^{n}; X)

JIE WU

Abstract. The homology with coefficients in a field of the configuration spaces C(M × R^{n}, M_{o} × R^{n}; X) is determined in this paper.

1. Introduction.

The purpose of this paper is to determine the homology (with coefficients in a field F) of configuration spaces C((M, M_{o}) × R^{n}; X). The homology groups

H_{∗}(C((M, M_{o}) × R^{n}; X); F)

are determined by dim M, H_{∗}(M, M_{o}; F) and H_{∗}(X; F). This answers a conjecture of F. Cohen and L. Taylor [10].

Let A be a topological space. The ordered configuration space C̃^{k}(A) is the subspace of A^{k} consisting of all k-tuples (a_{1}, ..., a_{k}) such that a_{i} ≠ a_{j} for i ≠ j, where A^{k} = A × ... × A with k copies. Define the configuration space

C(A, A_{o}; X) = ∐_{k=1}^{∞} C̃^{k}(A) ×_{Σ_{k}} X^{k}/ ≈

where A_{o} is a closed subspace of A, X is a space with non-degenerate base-point ∗, and ≈ is generated by

(a_{1}, ..., a_{k}; x_{1}, ..., x_{k}) ≈ (a_{1}, ..., a_{k−1}; x_{1}, ..., x_{k−1})

if a_{k} ∈ A_{o} or x_{k} = ∗.

The spaces in the title are given by C((A, A_{o}) × R^{n}; X) = C(A × R^{n}, A_{o} × R^{n}; X). The applications of configuration spaces can be found in [1, 3, 6, 7, 10, 11, 14, 17, 18]. The homology of configuration spaces for various cases can be found in [4, 5, 8, 9, 10, 11, 12, 13, 20]. We will consider the case that A is a manifold and A_{0} is a submanifold and n > 0.

In this paper, a space X will always mean a compactly generated weak Hausdorff space with nondegenerate base-point ∗ such that (X, ∗) is an NDR-pair. Homology groups are always taken with coefficients in the field F;S_{∗}(−) willalways mean the singular chain complex with coefficients in the field F. A manifold will always mean smooth triangulated manifold. For each finitedimensional graded F-module V_{∗} and a connected space X, define

Research at MSRI is supported in part by NSF grant DMS-9022140

C^{m}(V_{∗}; X) = ⨂_{q=0}^{m} ⨂^{β_{q}(V_{∗})} H_{∗}(Ω^{m−q}S^{m}X)

where β_{q}(V_{∗}) = dim_{F}(V_{q}) is the q-th Betti number of V_{∗} and Ω^{m−q}S^{m}X = ∗ if q > m.

If V_{q} = 0 for all q ≥ m, then each term H_{∗}(Ω^{m−q}S^{m}X) in C^{m}(V_{∗}; X) is an algebra with an algebra filtration induced by Ω^{m−q}S^{m}X ≃ C(R^{m−q}, S^{q}X). This filters C^{m}(V_{∗}; X) by the tensor product filtration. The natural filtration of configuration spaces will be given in section 2. Now our main theorems are as follows

Theorem A. Let M be a smooth triangulated compact manifold and let M_{o} be a smooth triangulated compact submanifold of M,then

(1): for each simply connected space X and n ≥ 1, there is an F-filtered module isomorphism

θ : C^{dim M+n}(H_{∗}(M, M_{o}); X) ⟶ H_{∗}C((M, M_{o}) × R^{n}, X)

(2): for each simply connected space X and n ≥ 2, there is an F-filtered algebra isomorphism

θ : C^{dim M+n}(H_{∗}(M, M_{o}); X) ⟶ H_{∗}C((M, M_{o}) × R^{n}; X)

Theorem B. Let M be a smooth triangulated compact manifold and let M_{o} be a smooth triangulated compact submanifold of M, then there exist isomorphisms

\overline{H} ∗ C((M, M_{o}) × R^{n}; X) ≈ ⨁_{k=1}^{∞} σ^{−2k}D_{k}^{m+n}(H_{∗}(M, M_{o}); S^{2}X)

as F-modules for any space X and n ≥ 1, where m =dim M, D_{k}^{m+n}(V_{∗}; X) =

F_{k}C^{m+n}(V_{∗}; X)/F_{k−1}C^{m+n}(V_{∗}; X)

and σ^{−t} denotes the t-th desuspension of graded modules.

