CONTROLLED TOPOLOGICAL EQUIVALENCE OF MAPS

IN THE THEORY OF

STRATIFIED SPACES AND APPROXIMATE FIBRATIONS

Bruce Hughes

Abstract. Ideas from the theory of topological stability of smooth maps are transported into the controlled topological category. For example, the controlled topological equivalence of maps is discussed. These notions are related to the classification of manifold approximate fibrations and manifold stratified approximate fibrations. In turn, these maps form a bundle theory which can be used to describe neighborhoods of strata in topologically stratified spaces.

1. Introduction

We explore some connections among the theories of topological stability of maps, controlled topology, and stratified spaces. The notions of topological equivalence of maps and locally trivial families of maps play an important role in the theory of topological stability of smooth maps. We formulate the analogues of these notions in the controlled topological category for two reasons. First, the notion of controlled topological equivalence of maps is a starting point for formulating a topological version of Mather’s theory of the topological stability of smooth maps. Recall that Mather proved that the topologically stable maps are generic for the space of all smooth maps (with the C^{∞} topology) between closed smooth manifolds (see Mather [22], Gibson, Wirthmüller, du Plessis, and Looijenga [9]). The hope is to identify an analogous generic class for the space of all maps (with the compact-open topology) between closed topological manifolds. Controlled topology at least gives a place to begin speculations. Second, the controlled analogue of local triviality for families of maps is directly related to the classification of approximate fibrations between manifolds due to Hughes, Taylor and Williams [17], [18]. We elucidate that relation in §8.

Another important topic in the theory of topological stability of smooth maps is that of smoothly stratified spaces (cf. Mather [21]). Quinn [26] initiated the study of topologically stratified spaces and Hughes [12], [13] has shown that ‘manifold stratified approximate fibrations’ form the correct bundle theory for those spaces. The classification of manifold approximate fibrations via controlled topology mentioned above extends to manifold stratified approximate fibrations; hence, we have another connection between controlled topology and stratified spaces. This classification of manifold stratified approximate fibrations is the main new result of this paper.

1991 Mathematics Subject Classification. Primary 57N80, 57N40, 58A35; Secondary 57R45, 55R65, 57N15.

Key words and phrases. stratified space, approximate fibration, controlled topology, topological stability of maps, strata, neighborhood germ, Thom’s isotopy lemmas.

Supported in part by NSF Grant DMS–9504759 and by the Vanderbilt University Research Council.

Presented at the Symposium on Surgery and Geometric Topology, Josai University, Sakado, Japan, September 17–21, 1996.

Two essential tools in stability theory are Thom’s two isotopy lemmas [21]. In §9 we formulate an analogue of the first of these lemmas for topologically stratified spaces. A non-proper version is also stated.

It should be noted that in his address to the International Congress in 1986, Quinn predicted that controlled topology would have applications to the topological stability of smooth maps [25]. In particular, controlled topology should be applicable to the problem of characterizing the topologically stable maps among all smooth maps. More recently, Cappell and Shaneson [1] suggested that topologically stratified spaces should play a role in the study of the local and global topological type of topologically smooth maps (the connection is via the mapping cylinder of the smooth map). While the speculations in this paper are related to these suggestions, they differ in that it is suggested here that controlled topology might be used to study a generic class of topological, rather than smooth, maps.

2. Topological equivalence and locally trivial families of maps

We recall some definitions from the theory of topological stability of smooth maps (see Damon [3], du Plessis and Wall [5], Gibson, Wirthmüller, du Plessis, and Looijenga [9], Mather [21], [22]).

Definition 2.1. Two maps p_{0} : X_{0} → Y_{0}, p_{1} : X_{1} → Y_{1} are topologically equivalent if there exist homeomorphisms h : X_{0} → X_{1} and g : Y_{0} → Y_{1} such that p_{1}h = gp_{0}, so that there is a commuting diagram:

X_{0} ⟶^{h} X_{1}
p_{0}↓  ↓^{p_{1}}
Y_{0} ⟶^{g} Y_{1}.

Definition 2.2. A smooth map p_{0} : M → N between smooth manifolds is topologically stable if there exists a neighborhood V of p_{0} in the space of all smooth maps C^{∞}(M, N) such that for all p_{1} ∈ V, p_{0} is topologically equivalent to p_{1}.

The space C^{∞}(M, N) is given the Whitney C^{∞} topology. Thom conjectured and Mather proved that the topologically stable maps are generic in C^{∞}(M, N); in fact, they form an open dense subset (see [9], [21], [22]). The proof yields a stronger result, namely that the strongly topologically stable maps are dense (see [9]).

Definition 2.3. A smooth map p_{0} : M → N between smooth manifolds is strongly topologically stable if there exists a neighborhood V of p_{0} in C^{∞}(M, N) such that for all p_{1} ∈ V, there exists a (topologically) trivial smooth one-parameter family p : M × I → N joining p_{0} to p_{1}. This means there exist continuous families {h_{t} : M → M | 0 ≤ t ≤ 1} and {g_{t} : N → N | 0 ≤ t ≤ 1} of homeomorphisms such that p_{0} = g_{t}^{−1} ∘ p_{t} ∘ h_{t} for all t ∈ I, so that there is a commuting diagram:

M ⟶^{h_{t}} M
p_{0}↓  ^{p_{t}}↓
N ⟶^{g_{t}} N.

The notion of triviality for the one-parameter family of maps in the definition above can be generalized to arbitrary families of maps. We now recall that definition and the related notion of local triviality (cf. [21]).

Definition 2.4. Consider a commuting diagram of spaces and maps:

E_{1} ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B ⟶^{id_{B}} B

(1) f is trivial over B if there exist spaces F_{1} and F_{2}, a map q : F_{1} → F_{2} and homeomorphisms h : E_{1} → F_{1} ×B, g : E_{2} → F_{2} ×B such that the following diagram commutes:

B ⟵^{p_{1}} E_{1}  ⟶^{f} E_{2}  ⟶^{p_{2}} B
id_{B} ↓  ^{h}↓  ^{g}↓  ↓^{id}_{B}
B ⟵^{proj} F_{1} × B ⟶^{q×id_{B}} F_{2} × B ⟶^{proj} B

(2) f is locally trivial over B if for every x ∈ B there exists an open neighborhood U of x in B such that f| : p_{1}^{−1}(U) → p_{2}^{−1}(U) is trivial over U.

(3) In either case, q : F_{1} → F_{2} is the model of the family f.

