Homotopy and homology of fibred spaces

Hans-Joachim Baues and Davide L. Ferrario

Abstract

We study fibred spaces with fibres in a structure category 𝔉 and we show that cellular approximation, Blakers–Massey theorem, Whitehead theorems, obstruction theory, Hurewicz homomorphism, Wall finiteness obstruction, and Whitehead torsion theorem hold for fibred spaces. For this we introduce the cohomology of fibred spaces.

AMS SC : 55R55 (Fiberings with singularities); 54H15 (Transformation groups and semigroups). 55R65 (Generalizations of fiber spaces and bundles); 55R70 (Fiberwise topology);

1 Introduction

A fibred space X is a map

p: X → X̄

for which fibres p^{−1}x, x ∈ X̄, are objects in a structure category 𝔉. For example a G-space X for a topological group G which acts properly on X yields the fibred space

p: X → X/G

where X/G is the orbit space. Here fibres are objects in the orbit category 𝔉_{G} consisting of G-orbits G/H and G-equivariant maps. Moreover fibre bundles [14, 31] and stratified fibre bundles [5] are examples of fibred spaces. In this paper we study the homotopy theory of fibred spaces. In particular we show that fundamental results like Whitehead theorems, cohomology, obstructions to the extension of maps, Wall-finiteness obstruction and Whitehead torsion are available for fibred spaces. These results form the bulk of the book [22] in the case of G-spaces. As it turns out homotopy theory of G-spaces to a large extent is just a specialization of the homotopy theory of fibred spaces.

The theory of 𝔉-fibred spaces yields a new flexibility since the structure category 𝔉 is a parameter of the theory. For example all constructions and obstructions are natural in 𝔉 and such naturality is useful similarly as in the case of naturality of coefficients in cohomology. By comparing the orbit category 𝔉_{G} with other structure categories 𝔉 one also obtains applications to the theory of transformation groups. On the other hand our theory can be applied to stratified bundles like stratified vector bundles which appear as tangent bundles of stratified manifolds, see [21, 4, 5]. Obstructions to non trivial sections of stratified vector bundles can be studied by use of the obstruction theory of fibred spaces (such problems were treated in [29]).

A self-contained proof of the results in this paper is highly elaborate and requires a rewriting of the book [22] (more than 400 pages) for fibred spaces. In this paper, however, we prove the results by use of the axiomatic approach of [3]. For this we only need to show that the Blakers–Massey theorem holds in the category of fibred spaces. This implies that we can use the methods in [3] leading, in particular, to a new construction of the cellular chain complex C_{∗}(X, A) which is simpler than the construction of Bredon [6, 7, 33, 22].

Our homotopy theory of fibred spaces with fibres in the (topological enriched) category 𝔉 is closely related to the theory of continuous 𝔉^{op}-diagrams of spaces (compare with the principal diagram in [5]). In case 𝔉 is discrete such diagrams are treated in chapter A of [3]. It is easy to transform the results in this paper to the theory of continuous diagrams by use of [3].

The methods of [3], in fact, yield further results on fibred spaces not treated in this paper. We leave it to the reader to study the “model lifting property” of the chain complex C_{∗}(X, A), the “obstruction for the realizability” of a chain complex, the “tower of categories” for the classification of homotopy classes of maps between fibred spaces and the associated spectral sequences.

On the other hand we refer to the recent book of Crabb and James [10] for a different perspective on spaces of fibres and for a detailed account of the areas of fibrewise topology and fibrewise homotopy theory.

2 Fibre families

Let 𝔉 be a small category together with a faithful functor F: 𝔉 → Top to the category Top of compactly generated Hausdorff spaces. Then 𝔉 is termed a structure category and F is the fibre functor on 𝔉. In many examples the functor F is actually the inclusion of a subcategory 𝔉 of Top so that in this case we need not to mention the fibre functor F.

A fibre family with fibres in 𝔉 (or a (𝔉, F)-family) is a topological space X, termed total space, together with a map p_{X}: X → X̄, termed projection to the base space X̄, and for every b ∈ X̄ a selected homeomorphism Φ_{b}: p_{X}^{−1}b ≈ F X_{b} where X_{b} is an object in 𝔉, called fibre, depending on b ∈ X̄. The homeomorphism Φ_{b} is termed chart at b. The family (p_{X}: X → X̄, X_{b}, Φ_{b}, b ∈ X̄) is denoted simply by X. A fibre family is also termed a fibred space with fibres in 𝔉.

Given two 𝔉-families X and Y a 𝔉-map from X to Y is a pair of maps (f, f̄) such that the following diagram

\xymatrix{X \ar[r]^{f} \ar[d]_{p_{X}} & Y \ar[d]^{p_{Y}} \\ X̄ \ar[r]^{f̄} & Ȳ ,}

commutes, and such that for every b ∈ X̄ the composition given by the dotted arrow of the diagram

\xymatrix{p_{X}^{−1}(b) \ar[r]^{f|p^{−1}(b)} & p_{Y}^{−1}(f̄b) \ar[d]^{Φ_{f̄b}} \\ F X_{b} \ar[u]^{Φ_{b}^{−1}} \ar@{.>}[r] & F Y_{f̄b}}

is a morphism in the image of the functor F. That is, there exists a morphism ϕ: X_{b} → Y_{f̄b} in 𝔉 such that the dotted arrow is equal to F (ϕ). We will often denote F X_{b} by X_{b} and it will be clear from the context whether X_{b} denotes an object in 𝔉 or a space in Top given by the functor F. If a 𝔉-map f = (f, f̄) is a 𝔉-isomorphism then f and f̄ are homeomorphisms but the converse need not be true.

If X is a 𝔉-family and Z is a topological space, then the product X × Z in Top is a 𝔉-family with projection p_{X×Z} = p × 1_{Z}: X × Z → X̄ × Z. The fibre over a point (b, z) ∈ X̄ × Z is equal to p_{X}^{−1}(b) × {z}; using the chart Φ_{b} : p_{X}^{−1}(b) → X_{b} the chart Φ_{(b,z)} is defined by (x, z) ↦ Φ_{b}(x) ∈ X_{b} where of course we set (X × Z)_{(b,z)} = X_{b}. In particular, by taking Z = I the unit interval we obtain the cylinder object IX = X ×I and therefore the notion of homotopy: two 𝔉-maps f_{0}, f_{1} : X → Y are 𝔉-homotopic (in symbols f_{0} ∼ f_{1}) if there is a 𝔉-map F: IX → Y such that f_{0} = F i_{0} and f_{1} = F i_{1}. Here i_{0} and i_{1} are the inclusions X → X × I at the levels 0 and 1 respectively.

Let 𝔉-Top be the category consisting of 𝔉-families p_{X} : X → X̄ in Top and 𝔉-maps. Homotopy of 𝔉-maps yields a natural equivalence relation ∼ on 𝔉-Top so that the homotopy category (𝔉-Top)/_{∼} is defined.

(2.1) Definition. If V is a fibre in 𝔉 and X̄ is a space in Top, then the projection onto the first factor p_{1} : X = X̄ ×V → X̄ yields the product family with fibre V; the charts Φ_{b}: {b} × V → V = X_{b} are given by projection and X_{b} = V for all b ∈ X̄. If X is a 𝔉-family 𝔉-isomorphic to a product family then X is said to be a trivial 𝔉-bundle. In general a 𝔉-bundle is a locally trivial family of fibres, i.e. a family X over X̄ such that every b ∈ X̄ admits a neighborhood U for which X|U is trivial. Here X|U is the restriction of the family X defined by U ⊂ X̄.

Given a family Y with projection p_{Y} : Y → Ȳ and a map f̄: X̄ → Ȳ, the pull-back X = f̄^{∗}Y is the total space of a family of fibres given by the vertical dotted arrow of the following pull-back diagram in Top.

\xymatrix{X = f̄^{∗}Y \ar@{.>}[r] \ar@{.>}[d] & Y \ar[d]^{p} \\ X̄ \ar[r]^{f̄} & Ȳ .} (2.2)

The charts are defined as follows: For every b ∈ X̄ let X_{b} = Y_{f̄b}, and Φ_{b} : p_{X}^{−1}(b) → X_{b} the composition p_{X}^{−1}(b) → p_{Y}^{−1}(f̄b) ≈ Y_{f̄b} = X_{b} where the map p_{X}^{−1}(b) → p_{Y}^{−1}(f̄b) is a homeomorphism since X is a pull-back.

A 𝔉-map i: A → Y is termed a closed inclusion if ī: Ā → Ȳ is an inclusion, īĀ is closed in Ȳ and the following diagram is a pull-back:

\xymatrix{i^{∗}Y = A \ar[r]^{i} \ar[d]_{p_{A}} & Y \ar[d]^{p_{Y}} \\ Ā \ar[r]^{ī} & Y} (2.3)

Hence a closed inclusion i: A → Y induces homeomorphisms on fibres.

The push-out construction can be extended to the category 𝔉-Top, provided the push-out is defined via a closed inclusion, see [5] (2.5).

(2.4) Lemma. Given 𝔉-families A, X, Y and 𝔉-maps f: A → X, i: A → Y with i a closed inclusion the push-out diagram in 𝔉-Top

\xymatrix{A \ar[r]^{f} \ar[d]_{i} & X \ar[d] \\ Y \ar[r] & Z}

exists and X → Z is a closed inclusion.

Let A ⊂ Y be a closed inclusion then also A × I ⊂ Y × I is a closed inclusion and the relative cylinder I_{A}Y is defined by the push-out

\xymatrix{A × I \ar[r]^{pr} \ar[d] & A \ar[d] \\ Y × I \ar[r] & I_{A}Y}

in 𝔉-Top. A map F: I_{A}Y → X is termed a homotopy f_{0} ∼ f_{1} rel A with f_{0} = F i_{0} and f_{1} = F i_{1}. Let g: A → X given. Then

[Y, X]_{𝔉}^{A} = [Y, X]_{𝔉}^{g} = {f: Y → X, f|A = g} /_{∼ rel A} (2.5)

denotes the set of homotopy classes relative A or under A. We call a closed inclusion (Y, Y′) a pair in 𝔉-Top and maps between pairs are defined as usual. Moreover for a closed inclusion A ⊂ Y′ and a map g: A → X′ we obtain the set [(Y, Y′), (X, X′)]_{𝔉}^{A} of homotopy classes of pair maps under A.

(2.6) Remark. All results in this paper remain true if Top is the category of all topological spaces and if the fibre functor F satisfies condition (∗): For every object V in 𝔉 the space F (V) is locally compact and Hausdorff. In [5] we assumed the fibres to be also second countable, in order to deal with metrizable spaces and avoid many unnecessary technicalities, which do not occur in homotopy theory of 𝔉-families.

3 𝔉-complexes

For an object V in 𝔉 the family F V → ∗ with base space a singleton is termed a 𝔉-point also denoted by V. A disjoint union of 𝔉-points is called a 𝔉-set. This is a 𝔉-family for which the base space has the discrete topology. Let D^{n} be the unit disc in ℝ^{n} and S^{n−1} its boundary with base point ∗ ∈ S^{n−1}. The complement e^{n} = D^{n} ∖ S^{n−1} is the open cell in D^{n}. A 𝔉-cell is a product family V × e^{n} → e^{n} with V ∈ 𝔉.