The article is organized as follows. In section 2, some basic properties of configuration spaces are studied. some product decompositions of certain configuration spaces are given in section 3. The proofs of Therem A and Theorem B will be given in section 4. The author would like to thank Professors Xueguang Zhou and Fred Cohen for their many helpful discussions and encouragement during the writing of the manuscript.The author also would like to thank Miguel Xicotencatl for his help to type the manuscript.

2. Basic properties of configuration spaces.

In this section, some basic properties of C(A, A_{o}; X) are recalled. Given an embedding (A, A_{o}) → (A′, A′_{o}) of space-pairs and a pointed map X → X′, there is an induced map C(A, A_{o}; X) ⟶ C(A′, A′_{o}; X′). Hence the homotopy type of C(A, A_{o}; X) is an invariant of the (relative) isotopy type of (A, A_{o}) and the homotopy type of (X, ∗). The length of a configuration induces a natural filtration of C(A, A_{o}; X) by the closed subspaces

F_{k}C = F_{k}C(A, A_{o}; X) = ∐_{j=1}^{k} C̃^{j}(A) ×_{Σ_{j}} X^{j}/ ≈

F_{o}C = ∗, F_{1}C = A/A_{o} ∧ X, see [3]. It is easy to see that C(A ⨿ A′, A_{o} ⨿ A′_{o}; X) is homeomorphic to the product

C(A, A_{o}; X) × C(A′, A′_{o}; X)

and the homeomorphism preserves the filtration. Hence C(A, A_{o}; X) is a filtered H-space if there is an embedding e : (A ⨿ A, A_{o} ⨿ A_{o}) ⟶ (A, A_{o}) such that e|_{first copy of (A,A_{o})} and e|_{second copy of (A,A_{o})} are (relatively) isotopic to the identity map of (A, A_{o}). In particular, C((A, A_{o}) × R^{n}; X) is a filtered H-space for each n ≥ 1.

Define

C̃^{k}(A|A_{o}) = {(a_{1}, ..., a_{k}) ∈ C̃^{k}(A)| some a_{j} ∈ A_{o}}

and

(X|∗)^{k} = {(x_{1}, ..., x_{k}) ∈ X^{k}| some x_{j} = ∗} .

If (A, A_{o}) is a relative CW-complex, there are Σ_{k}-equivalent cofibrations C̃^{k}(A|A_{o}) → C̃^{k}(A) and (X|∗)^{k} → X^{k} and therefore there are cofibrations

C̃^{k}(A|A_{o}) ×_{Σ_{k}} X^{k} ∪ C̃^{k}(A) ×_{Σ_{k}} (X|∗)^{k} ⟶ C̃^{k}(A) ×_{Σ_{k}} X^{k}

and F_{k−1}C → F_{k} with the same cofibre D_{k}(A, A_{o}; X) = F_{k}C/F_{k−1}C ≅

C̃^{k}(A) ×_{Σ_{k}} X^{k}/(C̃^{k}(A|A_{o}) ×_{Σ_{k}} X^{k} ∪ C̃^{k}(A) ×_{Σ_{k}} (X|∗)^{k})

see [2, pp 231-239; 14 pp 162-172 and Thm. 7.1]. We call D_{k}(A, A_{o}; X) the k-adic construction of C(A, A_{o}; X).

Proposition 2.1. If (A, A_{o}) is a relative CW-complex and k ≥ 1, then there is an isomorphism of F-modules

\overline{H}_{∗}D_{k}(A, A_{o}; X) ⟶ H_{∗} (S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o})) ⊗_{Σ_{k}} (\overline{H}_{∗}(X))^{⊗k})

The following lemma is useful.

Lemma 2.2. Let C be a free Σ_{k}-chain complex and let K and L be chain complexes. If the chain maps f, g : K → L are homotopic, then 1⊗f^{⊗k} and 1⊗g^{⊗k} : C ⊗L^{⊗k} ⟶ C ⊗ K^{⊗k} are Σ_{k}-equivariently homotopic, where Σ_{k} acts diagonally on C ⊗ L^{⊗k} and C ⊗ K^{⊗k}.

Proof. Let I be the unit chain complex with 0-simplexes \overline{0} and \overline{1} and 1-simplex \overline{I} and the differential ∂(\overline{I}) = \overline{1}−\overline{0}, and let D : I ⊗K → L the chain homotopy between f and g, the composite of Σ_{k}-equivariant chain maps

I ⊗ C ⊗ K^{⊗k} ⟶^{φ⊗1} I^{⊗k} ⊗ C ⊗ K^{⊗k} ≈ C ⊗ (I ⊗ K)^{⊗k} ⟶^{1⊗D^{⊗k}} C ⊗ L^{⊗k}

is the required Σ_{k}-equivariant chain homotopy, where the Σ_{k}-equivariant chain map φ : I ⊗ C → I^{⊗k} ⊗ C is defined as follows.

