K-theory of stratified vector bundles

Hans-Joachim Baues and Davide L. Ferrario

Abstract

We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold is such a stratified vector bundle.

1 Families of vector spaces

Let ℝ be the field of real numbers and ℝ^{n} = ℝ ⊕ ⋯ ⊕ ℝ be the standard n-dimensional ℝ-vector space. Let Vect denote the category of finite dimensional ℝ-vector spaces and linear maps and V a subcategory of Vect, termed the structure category. For example let V be the subcategory of the surjective maps in Vect. Or let G be a subgroup of the automorphism group GL_{n}(ℝ) of ℝ^{n}. Then G yields the subcategory G ⊂ Vect consisting of one object ℝ^{n} and morphisms given by elements in G.

The category Vect is a topological category. We say that V is a closed subcategory if for objects V, W ∈ V the inclusion of morphisms sets

hom_{V}(V, W) ⊂ hom_{Vect}(V, W)

is a closed subspace. Here hom_{Vect}(V, W) is homeomorphic to ℝ^{N} while hom_{V}(V, W) needs not to be a vector space.

Following Atiyah [1] a family of vector spaces in V is a topological space X together with a map p_{X} : X → X̄ and for every b ∈ X̄ a homeomorphism, named chart, Φ_{b} : p_{X}^{−1}(b) ≈ X_{b} where X_{b} is a suitable object of V depending upon b ∈ X̄. We denote the family (p_{X} : X → X̄, X_{b}, Φ_{b}, b ∈ X̄) simply by X (it will be clear from the context whether X is the underlying space or the family). The map p_{X} is termed the projection, the space X the total space of the family of vector spaces, the space X̄ the base space, and for every b the vector space X_{b} is termed fiber over b. A family X is also termed V-family, to make explicit the choice of V. The V-family X is discrete if X̄ is a discrete space. In this case X is a disjoint union of objects in V and we call X a V-set.

Given two V-families X and Y a V-map from X to Y (equivalently, a homomorphism or an admissible map controlled by V) is a pair of maps (f, f̄) such that the following diagram commutes

\xymatrix{X \ar[r]^{f} \ar[d]_{p_{X}} & Y \ar[d]^{p_{Y}} \\ X̄ \ar[r]^{f̄} & Ȳ ,} (1.1)

and such that for every b ∈ X̄ the composition given by the dashed arrow of the diagram

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

is a morphism of V. A V-isomorphism (f, f̄): X → Y induces homeomorphisms f: X → Y, f̄: X̄ → Ȳ, and f|X_{b}: X_{b} → Y_{f̄b} for every b ∈ X̄.

If V is a vector space in V and X̄ is a topological space, then the projection onto the first factor p_{1} : X = X̄ × V → X̄ is the projection map of the product family with fiber V; the charts Φ_{b} : {b} × V → V = X_{b} are given by projection and X_{b} = V for all b ∈ X̄.

If Y is a family V-isomorphic to a product family then Y is said to be a trivial vector bundle.

Given a family Y and a map f̄: X̄ → Ȳ, the pull-back X = f̄^{∗}Y is defined by the following pull-back diagram.

\xymatrix{X = f̄^{∗}Y \ar@{-->}[r] \ar@{-->}[d] & Y \ar[d]^{p} \\ X̄ \ar[r]^{f̄} & Ȳ .} (1.2)

If f̄ = i: X̄ ⊂ Ȳ is an inclusion, then i^{∗}Y is called the restriction of Y to X̄ and is denoted by Y |X̄.

A V-vector bundle or a V-bundle is a locally trivial family of vector spaces in V, i.e. a family X over X̄ such that every b ∈ X̄ admits a neighborhood U for which X|U is trivial.

If X is a V-family and Z is a topological space, then the cartesian product X × Z is a V-family with projection p_{X×Z} = p × 1_{Z}: X × Z → X̄ × Z. The fiber over a point (b, z) ∈ X̄ × Z is equal to p_{X}^{−1}(b) × {z} and 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} and Φ_{b} is the chart Φ_{b} : p_{X}^{−1}(b) → X_{b}.

In particular, if Z is the unit interval I one obtains the cylinder object X × I; this leads to the definition of homotopy: two V-maps f_{0}, f_{1} : X → Y are V-homotopic (in symbols f_{0} ∼ f_{1}) if there is a V-map F: X × I → 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 to X × {0} and X × {1} respectively.

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

A V-map i : A → Y is termed a closed inclusion if ī : Ā → Ȳ is an inclusion, īĀ is closed in Ȳ and ī^{∗}Y = A. A closed inclusion i : A → Y induces isomorphisms on fibers.

The push-out construction can be extended to the category V-Top, provided the push-out is defined by a closed inclusion.

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

\xymatrix{A \ar[r]^{f} \ar[d]_{i} & X \ar[d] \\ Y \ar[r] & Z} (1.4)

exists and X → Z is a closed inclusion.

Proof. See [3], lemma (2.5). q.e.d.

2 Stratified vector bundles

We introduce the notion of stratified space and stratified vector bundle. They are particular cases of stratified bundles as defined in [3], definition (4.1). In the next section we describe as an example the stratified tangent bundle.

Let A be a closed subset of a space X. We say that (X, A) is a CWpair if there exists a homeomorphism (X, A) ≈ (X′, A′) of pairs where X′ is a CW-complex and A′ a subcomplex of X′. A CW-space X is a space homeomorphic to a CW-complex.

(2.1) Definition. A stratified space is a space X̄ together with a filtration

X̄_{0} ⊂ X̄_{1} ⊂ ⋯ ⊂ X̄_{n} ⊂ ⋯ ⊂ lim_{n→∞} X̄_{n} = X̄

such that for every i ≥ 1 there is a CW-pair (M̄_{i}, Ā_{i}) and an attaching map h̄_{i}: Ā_{i} → X̄_{i−1} with the property that the subspace X̄_{i} is obtained by attaching M̄_{i} to X̄_{i−1} via the attaching map h̄_{i}, i.e. X̄_{i} = M̄_{i} ∪h̄_{i} X̄_{i−1}.

The complements X̄_{i} ∖ X̄_{i−1} are termed strata, while the filtration X̄_{0} ⊂ X̄_{1} ⊂ ⋯ ⊂ X̄_{n} ⊂ … is called stratification of X̄. In definition (2.1) the strata X̄_{i} ∖ X̄_{i−1} coincide with the complements M̄_{i} ∖ Ā_{i}. The pairs (M̄_{i}, Ā_{i}) are the attached spaces of X̄. A stratified space X̄ is Hausdorff and regular and if all the attaching maps h̄_{i} are cellular then X̄ is a CW-complex. If the number of non-empty strata of X̄ is finite, then X̄ is said finite. We always assume that a stratified space which is not finite has cellular attaching maps. A finite stratified space X̄ is compactly stratified if all the attached spaces M̄_{i} and X̄_{0} are compact.