Remarks 2.5.

(1) The model q : F_{1} → F_{2} is well-defined up to topological equivalence.

(2) Both p_{1} : E_{1} → B and p_{2} : E_{2} → B are fibre bundle projections with fibre F_{1} and F_{2}, respectively.

(3) For every x ∈ B, f_{x} = f| : p_{1}^{−1}(x) → p_{2}^{−1}(x) is topologically equivalent to q : F_{1} → F_{2}.

(4) One step in Mather’s proof that the topologically stable smooth maps form an open dense subset is to show that certain families of maps are locally trivial. Thom’s second isotopy lemma is used for this.

A fibre preserving map is a map which preserves the fibres of maps to a given space, usually a k-simplex or an arbitrary space B. Specifically, if ρ : X → B and σ : Y → B are maps, then a map f : X → Y is fibre preserving over B if σf = ρ.

There is a notion of equivalence for families of maps over B.

Definition 2.6.

(1) Two locally trivial families of maps over B

\begin{matrix}E_{1} & ⟶^{f} & E_{2} \\ ^{p_{1}}↓ & & ↓^{p_{2}} \\ B & ⟶^{id_{B}} & B \end{matrix} and \begin{matrix}E′_{1} & ⟶^{f′} & E′_{2} \\ ^{p′_{1}}↓ & & ↓^{p′_{2}} \\ B & ⟶^{id_{B}} & B \end{matrix}

are topologically equivalent provided there exist homeomorphisms

h_{1} : E_{1} → E′_{1} and h_{2} : E_{2} → E′_{2}

which are fibre preserving over B and f′h_{1} = h_{2}f; that is, the following diagram commutes:

B ⟵^{p_{1}} E_{1} ⟶^{f} E_{2} ⟶^{p_{2}} B
id_{B} ↓  ^{h_{1}}↓  ^{h_{2}}↓  ↓^{id}_{B}
B ⟵^{p′_{1}} E′_{1} ⟶^{f′} E′_{2} ⟶^{p′_{2}} B

(2) Let A_{1}(q, B) denote the set of topological equivalence classes of locally trivial families of maps over B with model q : F_{1} → F_{2}.

The set A_{1}(q, B) can be interpreted as a set of equivalence classes of certain fibre bundles over B as follows. Let TOP(q) be the topological group given by the pull-back diagram

TOP(q) ⟶ TOP(F_{2})
↓  ↓_{q^{♯}}
TOP(F_{1}) ⟶^{q_{♯}} Map(F_{1}, F_{2})

where q_{♯}(h) = q ∘ h and q^{♯}(g) = g ∘ q. That is,

TOP(q) = {(h, g) ∈ TOP(F_{1}) × TOP(F_{2}) | qh = gq}.

Note that TOP(q) is naturally a subgroup of TOP(F_{1} ⨿ F_{2}) via (h, g) ↦ h ⨿ g. Let A_{2}(q, B) denote the set of bundle equivalence classes of fibre bundles over B with fibre F_{1} ⨿ F_{2} and structure group TOP(q).

Proposition 2.7. There is a bijection α : A_{1}(q, B) → A_{2}(q, B). In particular, if B is a separable metric space, then there is a bijection A_{1}(q, B) → [B, BTOP(q)].

The function α is defined by sending a locally trivial family

E_{1} ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B ⟶^{id_{B}} B

to the fibre bundle p_{1} ⨿ p_{2} : E_{1} ⨿ E_{2} → B whose total space is the disjoint union of E_{1} and E_{2}. The fact that α is a bijection is fairly straightforward to prove. At any rate, it follows from a more general result in §5 (see Theorem 5.5 and the comments following it).

3. Controlled topological equivalence

We propose a definition of topological equivalence in the setting of controlled topology and use it to make some speculations about generic maps between topological manifolds.

The mapping cylinder of a map p : X → Y is the space

cyl(p) = (X × I ⨿ Y)/{(x, 1) ∼ p(x) | x ∈ X}.

There is a natural map π : cyl(p) → I defined by

\{ \begin{matrix}π([x, t]) = t, & if (x, t) ∈ X × I \\ π([y]) = 1, & if y ∈ Y . \end{matrix}

For clarification the map π will sometimes be denoted π_{p} : cyl(p) → I. If p : X → Y and p′ : X′ → Y′ are maps and π_{p} : cyl(p) → I and π_{p′} : cyl(p′) → I are the natural maps, then a homeomorphism h : cyl(p) → cyl(p′) is level if π_{p} = π_{p′}h. Let TOP^{level}(p) denote the simplicial group of level homeomorphisms from cyl(p) onto itself. That is, a k–simplex of TOP^{level}(p) consists of a Δ^{k}–parameter family of level homeomorphisms h : cyl(p) × Δ^{k} → cyl(p) × Δ^{k}. The group TOP(p) as defined in the previous section has a simplicial version (the singular complex of the topological group) and, as such, is a simplicial subgroup of TOP^{level}(p). For example, a pair of homeomorphisms (h : X → X, g : Y → Y) such that ph = gp induces a level homeomorphism

cyl(p) → cyl(p), \{ \begin{matrix}[x, t] ↦ [h(x), t], & if x ∈ X \\ [y] ↦ [g(y)], & if y ∈ Y \end{matrix}

Definition 3.1. Two maps p_{0} : X_{0} → Y_{0}, p_{1} : X_{1} → Y_{1} are controlled topologically equivalent if there exists a level homeomorphism h : cyl(p_{0}) → cyl(p_{1}).

Note that a level homeomorphism h : cyl(p_{0}) → cyl(p_{1}) induces (by restriction) a one-parameter family h_{t} : X_{0} → X_{1}, 0 ≤ t < 1, of homeomorphisms and a homeomorphism g : Y_{0} → Y_{1}. If all the spaces involved are compact metric, then

gp_{0} = lim_{t→1} p_{1}h_{t}

and such data is equivalent to having a level homeomorphism (cf. [16], [17], [19], [20]). This formulation should be compared with the formulation of topological equivalence in Definition 2.1.

Definition 3.2. Two maps p_{0} : X_{0} → Y_{0}, p_{1} : X_{1} → Y_{1} between compact metric spaces are weakly controlled topologically equivalent if there exist continuous families {h_{t} : X_{0} → X_{1} | 0 ≤ t < 1} and {g_{t} : Y_{0} → Y_{1} | 0 ≤ t < 1} of homeomorphisms such that p_{0} = lim_{t→1} g_{t}^{−1} ∘ p_{1} ∘ h_{t}.