Let B be a Σ_{k}-basis of C, define φ(\overline{0} ⊗ c) = \overline{0}^{k} ⊗ c, φ(\overline{1} ⊗ c) = \overline{1}^{k} ⊗ c for eachc ∈ B and define

φ(\overline{I} ⊗ σ · c) = σ · (\overline{0}^{k−1} ⊗ \overline{I} + ∑_{j=1}^{k−2} \overline{0}^{k−1−j} ⊗ \overline{I} ⊗ \overline{1}^{j} + \overline{I} ⊗ \overline{1}^{k−1}) ⊗ σ · c

for each σ ∈ Σ_{k}. Proof of Proposition 2.1. By Eilenberg-Zilber Theorem, there are isomorphisms

\overline{H}_{∗}D^{k}(A, A_{o}; X) ≈ H_{∗}(S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o})) ⊗_{Σ_{k}} S_{∗}(X^{k}, (X|∗)^{k}))
≈ H_{∗}(S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o})) ⊗_{Σ_{k}} (S_{∗}(X, ∗))^{⊗k}).

Since S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o}) is a free Σ_{k}-chain complex over F and S_{∗}(X, ∗) ≃ \overline{H}_{∗}(X) as chain complexes over F, where \overline{H}_{∗}(X) with trivial differential (see[19, Lemma VIII.3.1]). The assertion follows by the above Lemma.

Proposition 2.3. If (A, A_{o}) is a relative CW-complex, then

\overline{H}_{∗}D_{k}(A, A_{o}; X) ≈ σ^{−2k}\overline{H}_{∗}D_{k}(A, A_{o}; S^{2}X)

where σ^{−t} is the t-th desuspension.

Proof. By Proposition 2.1, there are isomorphisms

\overline{H}_{∗}D^{k}(A, A_{o}; X) ≈ H_{∗}(S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o})) ⊗_{Σ_{k}} (\overline{H}_{∗}(X))^{⊗k})

and

\overline{H}_{∗}D^{k}(A, A_{o}; S^{2}X) ≈ H_{∗}(S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o})) ⊗_{Σ_{k}} (\overline{H}_{∗}(S^{2}X))^{⊗k})
≈ σ^{2k}H_{∗}(S_{∗}(C̃^{k}(A), C̃^{k}(A|A_{o}) ⊗_{Σ_{k}} \overline{H}_{∗}(X)^{⊗k})

Proposition 2.4. If (A, A_{o}) is a relative CW-complex with an embedding

∐_{k=1}^{∞} C̃^{k}(A)/Σ_{k} → R^{∞},

then there is a stable equivalence

σ : C(A, A_{o}; X) ⟶ ⋁_{k=1}^{∞} D_{k}(A, A_{o}; X)

via stable equivalences

σ_{k} : F_{k}C(A, A_{o}; X) ⟶ ⋁_{j=1}^{k} D_{j}(A, A_{o}; X)

Proof. If (A, A_{o}) = (M, M_{o}) a manifold-pair, this is proved in [3, Prop. 3]. For the general cases, one proceeds in the same way.

Proposition 2.5. Let X be a path connected space and let (A, A_{o}) be a relative CWcomplex with an embedding ∐_{k=1}^{∞} C̃^{k}(A)/Σ_{k} ⟶ R^{∞}. There exists an isomorphism of F-modules

φ : H_{∗}C(A, A_{o}; ⋁_{α∈I} S^{n_{α}}) ⟶ H_{∗}C(A, A_{o}; X)

where \{n_{α}|α ∈ I\} is determinined by \overline{H}_{∗}(X).

Proof. Let \{x_{α}|α ∈ I\} be a basis of \overline{H}_{∗}(X) and let n_{α} = |x_{α}| for α ∈ I, we have an F-isomorphism

φ_{1} : \overline{H}_{∗}(∨_{α}S^{n_{α}}) → \overline{H}_{∗}(X)

Now the assertion follows by Proposition 2.1 and 2.3.