A map f̄: X̄ → X̄′ between stratified spaces is a filtration preserving map f̄ = {f̄_{n}}_{n≥0} together with commutative diagrams

[figure: X̄_{i−1}; f̄_{i−1}; Ā_{i}; M̄_{i}; ḡ_{i}; X̄′_{i−1}; Ā′_{i}; M̄′_{i}]

such that ḡ_{i} ∪ f̄_{i−1} = f̄_{i} for i ≥ 1. A map is termed stratum-preserving if for every i

f̄_{i}(X̄_{i} ∖ X̄_{i−1}) ⊂ X̄′_{i} ∖ X̄′_{i−1}.

Let Stra be the category of finite stratified spaces and stratum-preserving maps. Two maps f_{0} and f_{1} are homotopic in Stra if there exists a homotopy f_{t} such that the map f_{t} is a map in Stra for every t ∈ I. With this notion of homotopy the homotopy category Stra/_{∼} is defined. (2.2) Definition. A V-stratified bundle is a stratified space X̄ together with a V-family

X → X̄

with the following properties. For i ≥ 1 the restriction X_{i} = X|X̄_{i} is the push-out of V-maps

[figure: M_{i}; A_{i}; h_{i}; X_{i−1}; M̄_{i}; Ā_{i}; h̄_{i}; X̄_{i−1}]

where M_{i} → M̄_{i} is a V-vector bundle and A_{i} = M_{i}|Ā_{i}. Moreover, X_{0} → X̄_{0} is a V-vector bundle and X = lim_{i→∞} X_{i}. If (M̄_{i}, Ā_{i}) is a disjoint union of i-dimensional cells (D^{i}, S^{i−1}) then X̄ is a CW-complex and X is termed a V-complex.

A V-stratified map f: X → X′ between V-stratified bundles is given by sequences {f_{n}}_{n≥0} and {g_{n}}_{n≥1} of V-maps such that given the commutative diagrams

[figure: X_{i−1}; f_{i−1}; A_{i}; M_{i}; g_{i}; X′_{i−1}; A′_{i}; M′_{i}]

for every i we have g_{i} ∪ f_{i−1} = f_{i} for i ≥ 1. The V-map f is termed stratumpreserving if for every i

f(X_{i} ∖ X_{i−1}) ⊂ X′_{i} ∖ X′_{i−1}.

Let V-Stra denote the category of V-stratified bundles and stratum-preserving V-stratified maps. Of course a V-isomorphism in V-Stra is a stratumpreserving V-stratified map f with an inverse f′.

The following theorems are proved in [3] (theorem (4.7) and theorem (5.1)).

(2.3) Pull back theorem. Let X̄ and X̄′ be finite stratified spaces with attached spaces which are locally finite and countable CW-complexes. Let f̄: X̄ → X̄′ be a stratum-preserving map (in Stra) and X′ → X̄′ a Vstratified bundle. Then the pull-back f̄^{∗}X′ → X̄ is a V-stratified bundle.

(2.4) Bundle theorem. Let V ⊂ Vect be a subcategory such that all morphisms in V are isomorphisms (i.e. V is a groupoid). Then a V-stratified bundle X → X̄ is a V-bundle over X̄. Conversely, given a stratified space X̄ and a V-bundle X over X̄ then X is a V-stratified bundle.

In particular, a V-complex X → X̄ is a V-bundle over X̄ and, given a CW-complex X̄ and a V-bundle X over X̄, then X is V-isomorphic to a V-complex.

3 Stratified manifolds

In this section we describe special families of vector spaces which arise as “stratified tangent bundles“ of stratified manifolds or of manifolds with singularities. Such stratified tangent bundles are motivating examples of stratified vector bundles above. Concerning the general theory of stratification of manifolds we refer the reader for example to [5, 9, 12, 6, 10]. We here introduce stratified manifolds only since they show that stratified vector bundles are natural generalizations of vector bundles in a similar way as manifolds with singularities or stratifolds [5] generalize manifolds.

If M is a manifold with boundary ∂M, N a closed manifold, and h: ∂M → N a submersion, then the push-out of h and the inclusion ∂M ⊂ M is a stratified space X = M ∪_{h} N with stratification

X_{0} = N ⊂ X_{1} = X.

This is a stratifold in the sense of [5] and generalizes “manifolds with singularities”. See Rudyak [7], Baas [2], Botvinnik [4], Sullivan [8], Vershinin [11]. Below we introduce differentiable stratified spaces which we consider as the general form of stratified manifolds

If M is a differentiable manifold then the tangent bundle T M → M restricted to the boundary T M_{|∂M} is isomorphic to the sum of the tangent bundle T ∂M of the boundary of M and the normal bundle ν∂M = ∂M × ℝ as in the diagram

\xymatrix{T M_{|∂M} \ar[rr]^{≅} \ar[dr] & & T ∂M ⊕ ν∂M \ar[dl] \\ & ∂M} (3.1)

The sum decomposition of T M|∂M for example is obtained by a tubular neighborhood N of the boundary and an explicit homeomorphism N ≈ ∂M × ℝ_{+}. The sum decomposition yields the projection

p_{1} : T M_{|∂M} ≅ T ∂M ⊕ ν∂M → T ∂M. (3.2)

Now we define a stratified manifold X and the stratified tangent bundle 𝕋X by induction in the number of strata. (3.3) Definition. A stratified space X with finite stratification X_{0} ⊂ ⋯ ⊂ X_{n} = X and attached spaces (M_{i}, A_{i}) is termed differentiable if the following conditions hold. The spaces M_{i} are compact differentiable manifolds with boundary A_{i} = ∂M_{i}. Also X_{0} is a compact differentiable manifold with ∂X_{0} = ∅. Moreover for i = 1, …, n the restriction f_{i}^{∘} as in the diagram

[figure: M_{i−1} ∖ ∂M_{i−1}; f_{i}^{−1}(X_{i−1} ∖ X_{i−2}); f_{i}^{∘}; X_{i−1} ∖ X_{i−2}; ∂M_{i}; f_{i}; X_{i−1}]

is differentiable and a submersion. In particular, f_{1} : ∂M_{1} → X_{0} is a submersion. We assume that f_{i}^{−1}(X_{i−1} ∖ X_{i−2}) is dense in ∂M_{i} so that the map f_{i} is uniquely determined by f_{i}^{∘}.

We obtain inductively the tangent family 𝕋X_{i} in Vect-Top together with inclusions

T (X_{i} ∖ X_{i−1}) ⊂ 𝕋X_{i}

as follows. For i = 0 we have 𝕋X_{0} = T X_{0}. Assume T (X_{i−1} ∖ X_{i−2}) ⊂ 𝕋X_{i−1} is defined. Then we consider the diagram in Vect-Top

[figure: T (M_{i−1} ∖ ∂M_{i−1}); T f_{i}^{−1}(X_{i−1} ∖ X_{i−2}); df_{i}^{∘}; T (X_{i−1} ∖ X_{i−2}); T ∂M_{i}; df_{i}; 𝕋X_{i−1}] (3.4)

Since the inclusion in the first column of the diagram is dense there exists at most one Vect-map df_{i} for which the diagram commutes. In addition to the differentiability in (3.3) we assume inductively that df_{i} exists for i ≥ 2 and that df_{i} is surjective. Then 𝕋X_{i} is defined by the following push-out diagram, which exists by lemma (1.3).