The limit above is taken in the sup metric which is the metric for the compactopen topology. The space C(X, Y) of maps from X to Y is given the compact-open topology.

Definition 3.3. A map p_{0} : X → Y between compact metric spaces is weakly controlled topologically stable if there exists a neighborhood V of p_{0} in C(X, Y) such that for all p_{1} ∈ V and ϵ > 0, there exists a map p′_{1} : X → Y such that p_{0} is weakly controlled topologically equivalent to p′_{1} and p′_{1} is ϵ–close to p_{1}.

Many of the results in the theory of singularities have a mixture of smooth and topological hypotheses and conclusions. This is the case in Mather’s theorem on the genericness of topologically stable maps among smooth maps. One direction that controlled topology is likely to take is in finding the topological underpinnings in this area. The following speculation is meant to be a step towards formulating what might be true.

Speculation 3.4. If M and N are closed topological manifolds, then the weakly controlled topologically stable maps from M to N are generic in C(M, N).

This might be established by showing that the stratified systems of approximate fibrations are dense and also weakly controlled topologically stable (see Hughes [14] and Quinn [27] for stratified systems of approximate fibrations). As evidence for this line of reasoning, note that Chapman’s work [2] shows that manifold approximate fibrations are weakly controlled topologically stable.

Another line of speculation concerns polynomial maps between euclidean spaces. It is known that the classification of polynomial maps up to smooth equivalence differs from their classification up to topological equivalence (cf. Thom [32], Fakuda [8], Nakai [23]). What can be said about the classification of polynomial maps up to controlled topological equivalence?

4. Controlled locally trivial families of maps

Analogues in controlled topology of locally trivial families of maps are defined. In fact, we define a moduli space of all such families.

Definition 4.1. Consider a commuting diagram of spaces and maps:

E_{1} ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B ⟶^{id_{B}} B

(1) f is controlled trivial over B if there exist spaces F_{1} and F_{2}, a map q : F_{1} → F_{2} and a homeomorphism H : cyl(f) → cyl(q) × B such that the following diagram commutes:

B ⟵^{c} cyl(f) ⟶^{π_{f}} I
id_{B} ↓  ^{H}↓  ↓^{id}_{I}
B ⟵^{proj} cyl(q) × B ⟶^{π′_{q}} I

where c : cyl(f) → B is given by

\{ \begin{matrix}c([x, t]) = p_{1}(x) = p_{2}f(x), & if (x, t) ∈ E_{1} × I \\ c([y]) = p_{2}(y), & if y ∈ E_{2} \end{matrix}

and π′_{q} is the composition cyl(q) × B ⟶^{proj} cyl(q) ⟶^{π_{q}} I.

(2) f is controlled locally trivial over B if for every x ∈ B there exists an open neighborhood U of x in B such that f| : p_{1}^{−1}(U) → p_{2}^{−1}(U) is controlled trivial over U.

(3) In either case, q : F_{1} → F_{2} is the model of the family f.

Remarks 4.2.

(1) The model q : F_{1} → F_{2} is well-defined up to controlled topological equivalence.

(2) Both p_{1} : E_{1} → B and p_{2} : E_{2} → B are fibre bundle projections with fibre F_{1} and F_{2}, respectively.

(3) For every x ∈ B, f_{x} = f| : p_{1}^{−1}(x) → p_{2}^{−1}(x) is controlled topologically equivalent to q : F_{1} → F_{2}.

There is a notion of controlled equivalence for families of maps over B.

Definition 4.3. (1) Two controlled locally trivial families of maps over B

\begin{matrix}E_{1} & ⟶^{f} & E_{2} \\ ^{p_{1}}↓ & & ↓^{p_{2}} \\ B & ⟶^{id_{B}} & B \end{matrix} and \begin{matrix}E′_{1} & ⟶^{f′} & E′_{2} \\ ^{p′_{1}}↓ & & ↓^{p′_{2}} \\ B & ⟶^{id_{B}} & B \end{matrix}

are controlled topologically equivalent provided there exists a level homeomorphism

H : cyl(f) → cyl(f′)

which is fibre preserving over B in the sense that the following diagram commutes:

cyl(f) ⟶^{H} cyl(f′)
c↓  ↓_{c′}
B ⟶^{id_{B}} B

where c is given by

\{ \begin{matrix}c([x, t]) = p_{2}f(x) = p_{1}(x), & if (x, t) ∈ E_{1} × I \\ c([y]) = p_{2}(y), & if y ∈ E_{2} \end{matrix}

and c′ is given by

\{ \begin{matrix}c′([x, t]) = p′_{2}f′(x) = p′_{1}(x), & if (x, t) ∈ E′_{1} × I \\ c′([y]) = p′_{2}(y), & if y ∈ E′_{2}. \end{matrix}

(2) Let B_{1}(q, B) denote the set of controlled topological equivalence classes of locally trivial families of maps over B with model q : F_{1} → F_{2}.

In the next section we will show that the set B_{1}(q, B) can be interpreted as a set of equivalence classes of certain fibre bundles over B in analogy with Proposition 2.7 (see Theorem 5.5). But first we will define the moduli space of all controlled locally trivial families of maps over B with model q : F_{1} → F_{2}. This is done in the setting of simplicial sets as follows.

Define a simplicial set B_{1}(q, B) so that a typical k–simplex of B_{1}(q, B) consists of a commuting diagram

E_{1}  ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B × Δ^{k} ⟶^{id_{B×Δ^{k}}} B × Δ^{k}

which is a controlled locally trivial family of maps over B × Δ^{k} with model q : F_{1} → F_{2}. Thus, a vertex of B_{1}(q, B) is a controlled locally trivial family of maps over B with model q : F_{1} → F_{2}. (As in [17], [18] we also need to require that these spaces are reasonably embedded in an ambient universe, but we will ignore that technicality in this paper.) Face and degeneracy operations are induced from those on the standard simplexes. As in [18], this simplicial set satisfies the Kan condition.

Definition 4.4. The mapping cylinder construction μ takes a controlled locally trivial family of maps

E_{1} ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B ⟶^{id_{B}} B

to the mapping cylinder cyl(f) together with the natural map μ(f) : cyl(f) → B.