Proposition 2.6. Let M be a smooth compact manifold and let M_{o} and N be the smooth compact submanifolds of M with codimN = 0. If N/M_{o} ∩ N or X is path connected, then

C(N, N ∩ M_{o}; X) → C(M, M_{o}; X) → C(M, N ∪ M_{o}; X)

is a quasifibration.

[5, pp. 113].

Now let M be an m-manifold and let W be a m-manifold without boundary which contains M, e.g. W = M if M is closed, or W = M ∪ ∂M × [0, 1) if M has boundary. Let ξ be the principal O(m)-bundle of the tangent bundle of W. Let Γ_{ξ[S^{m}X]}(B, B_{o}) be the space of cross sections of ξ[S^{m}X] which are defined on B and take values at ∞ ∧ X on B_{o} for each subspace pair (B, B_{o}) in W, where ξ[S^{m}X] is the associated bundle and O(m) acts diagonally on S^{m}X = S^{m} ∧ X, trivially on X and canonically on S^{m} ≅ R^{m} ∪ {∞}.

Proposition 2.7. Let M be a smooth compact manifold and let M_{o} be a smooth compact submanifold of M. If M/M_{o} or X is path connected, then there is a (weak) homotopy equivalence

C(M, M_{o}; X) → Γ_{ξ[S^{m}X]}(W − M_{o}, W − M)

[5, Proposition 3.1 and 3, pp. 178]

Remark 2.8. By Proposition 2.6,there is a homotopy equivalence

C((M, M_{o}) × R^{n}; X) ≃ Ω^{n}C(M, M_{o}; S^{n}X)

if M/M_{o} or X is path connected.

3. Decomposition Theorems

In this section, assume that (M, M_{o}) is a smooth triangulated compact manifoldpair with m =dimM and W is a smooth m-manifold without boundary which contains M and ξ_{W} is the principal O(m)-bundle of the tangent bundle of W (see Proposition 2.6). Let \overline{W} = M if M is closed, or \overline{W} = M × [0, \frac{1}{2}] if M has boundary.

Lemma 3.1. If X is path connected, then there is a (weak) homotopy equivalence

C((M, M_{o}) × R; X) → Γ_{ξ_{W} [ΩS^{m+1}X]}(W − M_{o}, W − M)

where O(m) acts on ΩS^{m+1}X via the functor ΩS(−),i.e. by taking the homeomorphism ΩS(σ ∧ id|_{X}) : ΩS^{m+1}X ⟶ ΩS^{m+1}X for each σ ∈ O(m).

Proof. Notice that (M, M_{o}) × R is isotopic to (M, M_{o}) × I. By Proposition 2.6, it is easy to see that

C((M, M_{o}) × R; X) ≃ Γ_{ξ_{W ×R}[S^{m+1}X]}((W − M_{o}, W − M) × (I, ∂I))

Since ξ_{W ×R}[S^{m+1}X] = π^{∗}ξ_{W} [S^{m+1}X], where π : W ×R → W is the projection, there is

Γ_{ξ_{W ×R}[S^{m+1}X]}((W − M_{o}, W − M) × (I, ∂I))
≃ Γ_{ξ_{W} [ΩS^{m+1}X]}(W − M_{o}, W − M)

Lemma 3.2. Let (N, N_{o}) be a compact submanifold-pair in \overline{W} such that N ⊆ \overline{W} − ∂\overline{W}, then there is a (weak) homotopy equivalence

C((\overline{W} − ν(N_{o}), \overline{W} − ν(N)) × R; X) → Γ_{ξ_{W} [ΩS^{m+1}X]}(N, N_{o})

if X is path connected, where(ν(N), ν(N_{o})) is an open tubular neighborhood of (N, N_{o}) in \overline{W} − ∂\overline{W}.

Proof. By Lemma 3.1, there is a homotopy equivalence

C((\overline{W} −ν(N_{o}), \overline{W} −ν(N))×R; X) ≃ Γ_{ξ_{W} [ΩS^{m+1}X]}(W −(\overline{W} −ν(N)), W −(\overline{W} −ν(N_{o})))

≅ Γ_{ξ_{W} [ΩS^{m+1}X]}(ν(N), ν(N_{o}))
≃ Γ_{ξ_{W} [ΩS^{m+1}X]}(N, N_{o})

Lemma 3.3. Let (N, N_{o}) be a submanifold-pair in W. There is a homeomorphism

Γ_{ξ_{W} [ΩS^{mn+1}X]}(N, N_{o}) ≅ Γ_{ξ_{W^{n}}[ΩS^{mn+1}X]}(Δ_{n}(N), Δ_{n}(N_{o}))

for n ≥ 1, where Δ_{n} : W → W^{n} = W × ... × W the diagonal map, O(mn) acts on ΩS^{mn+1}X via the functor ΩS(−) ,see Lemma 3.1 as above, and O(m) acts on ΩS^{mn+1}X via the diagonal inclusion O(m) → O(mn), σ → diag(σ, σ, ..., σ).