We say that a 𝔉-family X is obtained from a 𝔉-family A by attaching n-cells if a 𝔉-set Z together with a 𝔉-map f is given, such that the following diagram

\xymatrix{Z × S^{n−1} \ar[r]^{f} \ar[d] & A \ar[d] \\ Z × D^{n} \ar[r]^{Φ} & X = A ∪_{f} (Z × D^{n})} (3.1)

is a push-out in 𝔉-Top. The inclusion Z × S^{n−1} → Z × D^{n} is a closed inclusion, therefore the push-out exists and the induced map A → X is a closed inclusion and X ∖ A = Z × e^{n} is a union of open 𝔉-cells. If Z is a 𝔉-point then we say that X is obtained from A by attaching a 𝔉-cell and Φ is the characteristic map of the 𝔉-cell.

(3.2) Definition. A relative 𝔉-complex (X, A) is a family X and a filtration

A = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ X_{n+1} ⊂ ⋯ ⊂ X

of 𝔉-families X_{n}, n ≥ −1, such that for every n ≥ 0 the 𝔉-family X_{n} is obtained from X_{n−1} by attaching n-cells and

X = lim_{n≥0} X_{n}.

The spaces X_{n} are termed n-skeleta of (X, A). If A is empty we call X a 𝔉-complex. Then X is a union of 𝔉-cells, that is, there are the 𝔉-sets Z_{n} of n-cells in X ∖ A such that X ∖ A is the union of Z_{n} × e^{n} for n ≥ 0.

(3.3) Example. Let G be a compact Lie group and let 𝔉 be the category of orbits of G, that is, 𝔉 is the subcategory of Top consisting of spaces G/H, where H is a closed subgroup of G, and G-equivariant maps G/H → G/H′. Then each G-CW-complex (see [33]) is a 𝔉-complex.

(3.4) Example. Each 𝔉-stratified fibre bundle with finitely many strata is 𝔉-homotopy equivalent to a 𝔉-complex [5].

A map f: (K, A) → (X, A) under A between relative 𝔉-complexes is termed cellular if f(K_{n}) ⊂ X_{n} for n ≥ 0.

(3.5) Theorem (Cellular approximation). Let (K, A) and (X, A) be relative 𝔉-complexes; let L be a subcomplex of K and g: K → X a 𝔉-map under A such that the restriction to L ⊂ K is cellular. Then there is a cellular map f: K → X extending g|L and a homotopy f ∼ g rel L.

Proof. The proof is identical to the classical case of CW-complexes, once lemma (5.1) is proved. Alternatively one can use IV.5.8 Baues [3] to obtain the result by use of the Blakers-Massey theorem (5.3) below. q.e.d.

(3.6) Remark. In [5] we have seen that 𝔉-complexes are stratified bundles which generalize the classical notion of bundle. The bundle theorem of [5] shows: Let 𝔉 be a structure category which is a groupoid. Then a 𝔉-complex X → X̄ is a 𝔉-bundle over X̄. Conversely, each 𝔉-bundle X → X̄ over a CW-complex X̄ is 𝔉-isomorphic to a 𝔉-complex.

(3.7) Remark. The definition of 𝔉-complex follows the lines of [2] (page 194). After Whitehead, Bredon defined G-complexes for G finite [6], independently Matumoto [23] and Illmann [16] extended the definition to the case G a compact Lie group. See [33] for a general approach of G-CW complexes and [22] for its consequences in K-theory of G-complexes. A notion of fibrewise CW-complex was presented by James in [18] (see also the decomposition of Bredon O(n)-manifolds W_{k}^{2n−1} there) in the framework of fibrewise topology [17]. While in the case of G-CW-complexes the structure category is evidently the orbit category of G, the structure category of a fibrewise complex [18] is the full subcategory of Top containing the fibres and the fibres are assumed to be compact.

4 Homotopy groups

A 𝔉-family X in 𝔉-Top is pointed if a 𝔉-map ∗_{V} : V → X is given, where V denotes the 𝔉-point given by the object V in 𝔉. Such a map is termed a base point of X. A pointed pair (X, Y, ∗_{V}) of 𝔉-families is given by a closed inclusion Y ⊂ X and a base point ∗_{V} : V → Y. As usual a base point is chosen in the sphere S^{n−1} so that the pair (D^{n}, S^{n−1}, ∗) is pointed in Top. For every n ≥ 0 and 𝔉-point V let S_{V}^{n} and D_{V}^{n+1} denote the pointed 𝔉-family S_{V}^{n} = V × S^{n} over S^{n} and D_{V}^{n+1} = V × D^{n+1} over D^{n+1} with S_{V}^{n} ⊂ D_{V}^{n+1}.

(4.1) Definition. For every pointed 𝔉-family (X, ∗_{V}) the homotopy group

π_{n}^{𝔉}(X; ∗_{V}) = [S_{V}^{n}, X]_{𝔉}^{V} (4.2)

is the set of homotopy classes relative V of 𝔉-maps f: S_{V}^{n} → X as in the commutative diagram (4.3) in 𝔉-Top.

[figure: V × ∗; V; ∗_{V}; S_{V}^{n}; V × S^{n}; f; X] (4.3)

Moreover the relative homotopy group

π_{n+1}^{𝔉}(X, Y; ∗_{V}) = [(D_{V}^{n+1}, S_{V}^{n}), (X, Y)]_{𝔉}^{V} (4.4)

is the set of homotopy classes relative V of pair maps f: (D_{V}^{n+1}, S_{V}^{n}) → (X, Y).

The homotopy groups above can also be described as ordinary homotopy groups of function spaces. For a fibre family X and V in 𝔉 let X^{V} be the space of all 𝔉-maps V → X with the compact-open topology in Top. For example the faithful fibre functor F yields the identification

W^{V} = hom_{𝔉}(V, W)

so that 𝔉 is a topological enriched category. Function spaces yield the functor

X^{∘}: 𝔉^{op} → Top (4.5)

which carries V to X^{V} and α: V → W to the induced map α^{∗}: X^{W} → X^{V} with α^{∗}(x) = x ∘ F (α).

We have canonical isomorphisms

π_{n}^{𝔉}(X; ∗_{V}) = π_{n}(X^{V}; ∗_{V}) (4.6)

π_{n}^{𝔉}(X, Y; ∗_{V}) = π_{n}(X^{V} , Y^{V}; ∗_{V})

where the right hand side denotes the homotopy groups in Top.

We will prove in section 8 the following Whitehead theorem in (𝔉-Top)^{A}.

(4.7) Theorem. A map f: (X, A) → (Y, A) under A between relative 𝔉complexes is a 𝔉-homotopy equivalence under A if and only if f induces isomorphisms

f_{∗}: \xymatrix{π_{n}^{𝔉}(X; ∗_{V}) \ar[r]^{≃} & π_{n}^{𝔉}(Y; f∗_{V})}

for all n ≥ 0, basepoints ∗_{V} : V → X, and objects V in 𝔉.

(4.8) Corollary. Let f be a 𝔉-map f: (X, A) → (Y, A) under A between relative 𝔉-complexes. Then the following statements are equivalent.

(i) The 𝔉-map f is a 𝔉-homotopy equivalence under A;

(ii) For every V in 𝔉 the induced map f^{V} : X^{V} → Y^{V} is a weak homotopy equivalence in Top.

Proof. It is clear that (i) ⟹ (ii). But if f^{V} is a weak equivalence in Top for every V, then by (4.7) (i) holds. q.e.d.

(4.9) Remark. It is clear that in (4.8) we can replace (ii) with the following: For every V in 𝔉 the induced map f^{V} : X^{V} → Y^{V} is a homotopy equivalence in Top. Thus we obtain a generalization to 𝔉-complexes of James–Segal theorem [19] for G-ANR’s. See also [13].

5 The Blakers-Massey theorem

We say that a pair (X, Y) is n-connected if for all basepoints ∗_{V} : V → Y, V ∈ 𝔉 and 1 ≤ r ≤ n the homotopy group π_{r}^{𝔉}(X, Y; ∗_{V}) = 0 are trivial and π_{0}^{𝔉}(Y; ∗_{V}) → π_{0}^{𝔉}(X; ∗_{V}) is surjective.

(5.1) Lemma. For a relative 𝔉-complex (X, A) the pair (X, X_{n}) is n-con-nected, n ≥ 0.

Proof. Let V ∈ 𝔉. As in the classical case it suffices to prove that for an attachment X = A∪_{α}(V ×D^{n}) as in (3.1) the pair (X, A) is (n−1)-connected. For this we use similar arguments and notations as in the proof of 13.5 of [12]. Let g: v × (B^{r}, S^{r−1}) → (X, A) be a map which represents an element in π_{r}^{𝔉}(X, A; ∗_{V}), with r < n. Using the classical map χ: (I^{r}, ∂I^{r}, J^{r−1}) → (B^{r}, S^{r−1}, ∗) we obtain the composite f′ = gχ which plays the role of f in the proof of 13.5 of [12]. The map f′ induces the commutative diagram

\xymatrix{V × (I^{r}, ∂I^{r}, J^{r−1}) \ar[r]^{f′} \ar[d] & (A ∪ (V × e^{n}), A, A_{∗}) \ar[d] \\ (I^{r}, ∂I^{r}, J^{r−1}) \ar[r]^{f} & (Ā ∪ e^{n}, Ā, ∗)}

Here ∗ ∈ Ā is determined by ∗_{V} and A_{∗} ∈ 𝔉 is the fibre of A over ∗. As in 13.5 of [12] we obtain for f the subspace Ū ⊂ I^{r} with f(Ū) ⊂ e^{n} and the homotopy h_{t}: Ū → e^{n} (t ∈ [0, 1]) relative ∂Ū with h_{0} = f|Ū. Hence we obtain by f′ the commutative diagram

\xymatrix{V × Ū \ar[r]^{f′|(V ×Ū)} \ar[d]^{q} & V × e^{n} \ar[d]^{q} \\ Ū \ar[r]^{f|Ū} & e^{n}}

The map f′|(V × Ū) is a 𝔉-map which is determined by (f|Ū)_{q} and the coordinate f′′: V × Ū → V. We define a 𝔉-homotopy

h′_{t}: V × Ū → V × e^{n}

by h′_{t} = (h_{t}p, f′′). We have h_{0} = f′|(V × Ū) and h′_{t} us a 𝔉-homotopy relative V × ∂Ū. Hence we can define a 𝔉-homotopy

H′_{t}: V × I^{r} → A ∪ (V × e^{n}) with H′_{0} = f′

which induces H_{t}: I^{r} → Ā ∪ e^{n} in the proof of 13.5 of [12]. We obtain H′_{t} by

H′_{t}(x, u) = \begin{cases}h′_{t}(x, u) & for u ∈ Ū, x ∈ V \\ f′(x, u) & for u ∈ I^{r} ∖ U, x ∈ V \end{cases}

Now we can choose p ∈ e^{n}(1/2) ⊂ e^{n} with p ∉ image(H_{1}); see 13.5 [12]. Hence the map H′_{1} has a factorization

H′_{1} : V × I^{r} → A ∪ (V × (e^{n} ∖ p)) ⊂ A ∪ (V × e^{n}).