Note that the controlled locally trivial condition on f means that μ(f) : cyl(f) → B is a fibre bundle with fibre cyl(q) and structure group TOP^{level}(q) where q is the model of f. If

E′_{1} ⟶^{f′} E′_{2}
p′_{1}↓  ↓^{p′_{2}}
B ⟶^{id_{B}} B

is another controlled locally trivial family of maps over B with model q, then to have a controlled topological equivalence H : cyl(f) → cyl(f′) as in Definition 4.3 means precisely to have a bundle isomorphism from μ(f) to μ(f′).

Proposition 4.5. There is a bijection π_{0}B_{1}(q, B) ≈ B_{1}(q, B).

Proof. In order to see that the natural function π_{0}B_{1}(q, B) → B_{1}(q, B) is welldefined, suppose

E_{1}  ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B × Δ^{1} ⟶^{id_{B×Δ^{1}}} B × Δ^{1}

is a locally trivial family of maps with model q : F_{1} → F_{2}. Then by the remarks above μ(f) : cyl(f) → B ×Δ^{1} is a fibre bundle with fibre cyl(q) and structure group TOP^{level}(q). Thus, there is a bundle isomorphism from the restriction of μ(f) over B ×{0} to the restriction of μ(f) over B ×{1}, and the remarks above further show that this isomorphism gives a controlled topological equivalence from

\begin{matrix}p_{1}^{−1}(B × \{0\}) & ⟶^{f|} & p_{2}^{−1}(B × \{0\}) \\ ^{p_{1}|}↓ & & ↓^{p_{2}|} \\ B × \{0\} & ⟶^{id_{B}} & B × \{0\} \end{matrix} to \begin{matrix}p_{1}^{−1}(B × \{1\}) & ⟶^{f|} & p_{2}^{−1}(B × \{1\}) \\ ^{p_{1}|}↓ & & ↓^{p_{2}|} \\ B × \{1\} & ⟶^{id_{B}} & B × \{1\} \end{matrix}

showing that the function is well-defined. The function is obviously surjective, so it remains to see that it is injective. To this end suppose that

\begin{matrix}E_{1} & ⟶^{f} & E_{2} \\ ^{p_{1}}↓ & & ↓^{p_{2}} \\ B & ⟶^{id_{B}} & B \end{matrix} and \begin{matrix}E′_{1} & ⟶^{f′} & E′_{2} \\ ^{p′_{1}}↓ & & ↓^{p′_{2}} \\ B & ⟶^{id_{B}} & B \end{matrix}

are controlled topologically equivalent with a level homeomorphism H : cyl(f) → cyl(f′) as in Definition 4.3. Let h_{0} : E_{1} → E′_{1} and h_{1} : E_{2} → E′_{2} be the restrictions of H to the top and bottom of the mapping cylinders, respectively. Then there is an induced commutative diagram

cyl(h_{0}) ⟶ cyl(h_{1})
↓  ↓
B × Δ^{1} ⟶ B × Δ^{1}

which is a 1–simplex in B_{1}(q, B) from f to f′. □

5. Bundles with mapping cylinder fibres

In this section we show that controlled locally trivial families of maps over B can be interpreted as fibre bundles over B with fibre the mapping cylinder of the model. Reduced structure groups are discussed as well as a relative situation in which the target bundle over B is fixed.

Let B be a fixed separable metric space. Let B_{2}(q, B) denote the set of bundle equivalence classes of fibre bundles over B with fibre cyl(q) and structure group TOP^{level}(q). Define B_{2}(q, B) to be the simplicial set whose k–simplices are fibre bundles over B × Δ^{k} with fibre cyl(q) and structure group TOP^{level}(q). The following result is well-known (cf. [17]).

Proposition 5.1. There are bijections

π_{0}B_{2}(q, B) ≈ B_{2}(q, B) ≈ [B, BTOP^{level}(q)].

The mapping cylinder construction of Definition 4.4 has the following simplicial version.

Definition 5.2. The mapping cylinder construction is the simplicial map

μ : B_{1}(q, B) → B_{2}(q, B)

defined by sending a diagram

E_{1}  ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B × Δ^{k} ⟶^{id_{B×Δ^{k}}} B × Δ^{k}

to cyl(f) → B × Δ^{k}. Note that the local triviality condition on f implies that cyl(f) → B × Δ^{k} is a fibre bundle projection with fibre cyl(q) and structure group TOP^{level}(q).

The first part of the following result is proved in [18]. The second part follows from the first part together with Propositions 4.5 and 5.1.

Theorem 5.3. The mapping cylinder construction defines a homotopy equivalence μ : B_{1}(q, B) → B_{2}(q, B). In particular, B_{1}(q, B) ≈ B_{2}(q, B) ≈ [B, BTOP^{level}(q)].

Reduced structure groups. Let G be a simplicial subgroup of TOP^{level}(q). We will now generalize the discussion above to the situation where the structure group is reduced to G.

Definition 5.4. Consider a controlled locally trivial family

E_{1} ⟶^{f_{1}} E_{2}
p_{1}↓  ↓^{p_{2}}
B ⟶^{id_{B}} B

with model q : F_{1} → F_{2}. Then f is G–locally trivial over B provided there exists an open cover U of B such that f is controlled trivial over U for each U ∈ U via a trivializing homeomorphism

H_{U} : cyl(f| : p_{1}^{−1}(U) → p_{2}^{−1}(U)) → cyl(q) × B.

These trivializing homeomorphisms are required to have the property that if U, V ∈ U and x ∈ U ∩ V, then

H_{V} ∘ H_{U}^{−1}| : cyl(q) × {x} → cyl(q) × {x}

is an element of G.

Let B_{1}(q, B, G) be the simplicial set whose k–simplices are the G–locally trivial families of maps over B × Δ^{k} with model q : F_{1} → F_{2}. For example,

B_{1}(q, B, TOP^{level}(q)) = B_{1}(q, B).

Definition 4.3 can be extended in the obvious way to define what it means for two G–locally trivial families to be G–controlled topologically equivalent (the homeomorphism H is required to be a family of homeomorphisms in the group G) and B_{1}(q, B, G) denotes the set of equivalence classes. In analogy with Proposition 4.5 there is a bijection

π_{0}B_{1}(q, B, G) ≈ B_{1}(q, B, G).

Likewise B_{2}(q, B, G) denotes the set of bundle equivalence classes of fibre bundles over B with fibre cyl(q) and structure group G, and B_{2}(q, B, G) is the simplicial set whose k–simplices are fibre bundles over B × Δ^{k} with fibre cyl(q) and structure group G. In analogy with Proposition 5.1 there are bijections

π_{0}B_{2}(q, B, G) ≈ B_{2}(q, B, G) ≈ [B, BG].