[figure: T M_{i}|∂M_{i}; p_{1}; T M_{i}; ∂M_{i}; f_{i}; M_{i}; T ∂M_{i}; df_{i}; X_{i−1}; X_{i}; 𝕋X_{i−1}; T M ∪_{df_{i}∘p_{1}} 𝕋X_{i−1} = 𝕋X_{i}; p_{i}]

A differentiable stratified space is termed a stratified manifold.

We call 𝕋X = 𝕋X_{n} the stratified tangent bundle of the differentiable stratified space X. As a set 𝕋X is the disjoint union of T (X_{i} ∖ X_{i−1}), that is of the tangent bundles T (M_{i} ∖ ∂M_{i}) for i = 0, …, n. Moreover, for every i, the pull-back of 𝕋X over the inclusion of the stratum X_{i} ∖ X_{i−1} ⊂ X is the tangent bundle of the stratum itself. But the topology of 𝕋X is not the topology of the disjoint union.

(3.5) Lemma. Different choices of projections p_{1} in the definition of 𝕋X yield isomorphic tangent families.

Proof. We prove it by induction. For n = 0 it is true, because 𝕋X_{0} = T X_{0}. So assume that 𝕋X_{i−1} does not depend upon the choice of p_{1}. Let p_{1} and p′_{1} be two projections p_{1}, p′_{1} : T M_{i}|∂M_{i} → T ∂M_{i} corresponding to two choices of the normal bundle

ν∂M_{i}, ν′∂M_{i} ⊂ T ∂M_{i}|∂M_{i}.

The two normal bundles are equivalent and there is a homotopy of bundles H_{t} : T M_{i}|∂M_{i} covering the identity of ∂M_{i} such that H_{1} is the identity, H_{t} is the identity whenever restricted to T ∂M_{i} and an isomorphism of bundles for every t ∈ I, and H_{0} sends ν∂M_{i} to ν′∂M_{i}. The fat homotopy (H_{t}, t): (T M_{i}|∂M_{i}) × I → (T M_{i}|∂M_{i}) × I defined by (x, t) ↦ (H_{t}(x), t) is therefore an isomorphism of V-families covering the identity of ∂M_{i} × I extending the V-isomorphism F = H_{0}. Moreover, there is a tubular neighborhood N ≈ ∂M_{i} × I of ∂M in M_{i}, where we identify ∂M_{i} with ∂M_{i} × {0} in N. Because T M_{i}|N = (T M_{i}|∂M_{i}) × I, the fat homotopy (H_{t}, t) induces a V-isomorphism on T M_{i}|N covering the identity of N, with the further property that it is the identity on T M_{i}|(∂M_{i} × {1}). Thus it is possible to extend F = H_{0} to a V-isomorphism F̃ on T M_{i}. By construction, p′_{1}F = p_{1}, and therefore the following diagram commutes and the V-equivalence G exists.

\xymatrix{T M_{i}|∂M_{i} \ar[rrr] \ar[dd]_{p′_{1}} &  &  & T M_{i} \ar[dddd] \\  & T M_{i}|∂M_{i} \ar[ul]^{F} \ar[d]^{p_{1}} \ar[r] & T M_{i} \ar[ur]^{F̃} \ar[dd] \\ T ∂M_{i} \ar[dd]^{df_{i}} & T ∂M_{i} \ar[l]^{1} \ar[d]^{df_{i}} \\  & 𝕋X_{i−1} \ar[r] \ar[dl]^{1} & 𝕋X_{i} \ar@{.>}[dr]^{G} \\ 𝕋X_{i−1} \ar[rrr] &  &  & 𝕋X_{i}}

Thus the isomorphism type of 𝕋X_{i} does not depend on the choice of the projection p_{i}, as claimed. q.e.d.

(3.6) Remark. The map f_{i} is proper since ∂M_{i} is compact and X_{i−1} is Hausdorff. Therefore the restriction map f_{i}^{∘}

f_{i}^{∘} : f_{i}^{−1}(X_{i−1} ∖ X_{i−2}) → X_{i−1} ∖ X_{i−2} = M_{i−1} ∖ ∂M_{i−1}

is a proper surjective submersion and hence by Ehresmann’s theorem it is a fiber bundle.

(3.7) Example. Let M be a differentiable manifold with boundary; let N = ∂M and h = 1_{∂M} : ∂M → ∂M = N. Then, the stratified space M ∪_{h} ∂M coincides with M. The tangent family p : 𝕋M → M defined above is not isomorphic to the tangent bundle T M → M (the fibers over ∂M in 𝕋M are vector spaces of dimension dim ∂M and not dim M). Actually the stratified tangent bundle 𝕋M has the quotient topology under the map ϕ : T M → 𝕋M induced by the push-out. For example let M = D^{n} the closed n-dimensional unit disc in ℝ^{n}, and N its boundary ∂M = S^{n−1}. M is a smooth manifold and the rays through the origin give rise to a normal bundle of ∂M in M. The identity map h = 1 : ∂M → S^{n−1} is of course differentiable, and the map dh ∘ p_{1} is nothing but the projection T M_{|∂M} = S^{n−1} × ℝ^{n} → T S^{n−1} defined by (x, v) ↦ (x, v − \frac{v·x}{x·x}x), where (x, v), with x ∈ S^{n−1} and v ∈ ℝ^{n}, orthogonal to x, are the usual coordinates for the tangent bundle of a sphere. Thus the stratified tangent bundle 𝕋M is defined by the quotient map ϕ : T M → 𝕋M which is the inclusion on T M|(M ∖ ∂M). Moreover two points (x, v) and (y, w) of T M|∂M = S^{n} × ℝ^{n} have the same image in 𝕋M if and only if v − \frac{v·x}{x·x}x = w − \frac{w·y}{y·y}y.

(3.8) Example. Let X be the standard cube in ℝ^{3}. Let X_{0} be the set of vertices of X, X_{1} the union of the edges, X_{2} the faces and X_{3} = X. Thus the filtration

X_{0} ⊂ X_{1} ⊂ X_{2} ⊂ X_{3} = X

yields a decomposition into strata of dimension 0,1, 2 and 3 respectively. It is not difficult to see that for every i = 1 … 3 there is a disjoint union of discs M_{i} = ∪D^{i} and a surjective map f_{i} : ∂M_{i} = ∪S^{i−1} → X_{i−1} such that X_{i} is obtained as attachment of M_{i} to X_{i−1} via f_{i}. The stratum X_{0} consists of the 8 vertices, the attaching manifold M_{1} of 12 edges (and the boundary of M_{1} consists of 24 disjoint points); in a similar way, M_{2} is the union of 6 discs, with boundary 6 circles S^{1}, and M_{3} is a unique disc with boundary S^{2}. The cube X is a differentiable stratified space.