Proof. The diagonal map Δ_{n} : W → W^{n} induces a bundle map \overline{Δ}_{n} : ξ_{W} → ξ_{W^{n}}. Consider the induced map on the total spaces of bundles \overline{Δ}_{n}×1 : Eξ_{W} ×ΩS^{mn+1}X ⟶ Eξ_{W^{n}} × ΩS^{mn+1}X.

For each (x,y)∈ Eξ_{W} × ΩS^{mn+1}X and σ ∈ O(m), notice that

\overline{Δ}_{n} × 1(xσ, σ^{−1}y) = ((x, …, x)σ̃, ΩS(σ^{−1} ∧ … ∧ σ^{−1} ∧ 1_{X})y)
= ((x, …, x)σ̃, σ̃^{−1}y)

where σ̃ = ( \begin{matrix}σ \\ & . & . & . \\ & & & & σ \end{matrix} ) ∈ O(mn).

Hence \overline{Δ}_{n} × 1 induces a map

Δ : E(ξ_{W} [ΩS^{mn+1}X]) = Eξ_{W} ×_{O(m)} ΩS^{mn+1}X ⟶ Eξ_{W^{n}} ×_{O(mn)} ΩS^{mn+1}X

= E(ξ_{W^{n}}[ΩS^{mn+1}X]).

Furthermore Δ induces a homeomorphism

Γ_{ξ_{W} [ΩS^{mn+1}X]}(N, N_{o}) ≅ Γ_{ξ_{W^{n}}[ΩS^{mn+1}X]}(Δ_{n}(N), Δ_{n}(N_{o})),

since the induced map is 1-1,onto and open.

The following lemma follows from the naturality of the Samelson products.

Lemma 3.4. There exists O(m)-map Φ_{n} : (S^{m}X)^{(n)} ⟶ ΩS^{m+1}X which represents the Samelson products [[E, E], …, E] for each n ≥ 2, where X^{(n)} is the reduced join of n copies of X and E : S^{m}X ⟶ ΩS^{m+1}X is the suspension.

Now we give some decomposition theorems.

Theorem C. Let (M, M_{o}) be a smooth triangulated compact manifold-pair, n+m+1 even and n ≥ 1. There exist a manifold-pair (M̃, M̃_{o}) so that dim M̃ = 2m and

C((M, M_{o}) × R; S^{n})_{(p)} ≃^{w} C((M̃, M̃_{o}) × R, S^{2n})_{(p)} × C(M, M_{o}; S^{n})_{(p)}

where m = dim M and p is an odd prime or zero and ≃^{w} means the (weak) homotopy equivalence

Proof. By Lemma 3.1, there is a homotopy equivalence

C((M, M_{o}) × R; S^{n}) ≃ Γ_{ξ_{W} [ΩS^{m+1+n}]}(W − M_{o}, W − M)

The inclusion j : S^{m+n} → ΩS^{m+1+n} is an O(m)-map. By Lemma 3.4, there is an O(m)-Samelson product [j, j] : S^{2m+2n} → ΩS^{m+1+n}, which induces an O(m)-H-map \overline{[j, j]} : ΩS^{2m+1+2n} → ΩS^{m+1+n}.

Now consider the O(m)-map

(\overline{[j, j]}, j) : ΩS^{2m+1+2n} × S^{m+n} ⟶ ΩS^{m+1+n},

which induces a bundle map

ξ_{W} [ΩS^{2m+1+2n} × S^{m+n}] ⟶ ξ_{W} [ΩS^{m+1+n}]

and therefore a map

Γ_{ξ_{W} [ΩS^{2m+1+2n}×S^{m+n}]}(W − M_{o}, W − M) ⟶ Γ_{ξ_{W} [ΩS^{m+1+n}]}(W − M_{o}, W − M)

Notice that

\begin{matrix}Γ_{ξ_{W} [ΩS^{2m+1+2n}×S^{m+n}]}(W − M_{o}, W − M) ≃ \\ ≃ Γ_{ξ_{W} [ΩS^{2m+1+2n}]}(W − M_{o}, W − M) × Γ_{ξ_{W} [S^{m+n}]}(W − M_{o}, W − M) \end{matrix}