Here the inclusion A → A ∪ (V × (e^{n} ∖ p)) is an 𝔉-homotopy equivalence. This completes the proof. q.e.d.

(5.2) Lemma. Let (X, A) be a relative 𝔉-complex and let (X, A) be nconnected, with n ≥ 0. Then there exists a relative 𝔉-complex (Y, A) with Y_{0} = Y_{1} = ⋯ = Y_{n} = A and a 𝔉-homotopy equivalence Y → X under A.

Proof. The map Y → X is obtained inductively by attaching “ball pairs” and collapsing the non-attached part of the boundary of the ball pair; compare the proof of (6.14) in [32]. q.e.d.

(5.3) Theorem. Let (X, A) a relative 𝔉-complex with subcomplexes X_{1},X_{2}, such that X = X_{1} ∪ X_{2}. Let A be the intersection A = X_{1} ∩ X_{2}. If (X_{1}, A) is n_{1}-connected and (X_{2}, A) is n_{2}-connected, with n_{1}, n_{2} ≥ 0, then for each basepoint ∗_{V} : V → A with V ∈ 𝔉 the induced map

i_{∗}: π_{r}^{𝔉}(X_{1}, A; ∗_{V}) → π_{r}^{𝔉}(X, X_{2}; ∗_{V})

is an isomorphism if r < n_{1} + n_{2} and is a surjection if r = n_{1} + n_{2}.

(5.4) Remark. Using the spaces (X^{V} , X_{1}^{V} , X_{2}^{V} , A^{V}) and the fact that A^{V} is a (𝔉, F^{V})-neighborhood deformation retract in X_{1}^{V} and X_{2}^{V} it is possible to prove theorem (5.3) also by using the adjoints of maps and the homotopy excision theorem of Spanier [30].

Proof. As in the proof of 16.27 of [12] it suffices to consider the case when A = X_{1} ∩ X_{2} and X_{1} and X_{2} are obtained from A by attaching a single cell, namely X = X_{1} ∪_{A} X_{2} with

X_{1} = A ∪_{β} (S × D^{m})

X_{2} = A ∪_{α} (R × D^{n})

with n = n_{1} + 1 and m = n_{2} + 1 and R, S ∈ 𝔉. In this case we generalize the proof of 13.6 of [12] as follows.

Let p ∈ e^{n} and q ∈ e^{m} and let R_{p} = R and S_{q} = S be the fibres of X over p and q respectively. Then we obtain the commutative diagram

\xymatrix{π_{r}^{𝔉}(X_{1}, A; ∗_{V}) \ar[r]^{i_{∗}} \ar[d]^{≅} & π_{r}^{𝔉}(X, X_{2}; ∗_{V}) \ar[d]^{≅} \\ π_{r}^{𝔉}(X ∖ R_{p}, X ∖ R_{p} ∖ S_{q}; ∗_{V}) \ar[r]^{j_{∗}} & π_{r}^{𝔉}(X, X ∖ S_{q}; ∗_{V})} (5.5)

in which the vertical arrows are isomorphisms. This is readily seen by applying (2.4). The diagram is the analogue of 13.7 of [12].

Next we consider an 𝔉-map

\xymatrix{V × I^{r} \ar[r]^{h′} \ar[d] & X = A ∪ (R × E^{n}) ∪ (e^{m} ∪ S) \ar[d] \\ I^{r} \ar[r]^{h} & X̄ = Ā ∪ e^{n} ∪ e^{m}}

which induces the map h in 13.9 of [12]. Using H_{t} in the proof of 13.9 of [12] we obtain a 𝔉-homotopy

\xymatrix{V × I^{r} \ar[d] \ar[r]^{H′_{t}} & X \ar[d] \\ I^{r} \ar[r]_{H_{t}} & X̄}

by defining for (x_{1}, …, x_{r}) ∈ I^{r}, v ∈ V

H′_{t}(x_{1}, …, x_{3}, v) = h′(x_{1}, …, x_{r−1}, λ, v)

with

λ = 1 − (1 − x_{r})(1 − tφ(x_{1}, …, x_{r−1}))

where φ is the map in the proof of 13.9 of [12]. One readily checks that (H′_{t}, H_{t}) is a well-defined 𝔉-homotopy.

We can now apply the arguments in the proof of 13.6 of [12]. We first show that if r ≤ m + n − 2, i_{∗} is surjective. Let

f′: V × (I^{r}, ∂I^{r}, J^{r−1}) → (X, X_{2}, X_{∗})

which represents an element {f′} in π_{r}^{𝔉}(X, X_{2}; ∗_{V}) as in the proof of (5.1) above. Hence f′ induces a map

f: (I^{r}, ∂I^{r}, J^{r−1}) → (X̄, X̄_{2}, ∗)

with X̄ = Ā ∪ e^{n} ∪ e^{m}. We obtain for f the homotopy f_{t} as in the proof of 13.6 of [12] and we define the 𝔉-homotopy

f′_{t}: V × (I^{r}, ∂I^{r}, J^{r−1}) → (X, X_{2}, X_{∗})

which induces f_{t} by

f′_{t}(v, u) = \begin{cases}h′_{t}(v, u) & for u ∈ Ū, v ∈ V \\ k′_{t}(v, u) & for u ∈ V̄, v ∈ V \\ f′(v, u) & for u ∈ I^{r} ∖ Ū ∖ V̄, v ∈ V. \end{cases}

Here h′_{t} is defined already in the proof of (5.1) and k′_{t} is defined in the same way as h′_{t}. Hence f′_{t}: f ≃ f′_{t} is a 𝔉-homotopy and f′_{1} represents {f′}. Moreover since H_{1}(I^{r}) ⊂ X̄ ∖ p we see that {f′} = {H′_{1}} is in the image of j_{∗} in (5.5). This shows that j_{∗}, and hence i_{∗}, is onto. In a similar way one follows the argument in the proof of 13.6 of [12] to show that i_{∗} is injective for n < n_{1} + n_{2}. q.e.d.

6 The category Π_{𝔉}(X) and Π_{𝔉}(X)-modules

Let X be a 𝔉-family and let x, y: V → X be 𝔉-points in X. Then we can consider homotopies F: x ≃ y with F: IV → X. Such a homotopy is termed a 𝔉-path in X. Homotopy classes of such 𝔉-paths relative the boundary V ∐ V of the cylinder IV are termed 𝔉-tracks {F }: x ⟹ y. Hence a 𝔉-track is an element

{F } ∈ [IV, X]^{V ∐ V} = [IV, X]^{(x,y)}.

Addition of homotopies yields the composite of tracks H, G denoted by H□G. Accordingly let Π_{𝔉}^{V}(X) be the following groupoid. Objects are 𝔉maps V → X and morphisms are 𝔉-tracks. For a closed inclusion A ⊂ X let

Π_{𝔉}^{V}(X, A) ⊂ Π_{𝔉}^{V}(X) (6.1)

be the full subgroupoid consisting of objects x: V → A ⊂ X.

The following category Π_{𝔉}(X) plays the role of the fundamental groupoid of a space X in classical homotopy theory.

(6.2) Definition. Let X be a 𝔉-family. First we define the category P_{𝔉}(X) as follows. Objects are pairs (V, x) where V is an object in 𝔉 and

x: V → X

is a 𝔉-point in X. A morphism (V, x) → (W, y) is a pair (a, α) where α: V → W is a morphism in 𝔉 and a: x ⟹ yα is a track. We associate to the morphism (a, α) the diagram

[figure: V; α; x; ⟹^{a}; W; y; X]

The composition of such morphisms according to the diagram

[figure: U; z; β; ⟹^{b}; V; x; α; ⟹^{a}; W; y; X; X; X]

is defined by

(a, α)(b, β) = ((aβ)□b, αβ).

Two morphisms (a, α) and (a′, α′): (V, x) → (W, y) are equivalent if there is a track

h: α ⟹ α′ in W^{V}

such that (yh)□a = a′. This is a natural equivalence relation on P_{𝔉}(X) so that we obtain the quotient category

Π_{𝔉}(X) = P_{𝔉}(X)/_{∼} (6.3)

which is termed the (discrete) fundamental category of X.

For a closed inclusion A ⊂ X in 𝔉-Top let

Π_{𝔉}(X, A) ⊂ Π_{𝔉}(X) (6.4)

be the full subcategory consisting of all objects (V, x) with x: V → A ⊂ X.

The cellular approximation theorem shows for a 0-connected relative 𝔉complex (X, A) that the inclusion X_{2} ⊂ X induces an isomorphism

Π_{𝔉}(X_{2}, A) = Π_{𝔉}(X, A). (6.5)

(6.6) Remark. For a space X in Top let Π(X) be the fundamental groupoid of X. Then we get the functor

Π(X^{∘}): 𝔉^{op} → Grd

which carries V to the fundamental groupoid Π(X^{V}) of the function space X^{V}. We have Π(X^{V}) = Π_{𝔉}^{V}(X). Now the category P_{𝔉}(X) above coincides with the “integration category” ∫_{𝔉} Π(X^{∘}), see [24] and compare with [22].

A left (resp. right) Π_{𝔉}(X)-module is a covariant (resp. contravariant) functor

M: Π_{𝔉}(X) → Ab

where Ab is the category of abelian groups. Let Mod(Π_{𝔉}(X)^{op}) be the category of right modules. Morphisms are natural transformations.

(6.7) Proposition. For n ≥ 2 the homotopy group yields a right module

π_{n}^{𝔉}(X): Π_{𝔉}(X)^{op} → Ab.

Moreover for n ≥ 3 relative homotopy groups of a pair (Y, X) yield the right module

π_{n}^{𝔉}(Y, X): Π_{𝔉}(X)^{op} → Ab.

Proof. An object (V, x) of Π_{𝔉}(X) yields a basepoint x: V → X, hence the abelian group π_{n}^{𝔉}(X; x) is defined for n ≥ 2. If (a, α): (V, x) → (W, y) is a morphism in P_{𝔉}(X) and n ≥ 2 then there is a canonical induced homomorphism of abelian groups defined by the composition

[figure: π_{n}(X^{W}; y); π_{n}(X^{V}; yα); π_{n}(X^{V}; x); π_{n}^{𝔉}(X; y); π_{n}(α^{∗}); π_{n}^{𝔉}(X; yα); a^{#}; π_{n}^{𝔉}(X; x)] (6.8)

where

α^{∗}(y) = yα: \xymatrix{V \ar[r]^{α} & W \ar[r]^{y} & X}

and a^{#} is the isomorphism of homotopy groups induced by the track a. It is not difficult to check that the structure of P_{𝔉}(X)-module induces a structure of Π_{𝔉}(X)-module, since the homomorphism in (6.8) does not depend on the representative (a, α) in P_{𝔉}(X) of a morphism in Π_{𝔉}(X). If n ≥ 3 the same argument can be applied to the relative group π_{n}^{𝔉}(Y, X; x). q.e.d.

Let Z be a 𝔉-set and let α: Z → X be a 𝔉-map. Hence (Z, α) is given by a family of 𝔉-maps α|V : V → X with V ∈ Z. We define the free right Π_{𝔉}(X)-module in Mod(Π_{𝔉}(X)^{op})

M[Z, α]: Π_{𝔉}(X)^{op} → Ab (6.9)

by the direct sum

M[Z, α] = ⨁_{V ∈Z} ℤ hom(−, (V, α|V)).