4 V-function spaces

Let Y be a V-family. For V ∈ V we denote by Y^{V} the space of all the Vmaps V → Y, endowed with the compact-open topology. A sub-basis of the topology in V^{Y} is given by all the sets of the form N_{K,U} = {f ∈ Y^{V} : fK ⊂ U} with K ⊂ V compact and U ⊂ Y open. For every V-family Y there is a standard projection Y^{V} → Ȳ sending a V-map a: V → Y to the point p_{Y} a(0) ∈ Ȳ, where 0 ∈ V is the zero of the vector space. The pre-image of a point b ∈ Ȳ under this projection is homeomorphic to hom_{V}(V, Y_{b}).

Given a V-map f: X → Y, let f^{V} denote the map f^{V} : X^{V} → Y^{V} defined by composing with f the V-maps V → X. Moreover for φ: V → W let X^{φ}: X^{W} → X^{V} be the map induced by φ. Given a V-map f: Z × V → X, its adjoint f̂ is the function f̂: Z → X^{V} defined by f̂(z)(v) = f(z, v) for every z ∈ Z and v ∈ V. Lemma (6.10) of [3] states that for every space Z a V-function f: Z×V → X is continuous if and only if the adjoint f̂: Z → X^{V} is continuous. As a consequence, the evaluation map X^{V} × V → X which sends (g, v) to g(v) is continuous and hence if f: X → Y is continuous then the induced function f^{V} : X^{V} → Y^{V} is continuous, and in the same way given φ: V → W the induced map X^{φ}: X^{W} → X^{V} is continuous.

(4.1) Theorem. Let (X, D) be a relative V-complex, where V is a closed subcategory of Vect. Let Φ : Z_{n} × D^{n} → X be its characteristic map of n-cells and let h_{n} denote the n-attaching map of X, i.e. the restriction of Φ to Z_{n} × S^{n−1}. For every n ≥ 0 and every V ∈ V the diagram

[figure: (Z_{n} × S^{n−1})^{V}; h_{n}^{V}; X_{n−1}^{V}; (Z_{n} × D^{n})^{V}; Φ^{V}; X_{n}^{V}]

is a push-out (in Top) with X_{−1} = D. Moreover, X^{V} = lim_{n→∞} X_{n}^{V}.

Proof. See corollary (10.16) of [3]. For this we need the following lemma. q.e.d.

A family K of compact sets of V is termed generating if for every V-family Y the subsets

N_{K,U} = {f ∈ Y^{V} , f(K) ⊂ U}

with K ∈ K and U open in Y are a sub-basis for the topology of the function space Y^{V}.

We say that a structure category V has the NKC-property (see [3], definition (6.5)) if for every object V ∈ V there is a generating family of compact sets K such that for every K ∈ K, every W ∈ V and every compact subset C ⊂ W the subspace

N_{K,C} ⊂ hom_{V}(V, W) = W^{V}

is compact.

(4.2) Lemma. Let V be a closed subcategory of Vect. Then V is a NKC category.

Since a closed subcategory of a NKC category is again a NKC category, it suffices to consider the case V = Vect. Thus lemma (4.2) is a consequence of lemma (4.7) below.

(4.3) Definition. A compact K ⊂ V containing a linear basis of the vector space V is termed spanning compact subset.

(4.4) Lemma. Let Y be a V-family, f: V → Y a V-map, U ⊂ Y an open subset, K ⊂ V a compact subset of V and f ∈ N_{K,U}. Then there is a spanning compact set K′ ⊂ V such that K′ ⊃ K, K′ and f ∈ N_{K′,U} ⊂ N_{K,U}. Thus the family K of spanning compact subsets of V is generating.

Proof. Being f continuous, f^{−1}U is open in V, and therefore there exists a basis B = {b_{1}, b_{2}, …, b_{d}} of V contained in f^{−1}U. Then take K′ = K ∪ B. By definition fB ⊂ U and hence f ∈ N_{K′,U}. Because K′ ⊃ K, N_{K′,U} ⊂ N_{K,U}. q.e.d.

(4.5) Lemma. Let V and W be finitely dimensional normed vector spaces, B = {b_{1}, b_{2}, …, b_{d}} a basis of V and R a constant. Let β: ℝ^{d} → V be an isomorphism sending the standard basis of ℝ^{d} to the basis B of V, and |β^{−1}| the norm of its inverse. If f: V → W is a linear map such that |f(b_{i})| ≤ R for i = 1, …, d, then |f| ≤ \sqrt{d}R|β^{−1}|.

Proof. The composition fβ: ℝ^{d} → W has norm

|fβ| = sup_{x∈ℝ^{d}} \frac{|fβ(x)|}{|x|}.

Write x ∈ ℝ^{d} as ∑_{j} x_{j}e_{j}, where e_{j} are the elements of the standard basis of ℝ^{d}; by assumption β(e_{j}) = b_{j} for every j = 1, …, d. Now consider the inequalities

\frac{| ∑_{j} x_{j}f(b_{j})|}{|x|} ≤ \frac{∑_{j} |x_{j}||f(b_{j})|}{\sqrt{∑_{j} x_{j}^{2}}} ≤ \sqrt{d}R.

This implies that |fβ| ≤ \sqrt{d}R, and hence |f| ≤ |fββ^{−1}| ≤ \sqrt{d}R|β^{−1}|, as claimed. q.e.d.

(4.6) Lemma. If X is a trivial Vect-family X = W × Y over an Hausdorff space Y then X^{V} is a closed subspace of Map(V, X). In particular, by taking Y = ∗, we obtain that for every V, W ∈ V, the function space W^{V} = Vect(V, W) is a closed subspace of Map(V, W).

Proof. Let p_{X} be the projection W × Y → Y and {b_{1}, …, b_{n}} a basis of the vector space V. First, assume that f_{0} ∈ Map(V, X) is a map not in X^{V} such that p_{X}f(V) ≠ p_{X}f(0) = y_{0}. Then there is y ∈ Y such that y ≠ y_{0} and p_{X}f(V) ∋ y. Since Y is Hausdorff, there are neighborhoods Ū_{0} and Ū_{1} in Y such that y_{0} ∈ Ū_{0}, y ∈ Ū_{1} and Ū_{0} ∩ Ū_{1} = ∅. Let use define U_{i} = W × Ū_{i} ⊂ X, K_{0} = {0} ⊂ V and K_{1} = {v}, with some v ∈ (p_{X}f)^{−1}y. We have fK_{i} ⊂ U_{i} and

N_{K_{0},U_{0}} ⋂ N_{K_{1},U_{1}} ⋂ X^{V} = ∅,

hence N_{K_{0},U_{0}} ⋂ N_{K_{1},U_{1}} is an open neighborhood of f contained in Map(V, X)∖ X^{V}.

On the other hand, assume that p_{X}f(V) = p_{X}f(0) = y_{0}. Thus f: V → p_{X}^{−1}y_{0} = W. Since f ∉ X^{V}, f is not linear, there exist a vector v ∈ V and n scalars λ_{i} in the field 𝕂 such that

f(∑_{i=1}^{n} λ_{i}b_{i}) ≠ ∑_{i=1}^{n} λ_{i}f(b_{i}).