By Lemma 3.3

Γ_{ξ_{W} [ΩS^{2m+1+2n}]}(W − M_{o}, W − M) ≃ Γ_{ξ_{W^{2}}[ΩS^{2m+1+2n}]}(Δ(W − M_{o}), Δ(W − M)) ≃ C((M̃, M̃_{o}) × R; S^{2n}) (Lemma 3.2)

where dim M̃ = 2m. Since (\overline{[j, j]}, j) is homotpy equivalent after p-localization, the assertion follows by induction on the handle decomposition of M.

Theorem D. Let (M, M_{o}) be a smooth triangulated compact manifold-pair. If X_{1}, …, X_{k} are connected spaces, there exist manifold pairs

(M_{ω}, M_{oω})

such that C((M, M_{o}) × R; X_{1} ∨ … ∨ X_{k}) is (weak) homotopy equivalent to the (weak) product

∏_{ω} C((M_{ω}, M_{oω}) × R; ω(X_{1}, …, X_{k}))

where ω runs over all addmissible words in x_{1}, …, x_{k} (the addmissible words see [21, pp. 511-514]) and ω(X_{1}, …, X_{k}) = X_{1}^{(a_{1})} ∧ … ∧ X_{k}^{(a_{k})} and X^{(n)} is the reduced join of n copies of X and a_{i} is the number of occurences of x_{i} in the word ω.

Proof. Denote X = X_{1} ∨ … ∨ X_{k}. By Lemma 3.1, there is a homotopy equivalence

C((M, M_{o}) × R; X) ≃ Γ_{ξ_{W} [ΩS^{m+1}X]}(W − M_{o}, W − M)

Consider the inclusions

i_{t} : S^{m}X_{t} → ΩS^{m+1}X for 1 ≤ t ≤ k

By Lemma 3.4, there are O(m)-Samelson products

ω(i_{1}, …, i_{k}) : S^{l(ω)·m}ω(X_{1}, …, X_{k}) ⟶ ΩS^{m+1}X

for each word ω, where l(ω), is the length of ω, which induces an O(m)-H-map

ω̃(i_{1}, …, i_{k}) : ΩS^{l(ω)·m}ω(X_{1}, …, X_{k}) ⟶ ΩS^{m+1}X

Consider the O(m)-map

f = ∏_{ω} ω̃(i_{1}, …, i_{k}) : ∏_{ω} ΩS^{l(ω)·m}ω(X_{1}, …, X_{k}) ⟶ ΩS^{m+1}X

which induces a bundle map

ξ_{W} [∏_{ω} ΩS^{l(ω)·m}ω(X_{1}, …, X_{k})] ⟶ ξ_{W} [ΩS^{m+1}X]

and therefore a map

f̃: Γ_{ξ_{W} [∏_{ω} ΩS^{l(ω)·m}ω(X_{1},... ,X_{k})]}(W − M_{o}, W − M)
⟶ Γ_{ξW [ΩS^{m+1}X]}(W − M_{o}, W − M).

By Hilton-Milnor Theorem, f is a (weak) homotopy equivalence. By induction on the handle decomposition of M,f̃ is a (weak) homotopy quivalence.

Notice that

Γ_{ξ_{W} [∏_{ω} ΩS^{l(ω)·m}ω(X_{1},... ,X_{k})]}(W − M_{o}, W − M)

≃ Γ_{ξ_{W} [∏_{ω} ΩS^{l(ω)+1}ω(X_{1},... ,X_{k})]}(\overline{W} − v(M_{o}), \overline{W} − v(M))

≅ ∏_{ω} Γ_{ξ_{W} [ΩS^{l(ω)+1}ω(X_{1},... ,X_{k})]}(\overline{W} − v(M_{o}), \overline{W} − v(M))

≅ ∏_{ω} Γ_{ξ_{W^{l(ω)}}[ΩS^{l(ω)+1}ω(X_{1},... ,X_{k})]}(Δ_{l(ω)}(\overline{W} − v(M_{o})), Δ_{l(ω)}(\overline{W} − v(M)))

≃ ∏_{ω} C((M_{ω}, M_{oω}) × R; ω(X_{1}, …, X_{k})) (Lemma 3.2)

where dim M_{ω} = dim W^{l(ω)} = l(ω) · m.

The assertion follows.