Moreover, the proof of Theorem 5.3 can be seen to give a proof of the following result (cf. [18, §2]).

Theorem 5.5. The mapping cylinder construction defines a homotopy equivalence μ : B_{1}(q, B, G) → B_{2}(q, B, G). In particular, B_{1}(q, B, G) ≈ B_{2}(q, B, G) ≈ [B, BG].

As an example, consider the group TOP(q) of §2. It was pointed out at the beginning of §3 that TOP(q) is naturally a subgroup of TOP^{level}(q). Note that B_{1}(q, B, TOP(q)) = A_{1}(q, B) and B_{2}(q, B, TOP(q)) = A_{2}(q, B), so that Proposition 2.7 follows directly from Theorem 5.5.

Fixed target bundle. There are also relative versions of the preceding results in which the bundle p_{2} : E_{2} → B is fixed. For example, B_{1}(q rel p_{2} : E_{2} → B) is the set of controlled locally trivial families of maps of the form

E_{1} ⟶^{f} E_{2}
p_{1}↓  ↓^{p_{2}}
B ⟶^{id_{B}} B.

Two such families f : E_{1} → E_{2} and f′ : E′_{1} → E_{2} are controlled topologically equivalent rel p_{2} if the homeomorphism H : cyl(f) → cyl(f′) of Definition 4.3 is required to be the identity on E_{2}. There are analogous definitions of the following:

(1) B_{1}(q rel p_{2} : E_{2} → B),

(2) B_{2}(q rel p_{2} : E_{2} → B),

(3) B_{2}(q rel p_{2} : E_{2} → B).

Definition 5.6. The group of controlled homeomorphisms of q is the subgroup TOP^{c}(q) of TOP^{level}(q) consisting of all level homeomorphisms h : cyl(q) × Δ^{k} → cyl(q) × Δ^{k} such that h|F_{2} × Δ^{k} = id_{F_{2}×Δ^{k}}.

Note that TOP^{c}(q) is the kernel of the restriction homomorphism

TOP^{level}(q) → TOP(F_{2}).

Let p̂_{2} : B → BTOP(F_{2}) be the classifying map for the bundle p_{2}. Thus, B_{2}(q rel p_{2} : E_{2} → B) is in one-to-one correspondence with the set of vertical homotopy classes of lifts of p̂_{2} : B → BTOP(F_{2}) to BTOP^{level}(q) → BTOP(F_{2}):

BTOP^{level}(q)
↓
B \underrightarrow{p̂_{2}} BTOP(F_{2}).

The following result follows from the proofs of the preceding results.

Proposition 5.7.

(1) π_{0}B_{1}(q rel p_{2} : E_{2} → B) ≈ B_{1}(q rel p_{2} : E_{2} → B),

(2) π_{0}B_{2}(q rel p_{2} : E_{2} → B) ≈ B_{2}(q rel p_{2} : E_{2} → B),

(3) the mapping cylinder construction defines a homotopy equivalence

μ : B_{1}(q rel p_{2} : E_{2} → B) → B_{2}(q rel p_{2} : E_{2} → B).

Reduced structure group and fixed target bundle. There are versions of these relative results when the structure groups are reduced to G as before. The sets and simplicial sets involved are denoted as follows:

(1) B_{1}(q, G rel p_{2} : E_{2} → B),

(2) B_{1}(q, G rel p_{2} : E_{2} → B),

(3) B_{2}(q, G rel p_{2} : E_{2} → B),

(4) B_{2}(q, G rel p_{2} : E_{2} → B).

The following result records the analogous bijections and homotopy equivalences.

Proposition 5.8.

(1) π_{0}B_{1}(q, G rel p_{2} : E_{2} → B) ≈ B_{1}(q, G rel p_{2} : E_{2} → B),

(2) π_{0}B_{2}(q, G rel p_{2} : E_{2} → B) ≈ B_{2}(q, G rel p_{2} : E_{2} → B),

(3) the mapping cylinder construction defines a homotopy equivalence

μ : B_{1}(q, G rel p_{2} : E_{2} → B) → B_{2}(q, G rel p_{2} : E_{2} → B).

6. Manifold stratified spaces

There are many naturally occurring spaces which are not manifolds but which are composed of manifold pieces, those pieces being called the strata of the space. Examples include polyhedra, algebraic varieties, orbit spaces of many group actions on manifolds, and mapping cylinders of maps between manifolds. Quinn [26] has introduced a class of stratified spaces called by him ‘manifold homotopically stratified sets’ with the objective ‘to give a setting for the study of purely topological stratified phenomena’ as opposed to the smooth and piecewise linear phenomena previously studied.

Roughly, the stratified spaces of Quinn are spaces X together with a finite filtration by closed subsets

X = X^{m} ⊇ X^{m−1} ⊇ ⋯ ⊇ X^{0} ⊇ X^{−1} = ∅,

such that the strata X_{i} = X^{i} \ X^{i−1} are manifolds with neighborhoods in X_{i} ∪ X_{k} (for k > i) which have the local homotopy properties of mapping cylinders of fibrations. These spaces include the smoothly stratified spaces of Whitney [35], Thom [31] and Mather [21] (for historical remarks on smoothly stratified spaces see Goresky and MacPherson [10]) as well as the locally conelike stratified spaces of Siebenmann [29] and, hence, orbit spaces of finite groups acting locally linearly on manifolds.

Cappell and Shaneson [1] have shown that mapping cylinders of ‘smoothly stratified maps’ between smoothly stratified spaces are in this class of topologically stratified spaces even though it is known that such mapping cylinders need not be smoothly stratified (see [1] and [32]). Hence, the stratified spaces of Quinn arise naturally in the category of smoothly stratified spaces. For a comprehensive survey of the classification and applications of stratified spaces, see Weinberger [34].

Smoothly stratified spaces have the property that strata have neighborhoods which are mapping cylinders of fibre bundles, a fact which is often used in arguments involving induction on the number of strata. Such neighborhoods fail to exist in general for Siebenmann’s locally conelike stratified spaces. For example, it is known that a (topologically) locally flat submanifold of a topological manifold (which is an example of a locally conelike stratified space with two strata) may fail to have a tubular neighborhood (see Rourke and Sanderson [28]). However, Edwards [6] proved that such submanifolds do have neighborhoods which are mapping cylinders of manifold approximate fibrations (see also [18]). On the other hand, examples of Quinn [24] and Steinberger and West [30] show that strata in orbit spaces of finite groups acting locally linearly on manifolds may fail to have mapping cylinder neighborhoods. In Quinn’s general setting, mapping cylinder neighborhoods may fail to exist even locally.