Since W × Y is Hausdorff, there are disjoint open sets U_{0} and U_{1} in W × Y such that f(v) ∈ U_{0} and ∑_{i=1}^{n} λ_{i}f(b_{i}) ∈ U_{1}. Now, the map W^{n} → W with sends n vectors v′_{i} to the linear combination ∑_{i=1}^{n} λ_{i}v′_{i} is continuous, hence there are n open sets U′_{i} ⊂ X, neighborhoods of f(b_{i}), such that if x_{i} ∈ U′_{i} for every i = 1, …, n and p_{X}(x_{i}) = p_{X}(x_{j}) for every i, j then ∑_{i=1}^{n} λ_{i}x_{i} ⊂ U_{1}. Consider the compact set K_{0} = {v}, K_{i} = {b_{i}} ⊂ V, i = 1, …, n. Consider the open subset A ⊂ Map(V, X) defined by

A = N_{K_{0},U_{0}} ⋂ N_{K_{1},U′_{1}} ⋂ ⋯ ⋂ N_{K_{n},U′_{n}}.

It consists of all maps g: V → X such that g(v) ⊂ U_{0} and g(b_{i}) ⊂ U′_{i}. By construction f ∈ A. Moreover, assume that g ∈ A ∩ X^{V}. Then

g(v) = ∑_{i=1}^{n} λ_{i}g(b_{i}).

But the left hand side of the equation belongs to U_{0}, while the right hand side belongs to U_{1}, hence g cannot belong to A ∩ X^{V}. Thus A ∩ X^{V} = ∅. q.e.d.

(4.7) Lemma. If K is a spanning compact subset of V, W is an object of V = Vect and C a compact subset of W then N_{K,C} is compact.

Proof. As in the proof of lemma (10.6) of [3] it is easy to see that N_{K,C} is closed. Now we prove that it is equicontinuous. By assumption C is compact and therefore bounded, and by assumption K contains a basis B = {b_{1}, …, b_{d}}; so there exists a constant R ≥ 0, R = |C|, such that if f ∈ N_{K,C} then |f(b_{j})| ≤ R for every j = 1, …, d. But because of lemma (4.5) this implies that there exists a constant R′ (depending only on B and R) such that |f| ≤ R′. This implies that for each v_{0} ∈ V there is a constant α such that for every f ∈ N_{K,C} and for every v ∈ V

|f(v) − f(v_{0})| ≤ α|v − v_{0}|,

hence N_{K,C} is equicontinuous. Now consider for every v_{0} ∈ V the space ω_{v_{0}} = {f(v_{0})}. By the same reason it is bounded in W, hence it has compact closure in W. Thus, by the Ascoli-Arzelà theorem, the closure of N_{K,C} in Map(V, W) is compact. But since N_{K,C} is a closed subset of Vect(V, W) which in turn is closed in Map(V, W) by lemma (4.6), it coincides with its closure in Map(V, W) and therefore is compact. q.e.d.

With lemma (4.7) we have actually proved lemma (4.2).

(4.8) Remark. The principal bundle theorem (6.17) in [3] shows that if V is a NKC category then the diagram X^{∘}: V^{op} → Top given by X^{∘}(V) = X^{V} is the “stratified principal bundle” associated to X. Moreover X is determined by X^{∘} since X = X^{∘} ⊗_{V} F where F: V → Top is the inclusion functor.

5 Algebraic constructions

In the following we will apply theorem (4.1).

Let V and W be two closed subcategories of Vect and let ψ: V → W be a continuous functor (that is, for every V_{1}, V_{2} ∈ V the induced map on hom-spaces ψ: hom_{V}(V_{1}, V_{2}) → hom_{W}(ψV_{1}, ψV_{2}) is continuous). Given a V-complex X we want to build an associated W-complex ψX over X̄ with fibers given by the images ψX_{b} of the fibers of X under ψ. At a set level, we know that X is the disjoint union

X = ∐(Z_{n} × e^{n}),

where e^{n} = D^{n} ∖ S^{n−1} is the open cell and Z_{n} is the discrete V-family of n-cells of X. Therefore it is possible to define ψX as a set by

ψX = ∐(ψ(Z_{n}) × e^{n}).

We now define the topology of ψX inductively. Furthermore, we show that for every V ∈ V the map ψ^{V} : X^{V} → (ψX)^{ψV} defined by sending a: V → X_{b} ⊂ X in X^{V} to ψ(a) ∈ hom(ψ(V), ψ(X_{b})), ψ(a): ψ(V) → ψ(X_{b}) = ψ(X)_{b}, is continuous. The topology of ψX_{0} is the topology of the W-set ψX_{0} = ψZ_{0}. Moreover, since ψ is a continuous functor we see that for every V ∈ V the induced map ψ_{0}^{V} : X_{0}^{V} → (ψX_{0})^{ψV} is continuous. Now, assume that the topology of ψX_{n−1} is defined and that the map ψ_{n−1}^{V}: X_{n−1}^{V} → (ψX_{n−1})^{ψV} is continuous. The adjoint of the attaching map h_{n} : Z_{n} × S^{n−1} → X_{n−1} is a continuous function

ĥ_{n}: S^{n−1} → X_{n−1}^{Z_{n}} = ∏_{V ∈Z_{n}} X_{n−1}^{V}

where the right hand side is the product space. By inductive assumption ψ_{n−1}^{Z_{n}}: X_{n−1}^{Z_{n}} → (ψX_{n−1})^{ψZ_{n}} is continuous because it is continuous on the factors of X_{n−1}^{Z_{n}} = ∏_{V ∈Z_{n}} X_{n−1}^{V}. Therefore the composition

\xymatrix{S^{n−1} \ar[r]^{ĥ_{n}} & X_{n−1}^{Z_{n}} \ar[r]^{ψ^{Zn}} & (ψX_{n−1})^{ψZ_{n}}}

is continuous and yields a W-map ψ(h_{n}): ψZ_{n} × S^{n−1} → ψX_{n−1}. We can therefore endow ψX_{n} with the topology of the push-out

[figure: ψZ_{n} × S^{n−1}; ψ(h_{n}); ψX_{n−1}; ψZ_{n} × D^{n}; ψX_{n}]

Now we need to show that ψ_{n}^{V} : X_{n}^{V} → (ψX_{n})^{ψV} is continuous for every V ∈ V. Consider the diagram (5.1).

\xymatrix{(ψZ_{n} × S^{n−1})^{V} \ar[ddd] \ar[rrr]^{ψ(h_{n})^{ψV}} & & & ψX_{n−1}^{ψV} \ar[ddd] \\ & (Z_{n} × S^{n−1})^{V} \ar@{-->}[ul]^{ψ̂} \ar[d] \ar[r]^{h_{n}^{V}} & X_{n−1}^{V} \ar[ur]^{ψ̂_{n−1}^{V}} \ar[d] \\ & (Z_{n} × D^{n})^{V} \ar[r] \ar[dl] & X_{n}^{V} \ar@{.>}[dr]^{ψ̂_{n}^{V}} \\ (ψZ_{n} × D^{n})^{V} \ar[rrr] & & & ψX_{n}^{ψV}} (5.1)

Because of theorem (4.1) the two square diagrams are push-out diagrams and therefore the induced map ψ̂_{n}^{V} is continuous.