4. Proofs of Theorems A and B.

Lemma 4.1. Let (N, N_{o}) ⊆ (M, M_{o}) be the smooth compact submanifold pair of (M, M_{o}) with dim N = dim M and let X be a simply connected space. If H_{∗}(N, N_{o}) → H_{∗}(M, M_{o}) is onto, then

H_{∗}C((N, N_{o}) × R^{n}; X) ⟶ H_{∗}C((M, M_{o}) × R^{n}; X)

is onto for n ≥ 1.

Proof. Since (M, M_{o}) × R^{n} is isotopic to (M × I^{n−1}, M_{o} × I^{n−1}) × R, it is sufficient to show that

H_{∗}C((N, N_{o}) × R; X) ⟶ H_{∗}C((M, M_{o}) × R; X)

is onto. Step1: Assume that X = S^{n} with n > 1.

If m + 1 + n is odd or F is of characteristic 2, this was proved in [5, Th. A], where m = dim M. Now assume that m+1+n is even and F is of characteristic ≠ 2. By Theorem C, there is a p-homotopy equivalence

C((M, M_{o}) × R; S^{n}) ≃ C((M̃, M̃_{o}) × R; S^{2n}) × C(M, M_{o}; S^{n})

and

C((N, N_{o}) × R; S^{n}) ≃ C((Ñ, Ñ_{o}) × R; S^{2n}) × C(N, N_{o}; S^{n})

where dim M̃ + 1 + 2n = dim Ñ + 1 + 2n = 2m + 2n + 1 and dim M + n = dim N + n = m + n. The assertion follows by [5, Th. A].

Step 2: Assume that X = S^{d_{1}} ∨ … ∨ S^{d_{k}} with d_{j} > 1.

By Theorem D, there is a (weak) homotopy equivalence

C((M, M_{o}) × R; X) ≃ ∏_{ω} C((M_{ω}, M_{oω}) × R; ω(X))

where ω(X) = S^{d_{1}a_{1}} ∧ … ∧ S^{d_{k}a_{k}} = S^{d_{1}a_{1}+···+d_{k}a_{k}}. The assertion follows from Step 1.

Step 3: Assume H_{∗}(X) is a finite dimensional F-vector space. The assertion follows

from Step 2 and Proposition 2.4. Step 4: General case.

There exist X_{α} such that H_{∗}(X) = lim_{α} H_{∗}(X_{α}) and H_{∗}(X_{α}) is finite dimensional F-vector space for each α. The assertion follows from Step 3.

Proof of Theorem A. First we will prove the absolute case M_{o} = ϕ by induction on on a handle decomposition of M. If M is a disjoint union M_{1} ⨿ M_{2}, then

C(M × R^{n}; X) ≅ C(M_{1} × R^{n}; X) × C(M_{2} × R^{n}; X)

Hence we can restrict to connected manifolds and start with M = I^{m}, the assertion is obvious for M = I^{m}. Assume that the assertion holds for M and \overline{M} = M ∪D with D ≅ I^{m} a handle of index q i.e. D ∩ M ≅ I^{m−q} × ∂I^{q}, q ≥ 1 since M is connected. There is a cofibration

M →^{i} \overline{M} →^{j} (\overline{M}, M) ≃ (I^{m}, I^{m−q} × ∂I^{q}) ≃ (S^{q}, ∗)

and alternative

I. H_{q}(\overline{M}) ⟶^{j_{∗}} H_{q}(S^{q}) is onto.

II. H_{q}(\overline{M}) ⟶ H_{q}(S^{q}) is zero. Case I. Consider the quasifibration

C(M × R^{n}; X) ⟶ C(\overline{M} × R^{n}; X) ⟶^{C(j)} C((\overline{M}, M) × R^{n}; X)

≃ Ω^{m+n−q}S^{m+n}X

Since j_{∗} is onto, C(j)_{∗} is onto and the Serre spectral sequence for the quasifibration above collapses. Hence there is a short exact sequence of Hopf algebras

H_{∗}C(M × R^{n}; X) ≻⟶ H_{∗}C(\overline{M} × R^{n}; X) ⟶≻ H_{∗}Ω^{m+n−q}S^{m+n}X

By Proposition 2.3, there is a commutative diagram

Σ^{∞}C(\overline{M} × R^{n}; X) →^{p}  Σ^{∞}D_{k}\overline{M} × R^{n}; X)
↓  ↓
Σ^{∞}C((\overline{M}, M) × R^{n}; X) →^{p} Σ^{∞}D_{k}((\overline{M}, M) × R^{n}; X)

for each k ≥ 1, where p is the projection.