Here ℤ hom(−, (V, α|V)) denotes the module which carries (W, y) to the free abelian group generated by the hom-set hom((W, y), (V, α|V)) of all morphisms (W, y) → (V, α|V) in Π_{𝔉}(X). A morphism

φ: M[Z, α] → M (6.10)

in Mod(Π_{𝔉}(X)^{op}) is uniquely determined by a family of elements

φ_{V} ∈ M(V, α|V) for V ∈ Z

such that φ(1_{(V,α|V)}) = φ_{V}. Here 1_{(V,α|V)} denotes the identity of the object (V, α|V) in Π_{𝔉}(X).

We now consider for (Z, α) above the pushout

[figure: Z × ∗; Z; α; X; Z × S^{n}; S_{Z}^{n}; X ∪_{α} S_{Z}^{n}]

The projection Z × S^{n} → Z induces the retraction

r_{X}: X ∪_{α} S_{Z}^{n} → X

which is the identity on X. For n ≥ 1 let

π_{n}^{𝔉}(X ∪_{α} S_{Z}^{n})_{X} = ker ((r_{X})_{∗}: π_{n}^{𝔉}(X ∪_{α} S_{Z}^{n}) → π_{n}^{𝔉}(X)) .

We define the partial suspension

E: π_{n}^{𝔉}(X ∪_{α} S_{Z}^{n})_{X} → π_{n+1}^{𝔉}(X ∪_{α} S_{Z}^{n+1})_{X} (6.11)

by the composition E = j^{−1}π_{∗}∂^{−1}

\xymatrix{π_{n+1}^{𝔉}(X ∪_{α} D_{Z}^{n+1}, X ∪_{α} S_{Z}^{n}) \ar[r]^{∂}_{≅} \ar[d]_{π_{∗}} & π_{n}^{𝔉}(X ∪_{α} S_{Z}^{n})_{X} \\ π_{n+1}^{𝔉}(X ∪_{α} S_{Z}^{n+1}, X) & π_{n+1}^{𝔉}(X ∪_{α} S_{Z}^{n+1})_{X} \ar[l]^{j}_{≅}}

Here π is induced by D^{n+1}/S^{n} = S^{n+1} and the isomorphisms ∂ and j are induced by the homotopy exact sequences of pairs. The Blakers-Massey theorem implies:

(6.12) Proposition. E is an isomorphism for n ≥ 2 and is surjective for n = 1.

(6.13) Proposition. There is an isomorphism of modules (n ≥ 2)

φ: M[Z, α] ≅ π_{n}^{𝔉}(X ∪_{α} S_{Z}^{n})_{X}

which is given by the family of maps

φ_{V} : S_{V}^{n} ⊂ S_{Z}^{n} → X ∪_{α} S_{Z}^{n}

with V ∈ Z. Moreover φ is compatible with E.

Proof. Without loss of generality, by additivity, we can assume that Z = W; moreover, by considering the mapping cylinder of α: W → X in 𝔉-Top we can assume that α: W → X is a closed inclusion (actually, a 𝔉-cofibration). Let (V, x) be an arbitrary object in Π_{𝔉}(X). We want to show that φ induces an isomorphism of groups

φ: M[W, α](V, x) = ℤ hom((V, x), (W, α)) → π_{n}^{𝔉}(X ∪_{α} S_{W}^{n}; x)_{X}. (6.14)

Let X_{x}^{V} denote the path-component in X^{V} containing x ∈ X^{V} and W_{x}^{V} = X_{x}^{V} ∩ α^{V} (W^{V}); let α_{x}: W_{x}^{V} ⊂ X_{x}^{V} be the inclusion. Moreover, let X̃_{x}^{V} = E_{X} be the universal covering space of X_{x}^{V}. We can assume that any space that we are considering in this proof has a universal cover by taking a CW-approximation weakly equivalent to it. The elements of X̃_{x}^{V} correspond bijectively with tracks x ⟹ ξ, where ξ: V → X is any 𝔉-map. The covering projection in this case is the evaluation of the track at 1. We can define E_{W} by the following pull-back diagram.

[figure: E_{W}; α̃_{x}; X̃_{x}^{V} = E_{X}; W_{x}^{V}; α_{x}; X_{x}^{V}] (6.15)

Then E_{X} ∪α̃_{x} (E_{W} × S^{n}) is the universal covering of X_{x}^{V} ∪_{α_{x}} (W_{x}^{V} × S^{n}). Consider the morphisms (b, β) in P_{𝔉}(X)((V, x), (W, α)). Then β and b are

elements in X̃_{x}^{V} and W_{x}^{V} respectively, compatible with the pull-back diagram (6.15). Hence there is a map λ as in the following diagram.

\xymatrix{P_{𝔉}(X)((V, x), (W, α)) \ar[r]^{λ} \ar@{->>}[d] & E_{W} \ar@{->>}[d] \\ Π_{𝔉}(X)((V, x), (W, α)) \ar@{.>}[r]^{λ̄}_{≅} & π_{0}(E_{W}).}

Here two morphisms (b, β) and (b′, β′) in P_{𝔉}(X)((V, x), (W, α)) have images under λ in the same path-component if and only if there is h: I → E_{W} such that h(0) = b, h(1) = b′. But this happens if and only if (b, β) and (b′, β′) belong to the same equivalence class in Π_{𝔉}(X), and thus there is an induced bijection λ̄ as in the diagram.

Now, using the exact homotopy sequence of the pair, the property of universal covering spaces, the homotopy excision, and the isomorphism induced by λ̄ we get the equations:

π_{n}^{𝔉}(X ∪_{α} S_{W}^{n}; x)_{X} = π_{n}^{𝔉}(X ∪_{α} S_{W}^{n} , X; x)

= π_{n}(X^{V} ∪_{α^{V}} S_{W^{V}}^{n} , X^{V}; x) = π_{n}(X_{x}^{V} ∪_{α_{x}^{V}} S_{W_{x}^{V}}^{n} , X_{x}^{V}; x)

= π_{n}(E_{X} ∪α̃_{x} (E_{W} × S^{n}), E_{X}; x) = π_{n}(E_{X} ∪α̃_{x} (E_{W} × S^{n}), E_{X})

≅ ⨁_{C⊂E_{W}} π_{n}(S^{n} × C, C)

= ℤπ_{0}(E_{W})

= ℤΠ_{𝔉}(X)((V, x), (W, α))

where the sum ⨁ ranges over the set of path-components C ⊂ E_{W}. We can apply the homotopy excision since E_{X} is simply connected and W = W × ∗ is a 𝔉-neighborhood deformation retract in S_{W}^{n}. Therefore we have the isomorphism

M[W, α](V, x) = ℤ hom((V, x), (W, α)) ≅ π_{n}^{𝔉}(X ∪_{α} S_{W}^{n}; x)_{X}.

It is not difficult, by chasing along the chain of equalities above, to see that this is the same homomorphism induced by φ as in (6.14) and that it commutes with the partial suspension E as claimed. Since φ is a morphism of Mod(Π_{𝔉}(X)^{op})-modules, the proof is complete. q.e.d.

7 Homology and cohomology

Let (X, A) be a relative 𝔉-complex. We say that (X, A) is normalized if the attaching maps

f_{n}: S_{Z_{n}}^{n−1} = Z_{n} × S^{n−1} → X_{n−1} (7.1)

of n cells in X (n ≥ 1) carry the basepoint ∗ of S^{n−1} to the 0-skeleton, that is f_{n}(Z_{n} × ∗) ⊂ X_{0} or equivalently if the attaching maps are cellular. Moreover (X, A) is reduced if X_{0} = A so that X is obtained from A by attaching 𝔉-cells of dimension ≥ 1.

(7.2) Lemma. Let (X, A) be a relative 𝔉-complex which is 0-connected. Then there exists a relative 𝔉-complex (Y, A) which is reduced and normalized together with a 𝔉-homotopy equivalence Y → X under A.

Proof. First we obtain a reduced 𝔉-complex (Y′, A) by (5.2). Using the cellular approximation theorem (3.5) the attaching maps in Y′ are 𝔉-homotopic to normalized attaching maps. This can be used to construct inductively Y. q.e.d.

If (X, A) is reduced and normalized we obtain for n ≥ 1 the commutative diagram

[figure: Z_{n} × S^{n−1}; S_{Z_{n}}^{n−1}; f_{n}; X_{n−1}; Z_{n} × ∗; Z_{n}; α_{n}; A] (7.3)

where α_{n} is the basepoint map associated to the attaching map f_{n}. We identify the closed cell D^{n} with the (reduced) cone over the pointed space S^{n−1} so that we obtain the pinch map under S^{n−1} (n ≥ 1)

\xymatrix{D^{n} \ar[r]^{μ̄} & D^{n} ∨ S^{n}.}

This map induces the coaction map

[figure: X_{n}; μ; X_{n} ∪_{α_{n}} S_{Z_{n}}^{n}; X_{n−1} ∪_{f_{n}} (Z_{n} × D^{n}); X_{n−1} ∪_{f_{n}} (Z_{n} × (D^{n} ∨ S^{n}))] (7.4) where the bottom row is 1_{X_{n−1}} ∪(Z_{n} ×μ̄). On the other hand we have for the 1 dimensional disk D^{1} with boundary points ∂D^{1} = {∂_{0}, ∂_{1}} and basepoint ∂_{0} = ∗ the map

\xymatrix{(S^{1}, ∂_{0}) \ar[r]^{μ̄_{0}} & (D^{1} ∪_{∂_{1}} S^{1}, ∂_{0})}

which is defined by composing obvious tracks ∂_{0} → ∂_{1} in D^{1}, ∗ ⟹ ∗ in S^{1} and ∂_{1} ⟹ ∂_{0} in D^{1}. If X is a normalized 𝔉-complex we have the 𝔉-set X_{0} = Z_{0} and the inclusion α_{0} : Z_{0} ⊂ X. Then μ̄_{0} induces the map

[figure: S_{Z_{1}}^{1}; μ_{0}; X ∪_{α_{0}} S_{Z_{0}}^{1}; Z_{1} × S^{1}; Z_{1} × (D^{1} ∪_{∂_{1}} S^{1})] (7.5)

Here the bottom row is Z_{1} × μ̄_{0} and the right hand side is f̄_{1} ∪ (S^{1} × α_{1}^{1}) where f̄_{1} is the characteristic map of 1-cells and α_{1}^{1} = f_{1}|(Z_{1} × ∂_{1}). The map μ_{0} restricted to Z_{1} × ∗ is given by α_{1} = f_{1}|(Z_{1} × ∂_{0}).