This inductive process yields a topology for every ψX_{n}. Let ψX = lim_{n≥0} ψX_{n}. Because ψ̂_{n}^{V} : X_{n}^{V} → (ψX_{n})^{ψV} ⊂ (ψX)^{ψX} is continuous for every V, the induced map ψ̂: X^{V} → (ψX)^{ψV} is continuous for every V ∈ V; here we use again theorem (4.1). Let V-CW be the full subcategory of V-Top consisting of V-complexes.

(5.2) Lemma. Let ψ be a continuous functor V → W where V and W are closed subcategories of Vect. The operation that sends each V-complex X to the W-complex ψX defined above yields a well-defined functor ψ: V-CW → W-CW.

This, in fact, is an example of the functor φ_{#} in [3] which carries a stratified bundle to an associated stratified bundle.

Proof. We only need to define the image ψ(f): ψX → ψY under ψ of a V-map f: X → Y. We define ψ(f) inductively on the skeleta of ψX. Let f_{n}: X_{n} → Y denote the restriction of f to the n-skeleton of X. On the zero-skeleton ψX_{0}, ψ(f) = ψ(f_{0}) is defined. Assume that ψ(f_{n−1}) is defined. Let Φ_{n} : Z_{n} × D^{n} → X_{n} be the n-characteristic map of X and consider the adjoint of the composition

(f_{n}Φ_{n}): Z_{n} × D^{n} → X → Y

{ \atop (f_{n}Φ_{n})}: D^{n} → Y^{Z_{n}}.

By construction the map ψ̂: Y^{Z_{n}} → (ψY)^{ψZ_{n}} is continuous. Therefore the composition ψ̂{ \atop (f_{n}Φ_{n})} is the adjoint of a continuous W-map ψZ_{n}×D^{n} → ψY. Furthermore, it is possible to check that this map fits into the outside square of the following diagram making it commutative.

\xymatrix{ψZ_{n} × S^{n−1} \ar[r] \ar[d] & ψX_{n−1} \ar[d] \ar[ddr]^{ψ(f_{n−1})} \\ ψZ_{n} × D^{n} \ar[r] \ar[drr] & ψX_{n} \ar@{.>}[dr]_{ψ(f_{n})} \\  &  & ψY}

Hence the W-map ψ(f_{n}) exists for every n ≥ 0. By taking the limit, we obtain the desired W-map ψ(f): ψX → ψY. q.e.d.

Now consider a continuous bifunctor ψ: V × V′ → W, a V-complex X over X̄ = B and a V′-complex X′ over the same base space X̄′ = B; assume that the cellular decompositions on B induced by X and X′ are the same and that V, V′ and W are closed. We are now going to define a W-complex ψ(X, X′) with base space B. If X = X_{0} and X′ = X′_{0} are a V-set and a V′- set respectively over the same base set B, then ψ(X, X′) is defined pointwise as ψ(V, V′), whenever V and V′ are fibers over the same point in B. In general, let Φ and Φ′ be the characteristic maps of X and X′. Using the decomposition into open cells of X and X′ it is possible to define ψ(X, X′) as a set as follows:

X = ∐_{n≥0} Z_{n} × e^{n}, X′ = ∐_{n≥0} Z′_{n} × e^{n},

ψ(X, X′) = ∐_{n≥0} ψ(Z_{n}, Z′_{n}) × e^{n}. (5.3)

We need to endow ψ(X, X′) with a suitable topology.

Consider a space Y and a space Y′ which are sources of maps to the space B. We denote by Y ×_{B} Y′ the pull-back

\xymatrix{Y ×_{B} Y′ \ar[r] \ar[d] & Y′ \ar[d] \\ Y \ar[r] & B.}

For every V ∈ V and V′ ∈ V′, consider the pull-back X^{V} ×_{B} X′^{V′} with respect to the maps X^{V} → X̄ and X′^{V′} → X̄. The pre-image of a point b ∈ X̄ = X̄′ under the natural projection X^{V} ×_{B} X′^{V′} → X̄ is equal to hom(V, X_{b}) × hom(V′, X′_{b}). Therefore we obtain a function

ψ̂: X^{V} ×_{B} X′^{V′} → ψ(X, X′)^{ψ(V,V′)}

which is continuous on fibers (by continuity of ψ) defined by

ψ̂: hom(V, X_{b}) × hom(V′, X′_{b}) → hom(ψ(V, V′), ψ(X_{b}, X′_{b})).

We want to give a topology to ψ(X, X′) so that ψ̂ is continuous. We can do it inductively on the skeleta of X and X′. If n = 0, then ψ(X_{0}, X′_{0}) is defined as above, and it is trivial to see that ψ̂_{0} : X_{0}^{V} ⊓ X′_{0}^{V′} → ψ(X_{0}, X′_{0})^{ψ(V,V′)} is continuous. More directly, we can obtain ψ(X, X′) by

ψ(X, X′) = ψ_{#}(X ×_{B} X′)

where the right hand side is defined by the associated bundle in [3].

(5.4) Remark. If X and X′ are V-vector bundles over a CW-complex X̄, then by the bundle theorem (5.1) of [3] X and X′ have a structure of Vcomplexes. In this case it is not difficult to show that the bundles ψ(X), ψ(X, X′) coincide with the corresponding constructions on vector bundles (see e.g. Atiyah [1], page 6).

6 Algebraic constructions on stratified vector bundles

Consider the closure V̄ in Vect of a subcategory V ⊂ Vect. Since composition maps are continuous, the closure V̄ is a closed subcategory of Vect. We say that a continuous functor ψ: V → W is regular if there is a continuous functor ψ̄: V̄ → W̄ such that ψ is the restriction of ψ̄. A similar definition holds for continuous bifunctors. As a slight generalization of lemma (5.2) we get.

(6.1) Lemma. Let ψ be a regular continuous functor V → W where V and W are subcategories of Vect. Then the operation that sends each Vcomplex X to the W-complex ψX defined above yields a well-defined functor ψ: V-CW → W-CW.

Proof. The embeddings of categories V ⊂ V̄ and W ⊂ W̄ induce embeddings i_{V} : V-CW ⊂ V̄-CW and i_{W} : W-CW ⊂ W̄-CW By lemma (5.2), the continuous functor ψ̄ yields a functor ψ̄: V̄-CW → W̄-CW. Since the image of the composition of functors ψ̄i_{V} is actually in W-CW, we obtain a functor ψ: V-CW → W-CW as claimed. q.e.d.