Thus

\overline{H}_{∗}D_{k}(\overline{M} × R^{n}) ⟶ \overline{H}_{∗}D_{k}((\overline{M}, M) × R^{n}; X)

is an epimorphism and F_{r}H_{∗}C(\overline{M} × R^{n}; X) ⟶ F_{r}H_{∗}Ω^{m+n−q}S^{m+n}X is an epimorphism by Proposition 2.3.

Hence there exists an F-map

φ : H_{∗}Ω^{m+n−q}S^{m+n}X ⟶ H_{∗}C(\overline{M} × R^{n}; X)

such that (1) φ preserves the filtration.

(2) C(j)_{∗} ∘ φ = id

Now the composite

H_{∗}C(M × R^{n}; X) ⊗ H_{∗}Ω^{m+n−q}S^{m+n}X ⟶ H_{∗}C(\overline{M} × R^{n}; X)^{⊗2}
⟶ H_{∗}C(\overline{M} × R^{n}; X)

is an isomorphism of filtered modules and (1) follows.

If n > 1, let φ : QH_{∗}(Ω^{m+n−q}S^{m+n}X) ⟶ QH_{∗}C(\overline{M} ×R^{n}; X) so that QC(j)_{∗}∘φ = id and φ preserves the filtration.

The homomorphism

QH_{∗}Ω^{m+n−q}S^{m+n}X ⟶ QH_{∗}C(\overline{M} × R^{n}; X) ⟶ H_{∗}C(\overline{M} × R^{n}; X)

induces an algebra map

H_{∗}Ω^{m+n−q}S^{m+n}X ⟶ H_{∗}C(\overline{M} × R^{n}; X)

since H_{∗}C(\overline{M} × R^{n}; X) is a commutative algebra and H_{∗}Ω^{m+n−q}S^{m+n}X is a free commutative algebra. Thus (2) follows. Case II. Consider the quasifibration

ΩC((\overline{M}, M) × R^{n}; X) ⟶ C(M × R^{n}; X) ⟶^{C(i)} C(\overline{M} × R^{n}; X)

Since i_{∗} is onto, C(i)_{∗} is onto and the Serre spectral sequence for the quasifibration above collapses. Hence

H_{∗}C(\overline{M} × R^{n}; X) ≈ H_{∗}C(M × R^{n}; X)//H_{∗}Ω^{m+n−q+1}S^{m+n}X

Both (1) and (2) follow. To treat the relative case, we can assume that M_{o} is part of a closed collar (see

[5]). We will prove by induction om a handle decomposition of M, and start with M = M_{o}, the assertion is obvious for M = M_{o} since F (Mo, M_{o}) ≃ ∗. Assume that the assertion holds for (M, M_{o}). If \overline{M} = M ∪ D with D ≅ I^{m} a hand of index q in M. Clearly we can assume that q ≥ 1. There is a cofibration

(M, M_{o}) → (\overline{M}, M_{o}) → (\overline{M}, M) ≃ (I^{m}, I^{m−q} × ∂I^{q})

and again an alternative:

III. H_{q}(\overline{M}, M_{o}) → H_{q}(\overline{M}, M) is onto. IV. H_{q}(\overline{M}, M_{o}) → H_{q}(\overline{M}, M) is zero. For Case III and IV, the assertion follows from Lemma 4.1 similar to Case I and

II respectively.

Proof of Theorem B. By Theorem A, there is an isomorphism of filtered F-modules

H_{∗}C((M, M_{o}) × R^{n}; S^{2}X) ≈ C^{m+n}(H_{∗}(M, M_{o}); S^{2}X).

Hence

\overline{H}_{∗}D_{k}((M, M_{o}) × R^{n}; S^{2}X) ≈ D_{k}^{m+n}(H_{∗}(M, M_{o}); S^{2}X).

By Proposition 2.2 and Proposition 2.3, there are isomorphisms of F-modules

\overline{H}_{∗}D_{k}((M, M_{o}) × R^{n}; X) ≈ σ^{−2k}\overline{H}_{∗}D_{k}((M, M_{o}) × R^{n}; S^{2}X)

and

\overline{H}_{∗}C((M, M_{o}) × R^{n}; X) ≈ ⨁_{k=1}^{∞} \overline{H}_{∗}D_{k}((M, M_{o}) × R^{n}; X)

The assertion follows.

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