(7.6) Definition. Let (X, A) be a 𝔉-complex which is reduced and normalized. Then we define the chain complex C_{∗}(X, A) in Mod(Π_{𝔉}(X, A)^{op}) as follows. For n ≤ 0 let C_{n}(X, A) = 0 and for n ≥ 1 let

C_{n}(X, A) = M[Z_{n}, α_{n}]

be defined as in (6.9) by the 𝔉-set Z_{n} of n-cells in (X, A) and the basepoint map α_{n}: Z_{n} → A ⊂ X. The differential

d: C_{n+1}(X, A) → C_{n}(X, A)

is defined for V ∈ Z_{n+1} by elements d_{V} obtained as follows. Let ∂_{V} be the composite (n ≥ 1)

∂_{V} : \xymatrix{S_{V}^{n} ⊂ S_{Z_{n+1}}^{n} \ar[r]^{f_{n+1}} & X_{n} \ar[r]^{μ} & X_{n} ∪_{α_{n}} S_{Z_{n}}^{n} ⊂ X ∪_{α_{n}} S_{Z_{n}}^{n}}

which for n ≥ 2 represents the element

d_{V} ∈ π_{n}^{𝔉}(X ∪_{α_{n}} S_{Z_{n}}^{n}; α_{n+1}|V)_{X} = M[Z_{n}, α_{n}](V, α_{n+1}|V).

Compare (6.6) and (6.9). For n = 1 we have the partial suspension

E: π_{1}^{𝔉}(X ∪_{α_{1}} S_{Z_{1}}^{1}; α_{2}|V)_{X} → π_{2}^{𝔉}(X ∪_{α_{1}} S_{Z_{1}}^{2}; α_{2}|V)_{X}

and we set d_{V} = E(∂_{V}).

If X is a normalized 𝔉-complex we define the chain complex C_{∗}X in Mod(Π_{𝔉}(X, X_{0})^{op}) by

C_{n}(X) = C_{n}(X, X_{0}) for n ≥ 1 and
C_{0}(X) = M[Z_{0}, α_{0}]

where Z_{0} = X_{0} is the 0-skeleton of X which is an 𝔉-set and α_{0} : Z_{0} ⊂ X is the inclusion. Moreover the differential

d: C_{n+1}(X) → C_{n}(X)

is defined as above for n > 1. For n = 1 the differential

d: C_{1}(X) → C_{0}(X)

is defined by d_{V} = E(μ_{0}|S_{V}^{1} ) for V ∈ Z_{1} where μ_{0} is the map in (7.5) and E is the partial suspension. We may consider C_{∗}(X, A) and C_{∗}(X) also as chain complexes in Mod(Π_{𝔉}(X)^{op}).

(7.7) Proposition. C_{∗}(X, A) and C_{∗}(X) are well defined chain complexes.

Proof. Since (𝔉-Top, T) is homological (B.3), the chain functor C_{∗} is defined as in V.2.4, page 255 of [3], and it is easy to see that it coincides with the chain functor C_{∗} defined above. q.e.d.

(7.8) Definition. Let (X, A) be a 0-connected relative 𝔉-complex. By lemma (7.2) we can assume that (X, A) is reduced and normalized. Given a right Π_{𝔉}(X)-module M we define the cohomology of (X, A) with coefficients in M by

H^{∗}(X, A; M) = H^{∗} (hom(C_{∗}(X, A), M)) ,

where hom is defined in the abelian category of right Π_{𝔉}(X)-modules.

In a similar way, if N is a left Π_{𝔉}(X)-module we can define the homology of (X, A) with coefficients in N by

H_{∗}(X, A; N) = H_{∗}(C_{∗}(X, A) ⊗ N),

where ⊗ stands for the tensor product of a right and a left Π_{𝔉}(X)-module. Here H^{∗}(X, A; M) and H_{∗}(X, A; M) are abelian groups.

Finally, the total homology of (X, A) is defined by

H_{∗}(X, A) = H_{∗}(C_{∗}(X, A)).

The total homology is the homology of a chain complex of right Π_{𝔉}(X)- modules, so that H_{∗}(X, A) is again a Π_{𝔉}(X)-module.

(7.9) Remark. This is a generalization of the Bredon–Illman cohomology [6] for G-spaces. See Bröcker for the singular Bredon homology [7]. See [26, 25, 24] for singular Bredon cohomology of discrete diagrams.

8 The Whitehead theorem

(8.1) Theorem. Let A be a 𝔉-family and let f: (X, A) → (Y, A) be a cellular map between normalized reduced relative 𝔉-complexes. Then f is a 𝔉-homotopy equivalence under A if and only if f induces an isomorphism

φ = Π(f): Π_{𝔉}^{V}(X, A) → Π_{𝔉}^{V}(Y, A)

of groupoids for all V ∈ 𝔉 and any one of the following cases is true:

(i) f induces a homotopy equivalence of Π_{𝔉}(X, A)-chain complexes

f_{∗}: C_{∗}(X, A) → C_{∗}(Y, A),

where we use the isomorphism φ to identify Π_{𝔉}(X, A) and Π_{𝔉}(Y, A).

(ii) The induced map between homology

f_{∗}: H_{∗}(X, A) → φ^{∗}H_{∗}(Y, A)

is an isomorphism.

(iii) For every Π_{𝔉}(Y, A)-module M the induced map

f^{∗}: H^{∗}(Y, A; M) → H^{∗}(X, A; φ^{∗}M)

is an isomorphism.

Proof. This is a consequence of the general homological Whitehead theorem V.7.1 of [3]. It is only needed to show that any reduced and normalized 𝔉-complex is a T-complex in the sense of IV.2.2 of [3], where T the theory of 𝔉-graphs (see definition (B.1)). This follows from lemma (B.2) in the appendix. q.e.d.

Proof of theorem (4.7 ). By (5.1) the T-complexes in 𝔉-Top are T-good in the sense of IV.3.7 [3]. Hence by (7.2) and IV.3.11 of [3] we know that the Whitehead theorem holds for 0-connected relative 𝔉-complexes (X′, A) and (Y′, A). This can be used to prove the general case in (4.7) as follows. Let f: (X, A) → (Y, A) be a Whitehead equivalence as in (4.7). We may assume that f is cellular. Hence f yields a restriction α: X_{0} → Y_{0} ⊂ Y which is surjective in π_{0}^{𝔉}. Let M_{α} = IX_{0} ∪_{α} Y be the mapping cylinder (in 𝔉-Top) of α, i.e. the push-out of i_{0} : X_{0} → IX_{0} and α (see (A.6)). Then i_{1} : X_{0} → IX_{0} yields a pair (M_{α}, X_{0}) which is 0-connected. Moreover Y → M_{α} is a 𝔉homotopy equivalence and the composite X → Y → M_{α} is by the homotopy extension property of X_{0} ⊂ X 𝔉-homotopic to a map g: X → M_{α} which is the identity on X_{0}. Hence g: (X, X_{0}) → (Mα, Y_{0}) is a Whitehead equivalence between 0-connected pairs and hence a 𝔉-homotopy equivalence. q.e.d.

9 The Whitehead sequence and obstruction theory

Let (X, A) a reduced normalized 𝔉-complex and Π_{𝔉}(X, A) the (restricted) fundamental category. Then the homotopy groups π_{n}^{𝔉}(X) and the homology groups H_{n}(X, A) are Π_{𝔉}(X, A)^{op}-modules and it is possible to define a Hurewicz homomorphism

h: π_{n}^{𝔉}(X) → H_{n}(X, A)

as follows.

Let x: V → X be a basepoint and α: S_{V}^{n} → X a representative of the element [α] ∈ π_{n}^{𝔉}(X; x). By (3.5) we can assume that αS_{V}^{n} ⊂ X_{n}. Consider the difference μα − iα of the two composites

[figure: S_{V}^{n}; α; α; X_{n}; μ; X_{n}; i; X_{n} ∪_{α_{n}} S_{Z_{n}}^{n}; j; X ∪_{α_{n}} S_{Z_{n}}^{n}]

where μ is the coaction map. Then j(μα − iα) is a cycle in C_{n}(X, A) that represents an element

h(α) = {j(μα − iα)} ∈ H_{n}(X, A).

It is not difficult to see that h(α) does not depend upon the 𝔉-homotopy class of α and that h is a Π_{𝔉}(X, A)-homomorphism, termed the Hurewicz homomorphism.

The groups Γ_{n}(X, A), if n ≥ 3, are defined as follows. If x: V → A is an object of Π_{𝔉}(X, A) then

Γ_{n}(X, A)(x) = Image [π_{n}^{𝔉}(X_{n−1}; x) → π_{n}^{𝔉}(X_{n}; x)]

For n = 1, 2 the definition is more complicate (see section V.5, page 262, of [3]).

(9.1) Theorem. Let (X, A) be a reduced and normalized 𝔉-complex Then the following sequence of Π_{𝔉}(X, A)^{op}-modules is exact

[figure: Γ_{n}(X, A); π_{n}^{𝔉}(X); h; H_{n}(X, A); Γ_{n−1}(X, A); …; Γ_{2}(X, A); π_{2}^{𝔉}(X); h; H_{2}(X, A); Γ_{1}(X, A); 0]

Furthermore, the sequence is natural in (X, A) in the category 𝔉-Top^{A} of 𝔉-families under A.

Proof. It is a direct consequence of theorem V.5.4, page 264 [3]. q.e.d.

The cohomology defined above is suitable for obstruction theory as follows.

(9.2) Theorem. Let (X, A) a normalized reduced relative 𝔉-complex and f: A → Y a 𝔉-map that admits an extension g: X_{n} → Y, for n ≥ 2. Then its restriction g|X_{n−1} has an extension g′: X_{n+1} → Y to the (n + 1)-skeleton if and only if the obstruction

O(g|X_{n−1}) ∈ H^{n+1}(X, A; g^{∗}π_{n}^{𝔉}Y)

vanishes, where g_{∗}: Π_{𝔉}(X, A) → Π(Y) induces the functor g^{∗}: Mod(Π_{𝔉}(Y)) → Mod(Π_{𝔉}(X, A)).

Proof. The obstruction O(g|X_{n−1}) is defined as in chapter V of [3]. See theorem V.4.4, page 262. q.e.d.

10 Wall finiteness obstruction

A 𝔉-complex X is finite if it has only finitely many cells or equivalently if X̄ is a finite CW-complex. A domination of Y in 𝔉-Top is given by 𝔉-maps

\xymatrix{Y \ar[r]^{f} & X \ar[r]^{g} & Y}

and a 𝔉-homotopy H: gf ∼ 1_{Y}. The domination has dimension ≤ n if the dimension of X is ≤ n and the domination is finite if X is a finite 𝔉-complex. For a ringoid R let K_{0}(R) denote the reduced projective class group of R. If C is a category let ℤC be the ringoid associated to C for which

hom_{ℤC}(X, Y) = ℤ[hom_{C}(X, Y)]

is the free abelian group generated by hom_{C}(X, Y).

(10.1) Theorem. Let Y be a 𝔉-complex which admits a finite domination in 𝔉-Top. Then the finiteness obstruction

[Y] = [C_{∗}(Y)] ∈ K_{0}(ℤΠ_{𝔉}(Y, Y_{0}))

is defined, with the property that the obstruction [Y] = 0 is trivial if and only if Y is 𝔉-homotopy equivalent to a finite 𝔉-complex. Moreover, if the domination of Y has dimension ≤ n and [Y] = 0 then Y is 𝔉-homotopy equivalent to a finite 𝔉-complex of dimension ≤ Max(3, n).

(10.2) Remark. A result like (10.1) was proved by different methods in the case of G-spaces in theorem 14.6 of Lück [22], without the dimension estimate.