Let V-CW ⊓ V′-CW denote the category with as objects pairs (X, X′) where X is a V-complex and X′ a V′-complex, such that X̄ and X̄′ coincide. A morphism in V-CW ⊓ V′-CW from (X, X′) to (Y, Y′) is a pair of maps (f, f′), where f: X → Y is a V-map and f′: X′ → Y′ is a V′-map, such that the induced maps on the base spaces coincide: f̄ = f̄′: X̄ → Ȳ. Given an object (X, X′) in V-CW⊓V′-CW the W-complex ψ(X, X′) has been defined above. Consider a morphism (f, f′): (X, X′) → (Y, Y′). This yields a map f ⊓ f′: X ⊓ X′ → Y ⊓ Y′.

More generally than (5.3) we get:

(6.2) Lemma. Let V, V′ and W be subcategories of Vect. For every regular continuous bifunctor ψ ψ: V × V′ → W the operation that sends (X, X′) to ψ(X, X′) and (f, f′): (X, X′) → (Y, Y′) to ψ(f, f′) is a well-defined functor

ψ: V-CW ⊓ V′-CW → W-CW.

Proof. The proof is identical as the proof of lemma (6.1). q.e.d.

(6.3) Theorem. Let ψ: V → W be a regular continuous functor, where V and W are subcategories of Vect. There is an induced functor ψ: V-Stra → W-Stra which sends a V-stratified vector bundle X to the space ψX over X̄ with fibers ψX_{b}, for b ∈ X̄.

Proof. By lemma (6.1) such functor is already defined for V-complexes. Using the same argument as in the proof of lemma (6.1), it suffices to consider the case of closed (and hence NKC) subcategories of Vect. Consider the subcategories V-Stra_{n} and W-Stra_{n} of V-stratified (resp. W-stratified) vector bundles of length at most n (i.e. with at most n strata). If X is an object in V-Stra_{1} then it is a vector bundle over a CW-complex, hence it is a V-complex by remark (5.4). Thus the objects of V-Stra_{1} and W-Stra_{1} are in V-CW and W-CW respectively, and the maps are of course V-maps and W-maps. This implies that

ψ: V-Stra_{1} → W-Stra_{1}

is well-defined. Assume by induction that ψ is defined on V-Stra_{n}, for some n ≥ 2. Let X be an object in V-Stra_{n+1}. Thus there are CW-pairs (M̄_{n}, Ā_{n}) a V-vector bundle M_{n} and a V-map h_{n}: A_{n} = M_{n}|Ā_{n} → X_{n−1} such that X = X_{n−1} ∪_{h_{n}} M_{n}. Since X_{n−1} and A_{n} are both objects of V-Stra_{n} we have that ψ(h_{n}): ψA_{n} → ψX_{n−1} is a well-defined W-map. Thus we can define ψX as the push-out ψX = ψX_{n−1} ∪_{ψ(h_{n})} M_{n}. The same argument can be applied to a stratified map f: X → X′, where X and X′ are objects in V-Stra_{n}. Hence ψ is well-defined on every V-Stra_{n}. Now, since every object in V-Stra belongs to V-Stra_{n} for some n, this implies that ψ is a well-defined functor from V-Stra to W-Stra. q.e.d.

Consider now a regular continuous bifunctor ψ: V × V′ → W. Let V-Stra⊓V′-Stra denote the category of pairs (X, X′), where X is an object in V-Stra and X′ an object in V′-Stra with the same base space X̄ = X̄′. A morphism in V-Stra ⊓ V′-Stra is a pair of maps (f, f′) such that f is a morphism in V-Stra and f′ a morphism in V′-Stra which induce the same map on the base space f̄ = f̄′: X̄ → X̄.

(6.4) Theorem. Every regular continuous bifunctor ψ: V × V′ → W induces a well-defined functor

ψ: V-Stra ⊓ V′-Stra → W-Stra,

which sends a pair of V-stratified vector bundles X, X′ over X̄ to the space ψ(X, X′) over X̄ with fibers ψ(p_{X}^{−1}b, p_{X′}^{−1}b), for b ∈ X̄.

Proof. First, assume that the structure categories are closed in Vect. As in the proof of theorem (6.3), let V-Stra_{n} ⊓ V′-Stra_{n} denote the subcategory of pairs of V-stratified vector bundles of length at most n. The same for W-Stra_{n}. We are going to show that ψ is defined on V-Stra_{n} ⊓ V′-Stra_{n} for every n ≥ 1. If n = 1 this is true as a consequence of lemma (6.2). So, assume that ψ is well-defined on V-Stra_{n−1} ⊓ V′-Stra_{n−1}. Let (M_{n}, A_{n}) be the attached pair for X, with attaching map h: A_{n} → X_{n−1}. Let (M′_{n}, A′_{n}) and h′_{n} the corresponding for X′. Since X̄ = X̄′, without loss of generality we can assume that M̄_{n} = M̄′_{n}, Ā_{n} = Ā′_{n} and that the induced maps h̄_{n} = h̄′_{n} coincide. Thus the induced map

ψ(h_{n}, h′_{n}): ψ(A_{n}, A′_{n}) → ψ(X_{n−1}, X′_{n−1})

is well-defined, since (X_{n−1}, X′_{n−1}) and (A_{n}, A′_{m}) belong to V-Stra_{n−1} ⊓ V′-Stra_{n−1}. The same for the inclusions (i, i′): (A_{n}, A′_{n}) → (M_{n}, M′_{n}). Thus it is possible to define ψ(X, X′) as the push-out

ψ(X, X′) = ψ(X_{n−1}, X′_{n−1}) ∪_{ψ(h_{n},h′_{n})} ψ(M_{n}, M′_{n}).

In the same way, given a pair of stratified maps (f, f′): (X, X′) → (Y, Y′) in V-Stra_{n} ⊓ V′-Stra_{n}, it is possible to define a W-map

ψ(f, f′): ψ(X, X′) → ψ(Y, Y′).

Now, if the categories are not close, it suffices to follow the same argument as in the proof of lemma (6.1). q.e.d.

7 K-theory of stratified spaces

Assume that the structure category V is closed under direct sum ⊕ of vector spaces. Then, if X̄ is any space, the set V(X̄) of isomorphism classes of V-vector bundles over X̄ is an abelian semigroup where the sum is the sum induced as in lemma (6.2) by the bifunctor ⊕. Let K_{0}^{V}(X̄) = K_{0}(V(X̄)) be the Grothendieck group of the abelian semigroup (V(X̄),⊕). If V is also closed under tensor product ⊗ then K_{0}^{V}(X̄) is a ring. This is the Atiyah-Hirzebruch K-theory of a space X̄.

We can do the same for the category V-Stra(X̄) of V-stratified bundles over the stratified base space X̄. Let X̄ be a stratified space in Stra. Then by theorem (6.4) the category V-Stra(X̄) of V-stratified vector bundles over X̄ and stratified V-maps over 1X̄ is an additive category with sum ⊕.

(7.1) Definition. Let K_{0}^{V}(X̄) denote the Grothendieck group K_{0}(V-Stra(X̄)).