The main result announced in [12] (and restated here in §8) gives an effective substitute for neighborhoods which are mapping cylinders of bundles. Instead of fibre bundles, we use ‘manifold stratified approximate fibrations,’ and instead of mapping cylinders, we use ‘teardrops’. This result should be thought of as a tubular neighborhood theorem for strata in manifold stratified spaces.

We now recall the concepts needed to precisely define the manifold stratified spaces of interest (see [26], [12], [15], [16]). A subset Y ⊆ X is forward tame in X if there exist a neighborhood U of Y in X and a homotopy h : U × I → X such that h_{0} = inclusion : U → X, h_{t}|Y = inclusion : Y → X for each t ∈ I, h_{1}(U) = Y, and h((U \ Y) × [0, 1)) ⊆ X \ Y.

Define the homotopy link of Y in X by

holink(X, Y) = {ω ∈ X^{I} | ω(t) ∈ Y iff t = 0}.

Evaluation at 0 defines a map q : holink(X, Y) → Y called holink evaluation.

Let X = X^{m} ⊇ X^{m−1} ⊇ ⋯ ⊇ X^{0} ⊇ X^{−1} = ∅ be a space with a finite filtration by closed subsets. Then X^{i} is the i-skeleton and the difference X_{i} = X^{i} \ X^{i−1} is called the i-stratum.

A subset A of a filtered space X is called a pure subset if A is closed and a union of components of strata of X. For example, the skeleta are pure subsets.

The stratified homotopy link of Y in X, denoted holink_{s}(X, Y) consists of all ω in holink(X, Y) such that ω((0, 1]) lies in a single stratum of X. The stratified homotopy link has a natural filtration with i–skeleton

holink_{s}(X, Y)^{i} = {ω ∈ holink_{s}(X, Y) | ω(1) ∈ X^{i}}.

The holink evaluation (at 0) restricts to a map q : holink_{s}(X, Y) → Y.

If X is a filtered space, then a map f : Z × A → X is stratum preserving along A if for each z ∈ Z, f({z} × A) lies in a single stratum of X. In particular, a map f : Z × I → X is a stratum preserving homotopy if f is stratum preserving along I.

Definition 6.1. A filtered space X is a manifold stratified space if the following four conditions are satisfied:

(1) Manifold strata. X is a locally compact, separable metric space and each stratum X_{i} is a topological manifold (without boundary).

(2) Forward tameness. For each k > i, the stratum X_{i} is forward tame in X_{i} ∪ X_{k}.

(3) Normal fibrations. For each k > i, the holink evaluation q : holink(X_{i} ∪ X_{k}, X_{i}) → X_{i} is a fibration.

(4) Finite domination. For each i there exists a closed subset K of the stratified homotopy link holink_{s}(X, X^{i}) such that the holink evaluation map K → X^{i} is proper, together with a stratum preserving homotopy

h : holink_{s}(X, X^{i}) × I → holink_{s}(X, X^{i})

which is also fibre preserving over X^{i} (i.e., qh_{t} = q for each t ∈ I) such that h_{0} = id and h_{1}(holink_{s}(X, X^{i})) ⊆ K.

7. Manifold stratified approximate fibrations

The definition of an approximate fibration (as given in [17]) was generalized in [12] to the stratified setting. Let X = X^{m} ⊇ ⋯ ⊇ X^{0} and Y = Y^{n} ⊇ ⋯ ⊇ Y^{0} be filtered spaces and let p : X → Y be a map (p is not assumed to be stratum preserving). Then p is said to be a stratified approximate fibration provided given any space Z and any commuting diagram

Z ⟶^{f} X
×0↓  ↓^{p}
Z × I ⟶^{F} Y

where F is a stratum preserving homotopy, there exists a stratified controlled solution; i.e., a map F̃ : Z × I × [0, 1) → X which is stratum preserving along I × [0, 1) such that F̃(z, 0, t) = f(z) for each (z, t) ∈ Z × [0, 1) and the function F̄ : Z ×I ×[0, 1] → Y defined by F̄|Z ×I ×[0, 1) = pF̃ and F̄|Z ×I ×{1} = F ×id_{{1}} is continuous.

A stratified approximate fibration between manifold stratified spaces is a manifold stratified approximate fibration if, in addition, it is a proper map (i.e., inverse images of compact sets are compact).

8. Teardrop neighborhoods

Given spaces X, Y and a map p : X → Y × ℝ, the teardrop of p (see [16]) is the space denoted by X ∪_{p} Y whose underlying set is the disjoint union X ⨿ Y with the minimal topology such that

(1) X ⊆ X ∪_{p} Y is an open embedding, and

(2) the function c : X ∪_{p} Y → Y × (−∞, +∞] defined by

c(x) = \{ \begin{matrix}p(x), & if x ∈ X \\ (x, +∞), & if x ∈ Y . \end{matrix}

is continuous. The map c is called the tubular map of the teardrop or the teardrop collapse. The tubular map terminology comes from the smoothly stratified case (see [4], [21], [33]). This is a generalization of the construction of the open mapping cylinder of

a map g : X → Y. Namely, cyl^{∘}(g) is the teardrop (X × ℝ) ∪_{g×id} Y.

Theorem 8.1. If X and Y are manifold stratified spaces and p : X → Y × ℝ is a manifold stratified approximate fibration, then X ∪_{p} Y is a manifold stratified space with Y a pure subset.

In this statement, Y × ℝ and X ∪_{p} Y are given the natural stratifications.

The next result from [12] is a kind of converse to this proposition. First, some more definitions. A subset Y of a space X has a teardrop neighborhood if there exist a neighborhood U of Y in X and a map p : U \ Y → Y × ℝ such that the natural function (U \ Y) ∪_{p} Y → U is a homeomorphism. In this case, U is the teardrop neighborhood and p is the restriction of the tubular map.

Theorem 8.2 (Teardrop Neighborhood Existence). Let X be a manifold stratified space such that all components of strata have dimension greater than 4, and let Y be a pure subset. Then Y has a teardrop neighborhood whose tubular map

c : U → Y × (−∞, +∞]

is a manifold stratified approximate fibration.

A complete proof of this result will be given in [13], but special cases are in [15] and [16].