(10.3) Lemma. Let X̃ be a finite domination of Y. Then there exists a finite domination X of Y for which f: Y → X induces an isomorphism π_{0}^{𝔉}(Y; ∗_{V}) ≅ π_{0}^{𝔉}(X; f∗_{V}) for every basepoint ∗_{V}.

Proof. We may assume that f̃: Y → X̃ and g̃: X̃ → Y are cellular maps which yield the composite λ = if̃_{0}g̃_{0} : X̃_{0} → X̃_{0} ⊂ X̃. Let X be obtained by attaching 1-cells to X̃ as in the push-out diagram

\xymatrix{X̃_{0} × ∂I \ar[r] \ar[d]^{(i,λ)} & X̃_{0} × I \ar[d] \\ X̃ \ar[r]^{j} & X}

where i: X̃_{0} → X̃ is the inclusion. Since g̃i ∼ g̃λ, there exists an extension g: X → Y of g̃ with gj = g̃. Hence for f = jf̃ we obtain the domination

\xymatrix{Y \ar[r]^{f} & X \ar[r]^{g} & Y}

with gf = g̃f̃ ∼ 1_{Y}. This shows that f_{∗}: π_{0}^{𝔉}(Y; ∗_{V}) → π_{0}^{𝔉}(X; f∗_{V}) is surjective. In fact, f is also surjective since an element of π_{0}^{𝔉}(X; f∗_{V}) can be represented by a 𝔉-map ξ: V → X_{0} = X̃_{0} via the cellular approximation theorem (3.5). For the inclusion j_{0} : X_{0} ⊂ X we get the following homotopy

j_{0} = jiξ ∼ jλξ = jif_{0}g_{0}ξ = j_{0}f_{0}(η) = f_{∗}(η),

where η = g_{0}ξ. q.e.d.

Proof of (10.1 ). Let X be a finite domination of Y as in (10.3). We can choose tg to be cellular and we obtain by restriction of g the map α = ig_{0} : X_{0} → Y_{0} ⊂ Y. We use α to construct the mapping cylinder Y′ = M_{α} = Y ∪_{α} IX_{0}, so that (M_{α}, X_{0}) is a 0-connected relative 𝔉-complex and M_{α} ≃ Y in 𝔉-Top. The domination X of Y now yields a domination X′ of M_{α} by the composites

\xymatrix{Y′ ≃ Y \ar[r]^{f} & X ≃ X′ ≃ X \ar[r]^{g} & Y ≃ Y′}

where the composites f′: Y′ → X′ and g′: X′ → Y′ can be chosen to be maps under X_{0} and where the 𝔉-homotopy g′f′ ∼ 1_{Y′} can be chosen to be a 𝔉homotopy relative X_{0}. Moreover we can assume by the approximation lemma (7.2) that (Y′, X_{0}) and (X′, X_{0}) are replaced by normalized and reduced relative 𝔉-complexes (Y′′, X_{0}) and (X′′, X_{0}). Hence we can apply theorem VII.2.5 of [3] which shows that a finiteness obstruction

[Y′′] = [C_{∗}(Y′′, X_{0})] ∈ K_{0}(ℤΠ_{𝔉}(X, A))

is defined and has the property that [Y′′] = 0 if and only if there exists a finite relative 𝔉-complex (Z, X_{0}) which is reduced and normalized with dim(Z) ≤ Max(3, n), where n = dim(X′′), and such that Z is 𝔉-homotopy equivalent to Y′′ relative X_{0}. Since we have homotopy equivalences of chain complexes C_{∗}(Y) ≃ C_{∗}(Y′) ≃ C_{∗}(Y′, X_{0}) ≃ C_{∗}(Y′′, X_{0}), and since by the finiteness of X_{0} we have [C_{∗}(Y′)] = [C_{∗}(Y′, X_{0})], we see that [C_{∗}(Y)] = [C_{∗}(Y′′, X_{0})]. This yields the results in (10.1). q.e.d.

11 Whitehead torsion

For n ≥ 0 let ℝ_{+}^{n+1} and ℝ_{−}^{n+1} be the subspaces of ℝ^{n+1} defined by the elements (x_{0}, …, x_{n}) ∈ ℝ^{n+1} with x_{0} ≥ 0 and x_{0} ≤ 0 respectively. Let D^{n+1} denote the unit sphere in ℝ^{n+1} and S^{n} its boundary. A ball pair is a tuple (□^{n+1}, S^{n}, P^{n}, Q^{n}) homeomorphic to the Euclidean ball pair

(D^{n+1}, S^{n}, S^{n} ∩ ℝ_{+}^{n+1}, S^{n} ∩ ℝ_{−}^{n+1}).

We can assume that the basepoint of D^{n+1} is in P^{n} ∩ Q^{n} = S^{n−1}. For every object W in 𝔉 and n ≥ 0 a ball pair in 𝔉-Top is defined as

(□_{W}^{n+1}, S_{W}^{n} , P_{W}^{n} , Q_{W}^{n}) = W × (□^{n+1}, S^{n}, P^{n}, Q^{n}),

and denoted simply by (□_{W}^{n+1}, Q_{W}^{n}), where □_{W}^{n+1} = D_{W}^{n+1}. If Z is a finite 𝔉-set, then we can define □_{Z}^{n+1} as the coproduct in 𝔉-Top of the □_{W}^{n+1} with W ∈ Z. The pair (□_{Z}^{n+1}, Q_{Z}^{n}) is also termed ball pair in 𝔉-Top, and P_{Z}^{n} is termed the complement of Q_{Z}^{n} in the boundary. It is easy to see that □_{Z}^{n+1} is a 𝔉-complex and P_{Z}^{n} and Q_{Z}^{n} are subcomplexes. There are 𝔉-cells only in dimensions 0, n − 1, n and n + 1: the 0-skeleton is Z, the (n − 1)-skeleton is ∂Q_{Z}^{n} = ∂P_{Z}^{n} = S_{Z}^{n−1}, the n-skeleton is ∂□_{Z}^{n+1} = S_{Z}^{n} = P_{Z}^{n} ∪_{S_{Z}^{n−1}} Q_{Z}^{n}, and the (n + 1)-skeleton is □_{Z}^{n+1} itself.

Now consider a ball pair (□_{Z}^{n+1}, Q_{Z}^{n}) with the complement P_{Z}^{n} in the boundary, a 𝔉-complex L and a 𝔉-map f: Q_{Z}^{n} → L with the property that f(Q_{Z}^{n}, S_{Z}^{n−1}) ⊂ (L_{n}, L_{n−1}). Let K be the push-out in 𝔉-Top

[figure: P_{Z}^{n}; f; L; i; □_{Z}^{n+1}; K]

It is easy to check that K is a 𝔉-complex with subcomplex L. As in [3], VIII.6, page 329, if K′ is a complex with L as subcomplex and 𝔉-isomorphic to K under L, then K′ is termed an elementary expansion of L; moreover, there is an associated canonical 𝔉-retraction r: K′ → L termed elementary collapse. If S is a subcomplex of L then we say that i (resp. r) is an elementary expansion (resp. collapse) relative S.

Now let K and L two finite 𝔉-complexes both containing a subcomplex S. If j: L → K is an expansion relative S we write L ↗ K rel S; is r: K → L is a collapse relative S, we write K ↘ L rel S. A formal deformation relative S (i.e. a finite composition of expansions and collapses relative S) is denoted by L ↷ L. A 𝔉-map f: L → K under S is a simple homotopy equivalence rel S if f is 𝔉-homotopic rel S to a formal deformation relative S.

Let L be a finite 𝔉-complex. Consider the set of all pairs (K, L) such that L is a subcomplex of K, K is a finite 𝔉-complex, and the inclusion L ⊂ K is a 𝔉-homotopy equivalence. Two pairs (K, L) and (K′, L) are equivalent if and only if K ↷ K′ rel L. Let [K, L] denote the class of the pair (K, L) and let Wh(L) denote the set of equivalence classes. It is possible to define an addition in Wh(L) by

[K, L] + [K′, L] = [K ∪_{L} K′, L]

so that (Wh(L), +) is an abelian group (lemma VIII.8.7, page 334 of [3]). Given a cellular 𝔉-map f: L → L′ between finite 𝔉-complexes, there is an induced map

f_{∗}: Wh(L) → Wh(L′)

defined by

f_{∗}[K, L] = [K ∪_{L} M_{f}, L′],

where M_{f} is the mapping cylinder of f in 𝔉-Top (see (A.6) below). Actually Wh is a functor from the category of finite 𝔉-complexes and 𝔉-homotopy classes of maps to the category of abelian groups.

Now consider a 𝔉-homotopy equivalence f: X → L of finite 𝔉-complexes; the Whitehead torsion of f is defined by

τ(f) = f_{∗}[M_{f}, X] = [M_{f} ∪_{X} L, L] ∈ Wh(L).

(11.1) Theorem. A 𝔉-homotopy equivalence f: X → L between finite 𝔉complexes is a simple homotopy equivalence if and only if the torsion vanishes τ(f) = 0 ∈ Wh(L). Moreover:

(i) For every pair of homotopy equivalences f and g the derivation property holds:

τ(gf) = τ(g) + g_{∗}τ(f).

(ii) Consider the following double push-out diagram

\xymatrix{K′_{0} \ar[rrr] \ar[ddd] &  &  & K′_{2} \ar[ddd] \\  & K_{0} \ar[ul]^{≃}_{f_{0}} \ar[r]^{i_{2}} \ar[d]^{i_{1}} \ar[dr]^{i_{0}} & K_{2} \ar[ur]^{f_{2}}_{≃} \ar[d] \\  & K_{1} \ar[dl]^{≃}_{f_{1}} \ar[r] & K \ar@{.>}[dr]^{f}_{≃} \\ K′_{1} \ar[rrr] &  &  & K′}

where f_{0}, f_{1} and f_{2} are 𝔉-homotopy equivalences and all the nondiagonal maps are inclusions of subcomplexes. Then the push-out map f is a 𝔉-homotopy equivalence and the addition formula holds:

τ(f) = τ(f_{1} ∪_{f_{0}} f_{2}) = i_{1}τ(f_{1}) + i_{2}τ(f_{2}) − i_{0}τ(f_{0}).

Proof. Let D denote the set of finite 𝔉-sets. Since (𝔉-Top, D) is a cellular I-category (see (B.3)), theorem (11.1) is a consequence of VIII.8.3, VIII.8.4 and VIII.8.5 (page 335) of [3]. q.e.d.

Following Ranicki [28], if A is a small additive category with sum denoted by ⊕, then the isomorphism torsion group K_{1}^{iso}(A) is the abelian group with one generator τ(f) for each isomorphism f: M → N in A, and relations

τ(gf) = τ(f) + τ(g)

τ(f ⊕ f′) = τ(f) + τ(f′)

for all isomorphisms f: M → N, f′: M′ → N′ and g: N → P in A. If R is a ringoid, then it is possible to define K_{1}^{iso}(R) as the isomorphism torsion group of the additive category A(R) consisting of finitely generated free Rmodules. A trivial unit in K_{1}^{iso}(ℤΠ_{𝔉}(K)) is represented by an automorphism of the free ℤΠ_{𝔉}(K)-module ℤΠ_{𝔉}(K)(−, (V, x)) that can be written as ±f_{∗} for some automorphism of (V, x) in Π_{𝔉}(K).