If V is closed under the tensor product ⊗ then K_{0}^{V}(X̄) is a ring. Now consider the full subcategory Stra_{0} ⊂ Stra of finite stratified spaces with attached spaces which are locally finite and countable CW-complexes.

(7.2) Theorem. For every subcategory V of Vect which is closed under direct sum ⊕ there is a functor

K_{0}^{V} : Stra_{0}^{op} → Ab,

where Ab denotes the category of Abelian groups. If V is closed under the tensor product ⊗ then the functor has its image in the category of rings. Moreover there are natural homomorphisms

\xymatrix{K_{0}^{V}(X̄) \ar[r] & K_{0}^{V}(X̄) \ar[d] \ar[r] & K_{0}^{V}(M̄_{i}) \\  & K_{0}^{V}(X̄_{0})}

where M̄_{i}, X̄_{0} are the attached spaces of X̄ ∈ Stra_{0}.

Proof. Let X̄, X̄′ be two stratified spaces in Stra_{0} and X′ → X̄′ an object of V-Stra(X̄′). By theorem (2.3), given a stratum-preserving map f̄: X̄ → X̄′ in Stra_{0}, the pull-back X = f̄^{∗}X′ is a well-defined object of V-Stra(X̄). Moreover, a morphism α in V-Stra(X̄′) (that is, a self-map α: X′ → X′ over the identity of X̄′) induces a morphism α^{∗}: X → X over the identity of X̄. Hence f̄^{∗}: V-Stra(X̄′) → V-Stra(X̄) is a functor. To show that this construction yields a homomorphism K_{0}^{V}(f̄): K_{0}^{V}(X̄′) → K_{0}^{V}(X̄) is sufficient to show that f̄^{∗}, as a functor, preserves the sum ⊕, i.e. that Y and Y′ are in V-Stra(X̄′) then

f̄^{∗}(Y ⊕ Y′) = f̄^{∗}Y ⊕ f̄^{∗}Y′. (7.3)

Let Y_{0} ⊂ Y_{1} ⊂ … and Y′_{0} ⊂ Y′_{1} ⊂ … be the filtrations of Y and Y′. We prove by induction that equation (7.3) holds for every stratified space X̄′ with at most n strata. If n = 1 then it is true, since in this case the V-stratified bundles are V-vector bundles. So assume that (7.3) is true for spaces with at most n − 1 strata. Let (M̄′_{n}, Ā′_{n}) be the pair attached via the map h̄′ to X̄′_{n−1} to obtain X̄′, where X̄′ has n exactly strata; in the same way, let (M̄_{n}, Ā_{n}) the corresponding CW-pair for X̄ and h̄: Ā_{n} → X̄_{n−1} the attaching map. Let (M′_{n}, A′_{n}), (M′′_{n}, A′′_{n}) be the attached bundles to Y_{n−1} and Y′_{n−1}. Since f̄ is a map in Stra_{0}, there is a map ḡ: M̄ → M̄′ such that f̄ = f̄_{n−1} ∪ ḡ. By proposition (7.4) of [3] we have the equality

f̄^{∗}(Y ⊕ Y′) = f̄^{∗}(Y_{n−1} ⊕ Y_{n−1}) ∪_{h} ḡ^{∗}(M_{n} ⊕ M′_{n}), (7.4) where h: ḡ^{∗}(A_{n} ⊕ A′_{n}) → f̄^{∗}(Y_{n−1} ⊕ Y′_{n−1}) is induced by the pull-back. Now, by induction

f̄^{∗}(Y_{n−1} ⊕ Y_{n−1}) = f̄^{∗}Y_{n−1} ⊕ f̄^{∗}Y_{n−1}
ḡ^{∗}(M_{n} ⊕ M′_{n}) = ḡ^{∗}M_{n} ⊕ ḡ^{∗}M′_{n},
ḡ^{∗}(A_{n} ⊕ A′_{n}) = ḡ^{∗}A_{n} ⊕ ḡ^{∗}A′_{n}.

But by definition the sum of the pull-backs f̄^{∗}Y ⊕ f̄^{∗}Y′ is equal to the pushout

\begin{aligned}f̄^{∗}Y ⊕ f̄^{∗}Y′ & = f̄^{∗}Y_{n−1} ⊕ f̄^{∗}Y_{n−1} ∪_{h} ḡ^{∗}M_{n} ⊕ ḡ^{∗}M′_{n} \\ & = f̄^{∗}(Y_{n−1} ⊕ Y_{n−1}) ∪_{h} ḡ^{∗}(M_{n} ⊕ M′_{n}). \end{aligned} (7.5)

Comparing equations (7.4) and (7.5) yields the desired result. q.e.d.

(7.6) Theorem. Consider a V-stratified vector bundle X′, a compactly stratified space X̄ and a homotopy of (stratum-preserving) maps f̄_{t}: X̄ → X̄′, t ∈ I. Then

f̄_{0}^{∗}X′ ≅ f̄_{1}^{∗}X′.

Proof. We prove it by induction on the number of strata of X̄′. If n = 1 then X′ is a V-vector bundle and hence the result is classical. Assume that it is true for X̄′ with at most n − 1 strata. Then if X̄ has less than n strata the claim is true. So let (M̄_{n}, Ā_{n}) be the CW-pair attached to X̄_{n−1}, (M′_{n}, A′_{n}) the CW-pair attached to X̄′_{n−1} and ḡ_{t}: (M̄_{n}, Ā_{n}) → (M̄′_{n}, Ā′_{n}) the homotopy such that f̄_{t} = f̄_{n−1,t} ∪ ḡ_{t}, with t ∈ I. Since X̄_{0} and M̄_{i} are compact for every i ≥ 1, we know that X̄_{i} is compact for every i. This implies that, by induction, f̄_{0,n−1}^{∗}X′_{n−1} ≅ f_{1,n−1}^{∗}X′_{n−1}, ḡ_{0}^{∗}A′_{n} ≅ g_{1}^{∗}A′_{n} and ḡ_{0}^{∗}M′_{n} ≅ g_{1}^{∗}M′_{n}. But by proposition (7.4) of [3]

f̄_{0}^{∗}X′ = f̄_{0,n−1}^{∗}X′_{n−1} ∪ ḡ_{0}^{∗}M′_{n},
f̄_{1}^{∗}X′ = f̄_{1,n−1}^{∗}X′_{n−1} ∪ ḡ_{1}^{∗}M′_{n},

and as a consequence f_{0}^{∗}X′ ≅ f_{1}^{∗}X′. q.e.d.

(7.7) Corollary. If Stra_{c}^{op} denotes the full subcategory of Stra^{op} consisting of compactly stratified spaces and stratum-preserving maps, the functor

K_{0}^{V} : Stra_{c}^{op} → Ab

induces a functor on the homotopy category

K_{0}^{V} : Stra_{c}^{op}/_{∼} → Ab.

Hence if X̄ is homotopy equivalent (in Stra_{c}) to Ȳ, then

K_{0}^{V}(X̄) = K_{0}^{V}(Ȳ).

References

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