The next result from [12] concerns the classification of neighborhoods of pure subsets of a manifold stratified space. Given a manifold stratified space Y, a stratified neighborhood of Y consists of a manifold stratified space containing Y as a pure subset. Two stratified neighborhoods X, X′ of Y are equivalent if there exist neighborhoods U, U′ of Y in X, X′, respectively, and a stratum preserving homeomorphism h : U → U′ such that h|Y = id. A neighborhood germ of Y is an equivalence class of stratified neighborhoods of Y.

Theorem 8.3 (Neighborhood Germ Classification). Let Y be a manifold stratified space such that all components of strata have dimension greater than 4. Then the teardrop construction induces a one-to-one correspondence from controlled, stratum preserving homeomorphism classes of manifold stratified approximate fibrations over Y × ℝ to neighborhood germs of Y.

9. Applications of Teardrop Neighborhoods

Teardrop neighborhoods can also be used in conjunction with the geometric theory of manifold approximate fibrations [11] to study the geometric topology of manifold stratified pairs. Examples of results proved using teardrop technology are stated in this section. Details will appear in [13].

Theorem 9.1 (Parametrized Isotopy Extension). Let X be a manifold stratified space such that all components of strata have dimension greater than 4, let Y be a pure subset of X, let U be a neighborhood of Y in X, and let h : Y ×Δ^{k} → Y ×Δ^{k} be a k-parameter stratum preserving isotopy. Then there exists a k-parameter isotopy h̃ : X × Δ^{k} → X × Δ^{k} extending h and supported on U × Δ^{k}.

This generalizes results of Edwards and Kirby [7], Siebenmann [29] and Quinn [26].

The next result is a topological analogue of Thom’s First Isotopy Theorem [31] and can be viewed as a first step towards a topological theory of topological stability.

Theorem 9.2 (First Topological Isotopy). Let X be a manifold stratified space and let p : X → ℝ^{n} be a map such that

(i) p is proper,

(ii) for each stratum X_{i} of X, p| : X_{i} → ℝ^{n} is a topological submersion, (iii) for each t ∈ ℝ^{n}, the filtration of X restricts to a filtration of p^{−1}(t) giving p^{−1}(t) the structure of a manifold stratified space such that all components of strata have dimension greater than 4.

Then p is a bundle and can be trivialized by a stratum preserving homeomorphism; that is, there exists a stratum preserving homeomorphism h : p^{−1}(0) × ℝ^{n} → X such that ph is projection.

Here is a non-proper version of the preceding result.

Theorem 9.3 (Non-proper First Topological Isotopy). Let X be a manifold stratified space and let p : X → ℝ^{n} be a map such that

(i) if ρ : X → [0, ∞) is a proper map and p′ = ρ × p : X → ℝ^{n} × [0, ∞), then the teardrop X ∪_{p′} ℝ^{n} is a manifold stratified space,

(ii) for each stratum X_{i} of X, p| : X_{i} → ℝ^{n} is a topological submersion, (iii) for each t ∈ ℝ^{n}, the filtration of X restricts to a filtration of p^{−1}(t) giving p^{−1}(t) the structure of a manifold stratified space such that all components of strata have dimension greater than 4.

Then p is a bundle and can be trivialized by a stratum preserving homeomorphism; that is, there exists a stratum preserving homeomorphism h : p^{−1}(0) × ℝ^{n} → X such that ph is projection.

10. Classifying manifold stratified approximate fibrations

Some applications of teardrop neighborhoods are combined with the material in §5 on bundles with mapping cylinder fibres in order to present a classification of manifold stratified approximate fibrations, at least when the range is a manifold, generalizing the classification of manifold approximate fibrations in [17] and [18].

For notation, let B be a connected i-manifold without boundary and let q : V → ℝ^{i} be a manifold stratified approximate fibration where all components of strata of V have dimension greater than 4. A stratified manifold approximate fibration p : X → B has fibre germ q if there exists an embedding ℝ^{i} ⊆ B such that p| : p^{−1}(ℝ^{i}) → ℝ^{i} is controlled, stratum preserving homeomorphic to q; that is, there exists a stratum preserving, level homeomorphism cyl(q) → cyl(p| : p^{−1}(ℝ^{i}) → ℝ^{i}) where the mapping cylinders have the natural stratifications.

The following result shows that fibre germs are essentially unique. For notation, let r : ℝ^{i} → ℝ^{i} be the orientation reversing homeomorphism defined by r(x_{1}, x_{2}, …, x_{i}) = (−x_{1}, x_{2}, …, x_{i}).

Theorem 10.1. Let p : X → B be a manifold stratified approximate fibration such that all components of strata have dimension greater than 4. Let g_{k} : ℝ^{i} → B, k = 1, 2, be two open embeddings. Then p| : p^{−1}(g_{0}(ℝ^{i})) → g_{0}(ℝ^{i}) is controlled, stratum preserving homeomorphic to either p| : p^{−1}(g_{1}(ℝ^{i})) → g_{1}(ℝ^{i}) or p| : p^{−1}(g_{1}(ℝ^{i})) → rg_{1}(ℝ^{i}).

Proof. The proof follows that of the corresponding result for manifold approximate fibrations in [17, Cor. 14.6]. The stratified analogues of the straightening phenomena are consequences of the teardrop neighborhood results [12], [13]. The use of Siebenmann’s Technical Bundle Theorem is replaced with the non-proper topological version of Thom’s First Isotopy Lemma in §9. □

There is a moduli space MSAF(B)_{q} of all manifold stratified approximate fibrations over B with fibre germ q. It is defined as a simplicial set with a typical k–simplex given by a map p : X → B × Δ^{k} such that for each t ∈ Δ^{k}, p| : p^{−1}(t) → B × {t} is a manifold stratified approximate fibration with fibre germ q and there exists a stratum preserving homeomorphism p^{−1}(0)×Δ^{k} → X which is fibre preserving over Δ^{k}. (There is also a technical condition giving an embedding in an ambient universe; cf. [17]).

The proof of the next proposition follows that of the corresponding result for manifold approximate fibrations in [17]. The necessary stratified versions of the manifold approximate fibration tools are in [12] and [13] and follow from teardrop technology.

Proposition 10.2. π_{0} MSAF(B)_{q} is in one-to-one correspondence with the set of controlled, stratum preserving homeomorphism classes of stratified manifold approximate fibrations over B with fibre germ q.