(11.2) Theorem. Let K be a finite 𝔉-complex. Then there is an isomorphism

τ: Wh(K) ≅ K_{1}^{iso}(ℤΠ_{𝔉}(K))/T

between the Whitehead group of K and the quotient of K_{1}^{iso}(ℤΠ_{𝔉}(K)) by the subgroup T generated by all trivial units. It is defined by associating to [X, K] ∈ Wh(K) the torsion of the contractible Π_{𝔉}(K)-chain complex C_{∗}(X′, K), where X′ is a normalization of X rel K.

Proof. It is a consequence of theorem VIII.12.7, page 348, of [3]. q.e.d.

In case of transformation groups this is theorem 14.16, page 286, of [22] (in the same book the equivariant finiteness obstruction can be found). Compare with the equivariant Whitehead torsion of Hauschild [13], Dovermann and Rothenberg [11], Illman [15] and Anderson [1].

Appendix A. Basic homotopy theory in 𝔉-Top

We proved several results above by refering to [3]. We now describe some results that are needed in order to apply the abstract theory of [3] to 𝔉-Top.

(A.1) Definition. A 𝔉-map i: Y → X is a 𝔉-cofibration if i is a closed inclusion and the following homotopy extension property holds: for every commutative diagram in 𝔉-Top

[figure: Y; i; i_{0}; X; i′_{0}; f; Y × I; i×1_{I}; H; X × I; G; Z]

there is a 𝔉-map G: X × I → Z such that G ∘ i′_{0} = f and G ∘ (i × 1_{I}) = H; in the diagram i_{0} is the map defined by y ↦ (y, 0) for every y ∈ Y, i′_{0} the same for X.

Equivalently a closed inclusion i: Y → X in 𝔉-Top is a 𝔉-cofibration if and only if the 𝔉-map (Y × I) ∪_{Y} X → X × I admits a retraction (that is a left inverse). This implies that Ȳ → X̄ and Y → X are cofibrations in Top.

(A.2) Lemma. For a cofibration i: A → M in 𝔉-Top and a 𝔉-map h: A → Y, in the push-out diagram (which exists by (2.4 ))

\xymatrix{A \ar[r]^{h} \ar[d]^{i} & Y \ar[d]^{ī} \\ M \ar[r] & X = M ∪_{A} Y}

the induced 𝔉-map ī is also a cofibration. Moreover, I carries this push-out diagram into a push-out diagram (i.e. (M ∪_{A} Y)×I = (M ×I)∪_{A×I} (Y ×I)).

Proof. The proof is formally identical to the proof for compactly generated spaces with no structure group (see (5.1) and (5.4) of [34], pages 22–23); or, directly applying the homotopy extension property (A.1) and the assumptions on the fibres, is the same as the proof in Top (see e.g. lemma 2.3.6, page 56 of [27]). q.e.d.

The next result shows that basic homotopy theory is available in 𝔉-Top. For the axioms and properties of I-categories and cofibration categories we refer the reader to [3]. As in section 1 the cylinder in 𝔉-Top is given by IX = X × I.

(A.3) Theorem. The category 𝔉-Top with the cylinder IX is an I-category. Moreover, it is a cofibration category, with cofibrations the 𝔉-cofibrations and weak equivalences the 𝔉-homotopy equivalences. All objects are fibrant and cofibrant in 𝔉-Top.

Proof. To show that the axioms of I-category are fulfilled, since lemma (A.2) holds, it suffices to use the argument of [3], proposition (8.2), with the same homeomorphism α: I^{2} → I^{2}, to prove the relative cylinder axion, see page 222 of [3]. Then we can apply Theorem (7.4) of [3], page 223. q.e.d.

The theorem implies that the following lemmata are true in 𝔉-Top. Corollaries (A.4), (A.5) and (A.6) are immediate consequences of the first three axioms of a cofibration category.

(A.4) Corollary (Composition). Isomorphisms in 𝔉-Top are 𝔉-homotopy equivalences and also cofibrations. For two maps

\xymatrix{A \ar[r]^{f} & B \ar[r]^{g} & C}

if any two of f, g and gf are 𝔉-homotopy equivalences then so is the third. The composite of 𝔉-cofibrations is a 𝔉-cofibration.

(A.5) Corollary (Pushout). Let i: A → M be a 𝔉-cofibration and f: A → Y be a 𝔉-map. Consider the pushout in 𝔉-Top,

[figure: A; i; f; Y; ī; M; f̄; Y ∪_{f} M]

where ī is a 𝔉-cofibration by (A.2 ). If f is a 𝔉-homotopy equivalence, then so is f̄; if i is a 𝔉-homotopy equivalence, then so is ī.

(A.6) Corollary (Factorization). Every 𝔉-map f: X → Y can be factorized as

[figure: X; f; i; Y; Y′; ≃; g]

where g is a 𝔉-homotopy equivalence and i a 𝔉-cofibration. The 𝔉-family Y′ is the mapping cylinder (in 𝔉-Top) of f, i.e. the push-out of i_{0} : X → IX and f.

(A.7) Corollary (Gluing lemma). Consider the following diagram,

[figure: A′; h′; Y′; A; ≃; h; Y; ≃; M; ≃; Y ∪_{A} M; ≃; M′; Y′ ∪_{A′} M′]

where the two squares are push-outs in 𝔉-Top, and A → M and A′ → M′ are 𝔉-cofibrations. The dotted map exists as a consequence of (A.2 ). If the arrows A → A′, M → M′ and Y → Y′ are 𝔉-homotopy equivalences, then so is the push-out map Y ∪_{A} M → Y′ ∪_{A′} M′.

Proof. See lemma (1.2) of [2], page 84. q.e.d.

(A.8) Corollary (Lifting lemma). Consider a 𝔉-cofibration i: A → M, a 𝔉-homotopy equivalence p: X → Y, and two maps f: A → X, g: M → Y such that gi = pf. Then there is a 𝔉-map h: M → X such that hi = f and ph ∼ g rel A. The map h is termed the lifting of the following diagram, and is unique up to homotopy rel A.

[figure: A; f; i; X; p; ≃; M; h; g; X]

Proof. See lemma (1.11) of [2], page 90. q.e.d.

The following is Dold’s theorem in 𝔉-Top.

(A.9) Corollary. Consider the commutative diagram.

[figure: X; g; Y; A]

If g is a 𝔉-homotopy equivalence, then g is a 𝔉-homotopy equivalence under A, that is, there is a 𝔉-map f: Y → X such that gf ∼ 1_{Y} rel A and fg ∼ 1_{X} rel A.

Proof. See corollary (2.12) of [2], page 96. q.e.d.

(A.10) Remark. Let 𝔉 be a structure category with fibre functor F and i: A → X a (𝔉, F)-cofibration. Then ī: Ā → X̄ and i: A → X are (closed) cofibrations in Top. Moreover, since the fibre functor F is faithful, the induced map i^{V} on function spaces

i^{V} : A^{V} → X^{V}

is a (𝔉, F^{V})-cofibration for every V ∈ 𝔉.

Proof. Since i: A → X is a (𝔉, F)-cofibration, there is a (𝔉, F)-retraction

r: X × I → A × I ∪ X × {0}.

By definition of 𝔉-map, this implies that i and ī are cofibrations in Top, where clearly A ⊂ X and Ā ⊂ X̄ are closed. Moreover, assume that F is faithful, so that the function spaces A^{V}, X^{V} are defined for every V ∈ 𝔉. Then there exists the induced map

r^{V} : (X × I)^{V} → (A × I ∪ X × {0})^{V} .

Since (X × I)^{V} ≅ X^{V} × I and

(A × I ∪ X × {0})^{V} ≅ A^{V} × I ∪ X^{V} × {0},

and the homeomorphisms are actually (𝔉, F^{V})-isomorphisms, there is a (𝔉, F^{V})- retraction

X^{V} × I → A^{V} × I ∪ X^{V} × {0}.

Since i^{V} : A^{V} → X^{V} is a closed inclusion, it is a (𝔉, F^{V})-cofibration. q.e.d.

We recall now the definition of 𝔉-deformation retract, following Whitehead [34], page 24. A 𝔉-family A is a 𝔉-deformation retract of X ⊃ A if there is a 𝔉-homotopy F: X × I → X such that F (x, t) = x for each (x, t) ∈ A × I ∪ X × {0} and F (X × {1}) ⊂ A.

(A.11) Lemma. Let X be a (𝔉, F)-family and A ⊂ X a closed inclusion. Then for every V ∈ 𝔉 the induced map A^{V} → X^{V} is a closed inclusion of (𝔉, F^{V})-families. Moreover, if A is a (𝔉, F)-deformation retract of X, then A^{V} is a (𝔉, F^{V})-deformation retract of X^{V}. A is a deformation retract of X in Top and Ā is a deformation retract of X̄ in Top.

Proof. The inclusion A^{V} → X^{V} is a closed inclusion since A^{V} ⊂ X^{V} is equal to the pre-image of Ā under the map X^{V} → X̄. Furthermore, any (𝔉, F)-deformation retraction F: X × I → X yields a (𝔉, F^{V})-deformation retraction F^{V} : X^{V} × I → X^{V}. q.e.d.

Appendix B. The category of coefficients

We here describe the coefficient functor for 𝔉-Top which is a basic ingredient of the theory in [3].

(B.1) Definition. Let A be a 𝔉-space. Consider pairs of 𝔉-maps x, y: Z → A where Z is a 𝔉-set. We associate with x, y the 𝔉-space X(x, y) given by the push-out in 𝔉-Top

\xymatrix{Z × \{0, 1\} \ar[r]^{(x,y)} \ar[d] & A \ar[d] \\ Z × I \ar[r] & X(x, y)}

Hence (X(x, y), A) is a reduced 1-dimensional relative 𝔉-complex. We call it a 𝔉-graph under A. Let T denote the category of 𝔉-graphs: Objects are 𝔉-graphs X = (X, A) = (X(x, y), A) and morphisms are 𝔉-homotopy classes rel A of 𝔉-maps, so that

T(X, Y) = [X, Y]_{𝔉}^{A}

for 𝔉-graphs X, Y under A. The coproduct X ∨_{A} Y = X ∪_{A} Y in T is defined by the push-out of X ← A → Y in 𝔉-Top. Since finite coproducts exists in T, T is a theory. Let T^{♯} denote the full subcategory of T consisting of 𝔉-graphs X(x, y) for which the 𝔉-set Z above is finite. Since 𝔉 is a small category, we see that T^{♯} is a small category. Let model(T^{♯}) denote the category of models of 𝔉-graphs, that is the category of functors (T^{♯})^{op} → Set which carry coproducts in T^{♯} to products in Set.

(B.2) Lemma. A T-complex in 𝔉-Top^{A} as defined in [3] is the same as a relative 𝔉-complex (X, A) which is reduced and normalized.

Proof. According to IV.2.2 of [3] a T-complex X_{≥1} in 𝔉-Top is given by a sequence of 𝔉-cofibrations

X_{≥1} = (X_{1} ⊂ X_{2} ⊂ …)

where X_{1} is an object in T and (X_{n+1}, X_{n}) is a principal cofibration with attaching map ∂_{n+1} : Σ^{n−1}C_{n+1} → X_{n}, n ≥ 1, where C_{n+1} is a cogroup in T (that is, a 1-dimensional spherical object S_{α}^{1} under A). Since Σ^{n−1}S_{α}^{1} = S_{α}^{n}, we have the result. q.e.d.

(B.3) Remark. The category 𝔉-Top is a cofibration category under T that satisfies the Blakers–Massey property (see [3], page 245). Hence it follows from proposition 1.2, page 250 of [3] that 𝔉-Top is a homological cofibration category under T. Moreover 𝔉-Top is a cellular I-category since the ball pair axiom holds, as a consequence of the ball pair axiom in Top and the cellular approximation theorem (3.5) (see page 327 of [3]).

We recall that a track category, i.e. a category enriched in groupoids, is a 2-category all of whose 2-cells are invertible (see [20] for details). In particular the category of groupoids Grd is a track category. Objects are groupoids, morphisms are functors between groupoids and tracks are natural transformations. A track functor, or else a 2-functor, between track categories is a groupoid enriched functor.

Let 𝔉 be a topological enriched category. Then there is a canonical track category ⟦𝔉⟧ associated to 𝔉 defined as follows. Objects and morphisms are the same as 𝔉 and 2-cells are tracks b: α ⟹ α′ in W^{V} = hom_{𝔉}(V, W) with α, α′: V → W and V, W objects in 𝔉

A ⟦𝔉^{op}⟧-diagram is a track functor ⟦𝔉^{op}⟧ → Grd. A morphism between ⟦𝔉^{op}⟧-diagrams is a natural transformation of track functors. An example of a track functor is given by

⟦Π_{𝔉}(X^{∘}, A^{∘})⟧: ⟦𝔉⟧ → Grd,

with (X, A) a pair of 𝔉-families. The image of an object V ∈ ⟦𝔉⟧ is the groupoid Π(X^{V} , A^{V}). The image of a morphism λ: V → W in ⟦𝔉⟧ is the morphism of groupoids λ^{∗}: Π(X^{W} , A^{W}) → Π(X^{V} , A^{V}) induced by the map X^{λ}: X^{W} → X^{V}. The image of a track b: λ ⟹ λ′ in ⟦𝔉⟧ is given by the natural transformation

\xymatrix{Π(X^{W} , A^{W}) \ar[rr]^{Π(λ)} \ar[rr]_{Π(λ′)} & ⇓ b^{∗} & Π(X^{V} , A^{V})}

in Grd.

With the aid of the track category ⟦𝔉⟧ we define the category (𝔉, A)-Grd as follows. Objects are track functors F: ⟦𝔉^{op}⟧ → Grd under ⟦Π(A^{∘})⟧ such that the natural transormation

ι: ⟦Π(A^{∘})⟧ → F

for each V ∈ 𝔉 yields a functor

ι: Π(A^{V}) → F^{V}

which is the identity on objects. A morphism in (𝔉, A)-Grd is a natural transformation of track functors under ⟦Π(A^{∘})⟧.

(B.4) Theorem. Let T^{♯} denote the theory of finite 𝔉-graphs under A. Then there is an equivalence of categories

model(T^{♯}) ≃ (𝔉, A)-Grd.

Proof. Consider the following functor

ξ: model(T^{♯}) → (𝔉, A)-Grd.

Let M: (T^{♯})^{op} → Set be a model of T^{♯}. Then ξ(M) is the ⟦𝔉^{op}⟧-diagram of groupoids under ⟦Π(A^{∘})⟧ defined as follows. Let V be an object of 𝔉.

Objects of the groupoid ξ(M)(V) are the same objects as Π_{𝔉}^{V}(A) = Π(A^{V}), i.e. 𝔉-points of type x_{V} : V → A. For every pair of objects x_{V} , y_{V} in ξ(M)(V) the homset is given by the set M(X(x_{V} , y_{V})). Let x_{V} , y_{V} and z_{V} be points in A^{V}. The natural 𝔉-map

X(x_{V} , z_{V}) → X(x_{V} ∪ y_{V} , y_{V} ∪ z_{V}) = X(x_{V} , y_{V}) ∨ X(y_{V} , z_{V})

induces under M a function

[figure: M(X(x_{V} ∪ y_{V} , y_{V} ∪ z_{V})); M(X(x_{V} , y_{V})) × M(X(y_{V} , z_{V})); M(X(x_{V} , z_{V}))]

which yields the composition law for morphisms in ξ(M). Since there is a natural 𝔉-isomorphism X(x_{V} , y_{V}) ≃ X(y_{V} , x_{V}), we obtain that ξ(M)(V) is a groupoid.

Now let λ: V → W be a morphism in 𝔉. The induced 𝔉-map A^{W} → A^{V} is a function on objects of ξ(M). Given x_{W} and y_{W} in A^{W} the induced 𝔉-map

X(x_{W} λ, y_{W} λ) → X(x_{W} , y_{W})

has an image under M

[figure: ξ(M)(W)(x_{W} , y_{W}); ξ(M)(V)(x_{W} λ, y_{W} λ); M(X(x_{W} , y_{W})); M(X(x_{W} λ, y_{W} λ))]

Hence ξ(M)(λ) is a morphism of groupoids, and ξ(M) is a functor 𝔉^{op} → Grd. It is not difficult to show that it is a functor under Π(A^{∘}).

It is left to define the values of ξ(M) on the tracks of ⟦𝔉^{op}⟧. A track b: λ ⟹ λ′ in ⟦𝔉^{op}⟧ is represented by a 𝔉-map

b: V × I → W.

Hence for every x_{W} : W → A the track b induces a 𝔉-map x_{W} b: V × I → A, and hence a 𝔉-map

X(x_{W} λ, x_{W} λ′) → A.

By taking the values of the model M we obtain a function

∗ = M(A) → M(X(x_{W} λ, x_{W} λ′)),

hence we obtain an element in M(X(x_{W} λ, x_{W} λ′)) which is therefore a morphism in ξ(M)(V)(x_{W} λ, x_{W} λ′). This defines the natural transormation induced by b. By the elementary properties of M it is easy to prove that ξ is a track functor.

Conversely, let C be an object in (𝔉, A)-Grd. Then to each 𝔉-graph X(x_{Z}, y_{Z}) with x_{Z}, y_{Z}: Z → A we can associate the set

ξ′(C)(X(x_{Z}, y_{Z}) = Mor (⟦Π(X(x_{Z}, y_{Z})^{∘}, A^{∘})⟧, C)

of all morphisms in (𝔉, A)-Grd from ⟦Π(X(x_{Z}, y_{Z})^{∘}, A^{∘})⟧ to C. It is clear that ξ′(C) is a functor (T^{♯})^{op} → Set. It is also a model by (B.8) below. It is possible to prove that this construction yields a functor

ξ′: (𝔉, A)-Grd → model(T^{♯}).

The functors ξ and ξ′ are equivalences of categories (the details are left to the reader). q.e.d.

We define the coefficient functors c and π_{𝔉}

\xymatrix{ & model(T^{♯}) \ar[d]^{≃}_{ξ} \\ (𝔉-Top)^{A}/_{≃ rel A} \ar[r]^{π_{𝔉}} \ar@{-->}[ur]^{c} & (𝔉, A)-Grd}

(B.5)

as follows. For every 𝔉-family (X, A) under A (where A → X is a 𝔉cofibration) let π_{𝔉}(X, A) = ⟦Π(X^{∘}, A^{∘})⟧. Furthermore, let c(X, A) be the model in model(𝕋^{♯}) that associates to the 𝔉-graph X(x, y) with x, y: Z → A the set of 𝔉-homotopy classes rel A of 𝔉-maps extending 1_{A}

X(x, y) ↦ [X(x, y), X]_{𝔉}^{A}.

(B.6) Proposition. The diagram (B.5 ) is commutative:

ξc = π_{𝔉}.

Proof. The object ξc(X, A) in (𝔉, A)-Grd associates to V ∈ ⟦𝔉^{op}⟧ the groupoid ξc(X, A)^{V} with objects the points in A^{V}; the morphisms are the sets

c(X, A)(X(x, y)) = [X(x, y), X]_{𝔉}^{A}

where x, y: V → A are 𝔉-points. The morphism set [X(x, y), X]_{𝔉}^{A} is naturally isomorphic to Π(X^{V} , A^{V})(x, y). The same argument can be applied to tracks, so that we get ξc = π_{𝔉} as claimed. q.e.d.

Now we define the category (𝔉, A)-Set. Objects are pairs of 𝔉-maps x, y: Z → A where Z is a 𝔉-set. A morphism (x, y) → (x′, y′) is a 𝔉-map u: Z → Z′ such that x′u = x and y′u = y.

Consider now an object F: ⟦𝔉^{op}⟧ → Grd in (𝔉, A)-Grd. For every object V ∈ 𝔉 we denote by F^{V} the groupoid associated to V. For every V ∈ 𝔉 consider the two maps

\xymatrix{F_{1}^{V} \ar[r]^{∂_{0}} \ar[r]_{∂_{1}} & A^{V}}

from the set of arrows of F^{V} to A^{V} given by the source ∂_{0} and the target ∂_{1} in the groupoid F^{V}. This yields an object x_{V} , y_{V} : Z_{V} = V × F_{1}^{V} → A in (𝔉, A)-Set. The coproduct of all such objects is an object Φ(F) = x, y: Z → A. This defines the forgetful functor

Φ: (𝔉, A)-Grd → (𝔉, A)-Set.

The left-adjoint of Φ denoted by <, > sends an object x, y: Z → A in (𝔉, A)-Set to the free object < x, y > in (𝔉, A)-Grd.

(B.7) Proposition. For every x, y: Z → A, we have a natural isomorphism

< x, y >= π_{𝔉}(X(x, y), A).

Moreover a result corresponding to the Seifert–van Kampen theorem holds. The proof is a modification of the Brown’s proof [8, 9] of the Seifert– van Kampen theorem for groupoids.

(B.8) Proposition. If X_{1} and X_{2} are 𝔉-complexes relative A both containing a copy of a relative 𝔉-complex (X_{0}, A) as a subcomplex then

π_{𝔉}(X_{1} ∪_{X_{0}} X_{2}) = π_{𝔉}(X_{1}) ∪_{π_{𝔉}(X_{0})} π_{𝔉}(X_{2}).

Here the right hand side denotes the push-out in (𝔉, A)-Grd.

(B.9) Remark. In [3] the coefficient functor c maps to the category Coef which can be identified with

(𝔉, A)-Grd ≃ model(T^{♯}) ≃ Coef

such that the coefficient functor c in [3] mapping to Coef coincides with π_{𝔉} and c in (B.5). Moreover the enveloping functor U of [3] (pages ≥ 150) can be identified with the composite

U: \xymatrix{Coef \ar[r]^{≃} & (𝔉, A)-Grd \ar[r]^{Ū} & Ringoids}

where Ū carries F to the category ℤ [(∫_{𝔉} F)/_{∼}]. Here the equivalence relation ∼ for the integration category ∫_{𝔉} F is defined as in (6.3) above. Hence for a reduced relative 𝔉-complex (X, A) we get

Uc(X, A) = Ū(π_{𝔉}(X, A)) = ℤΠ_{𝔉}(X, A)

where we use the discrete fundamental category in (6.3).

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