Let TOP_{s}^{level}(q) denote the simplicial group of self homeomorphisms of the mapping cylinder cyl(p) which preserve the mapping cylinder levels and are stratum preserving with respect to the induced stratification of cyl(q). Note that there is a restriction homomorphism TOP_{s}^{level}(q) → TOP_{i}.

Let τB → B denote the topological tangent bundle of B. Consider τB as an open neighborhood of the diagonal in B × B so that τB → B is first coordinate projection. As in §5 we can form the simplicial set B_{1}(q, TOP_{s}^{level}(q) rel τB → B) which we denote simply by B_{1}(q, TOP_{s}^{level}(q) rel τB).

The differential

d : MSAF(B)_{q} → B_{1}(q, TOP_{s}^{level}(q) rel τB)

is a simplicial map whose definition is illustrated on vertices as follows (for higher dimensional simplices, the construction is analogous; cf. [17]). If p : X → B is a vertex of MSAF(B)_{q}, then form

id_{B} ×p : B × X → B × B

and let

p̂ = p| : E = p^{−1}(τB) → τB.

Thus, there is a commuting diagram

E ⟶^{p̂} τB
↓  ↓
B ⟶^{id_{B}} B.

It follows from the stratified straightening phenomena [13] that the local triviality condition is satisfied, so that the diagram is a vertex of

B_{1}(q, TOP_{s}^{level}(q) rel τB).

Once again the proof of the following result follows that of the corresponding manifold approximate fibration result in [17] using the stratified results of [12] and [13].

Theorem 10.3 (MSAF Classification). The differential

d : MSAF(B)_{q} → B_{1}(q, TOP_{s}^{level}(q) rel τB)

is a homotopy equivalence.

Corollary 10.4. Controlled, stratum preserving homeomorphism classes of stratified manifold approximate fibrations over B with fibre germ q are in one-to-one correspondence with homotopy classes of lifts of the map τ : B → BTOP_{i} which classifies the tangent bundle of B, to BTOP_{s}^{level}(q):

BTOP_{s}^{level}(q)
↓
B ⟶^{τ} BTOP_{i} .

Proof. Combine Theorem 10.3, Proposition 10.2 and Proposition 5.8. □

Finally, observe that Corollary 10.4 can be combined with Theorem 8.3 to give a classification of neighborhood germs of B with fixed local type.

References

1. S. Cappell and J. Shaneson, The mapping cone and cylinder of a stratified map, Prospects in topology; Proceedings of a conference in honor of William Browder (F. Quinn, ed.), Ann. of Math. Studies, vol. 138, Princeton Univ. Press, Princeton, 1995, pp. 58–66.

2. T. A. Chapman, Approximation results in topological manifolds, Mem. Amer. Math. Soc. 34 no. 251 (1981).

3. J. Damon, Applications of singularity theory to the solutions of nonlinear equations, Topological Nonlinear Analysis: Degree, Singularity and Variations (M. Matzeu and A. Vignoli, eds.), Progress in Nonlinear Differential Equations and Their Applications, vol. 15, Birkhäuser, Boston, 1995, pp. 178–302.

4. K. H. Dovermann and R. Schultz, Equivariant surgery theories and their periodicity properties, Lecture Notes in Math. 1443, Springer-Verlag, New York, 1990.

5. A. du Plessis and T. Wall, The geometry of topological stability, London Math. Soc. Mono. New Series 9, Oxford Univ. Press, Oxford, 1995.

6. R. D. Edwards, TOP regular neighborhoods, handwritten manuscript (1973).

7. R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88.

8. T. Fakuda, Types topologiques des polynomes, Publ. Math. Inst. Hautes Études Sci. 46 (1976), 87–106.

9. C. G. Gibson, K. Wirthmüller, A. A. du Plessis and E. J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Math. 552, Springer-Verlag, New York, 1976.

10. M. Goresky and R. MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb. (3) 14, Springer-Verlag, New York, 1988.

11. B. Hughes, Approximate fibrations on topological manifolds, Michigan Math. J. 32 (1985), 167–183.

12. , Geometric topology of stratified spaces, ERA-AMS 2 (1996), (to appear), http:// www.ams.org/journals/era/.

13. , The geometric topology of stratified spaces (in preparation).

14. , Stratified path spaces and fibrations (in preparation).

15. B. Hughes and A. Ranicki, Ends of complexes, Cambridge Tracts in Math. 123, Cambridge Univ. Press, Cambridge, 1996.

16. B. Hughes, L. Taylor, S. Weinberger and B. Williams, Neighborhoods in stratified spaces with two strata (preprint).

17. B. Hughes, L. Taylor and B. Williams, Bundle theories for topological manifolds, Trans. Amer Math. Soc. 319 (1990), 1–65.

18. , Manifold approximate fibrations are approximately bundles, Forum Math. 3 (1991), 309–325.

19. , Bounded homeomorphisms over Hadamard manifolds, Math. Scand. 73 (1993), 161– 176.

20. , Rigidity of fibrations over nonpositively curved manifolds, Topology 34 (1995), 565– 574.

21. J. Mather, Notes on topological stability, Harvard Univ., Cambridge, 1970 (photocopied).

22. , Stratifications and mappings, Dynamical Systems (M. M. Peixoto, ed.), Academic Press, New York, 1973, pp. 195–232.

23. I. Nakai, On topological types of polynomial mappings, Topology 23 (1984), 45–66.

24. F. Quinn, Ends of maps. II, Invent. Math. 68 (1982), 353–424.

25. , Applications of topology with control, Proceedings of the International Congress of Mathematicians, Berkeley 1986, Amer. Math. Soc., Providence, R.I., 1986, pp. 598–606.

26. , Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), 441–499.

27. , Speculations on Gromov convergence of stratified sets, and Riemannian volume collapse, Prospects in topology; Proceedings of a conference in honor of William Browder (F. Quinn, ed.), Ann. of Math. Studies, vol. 138, Princeton Univ. Press, Princeton, 1995, pp. 301– 313.

28. C. Rourke and B. Sanderson, An embedding without a normal bundle, Invent. Math. 3 (1967), 293–299.

29. L. Siebenmann, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 47 (1971), 123–165.

30. M. Steinberger and J. West, Equivariant h–cobordisms and finiteness obstructions, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 217–220.

31. R. Thom, Ensembles et morphismes stratifies, Bull. Amer. Math. Soc. 75 (1969), 240–282.

32. , La stabilite des applications polynomials, L’Enseignement Mathematique II 8 (1969), 24–33.

33. A. Verona, Stratified mappings – structure and triangulability, Lecture Notes in Math. 1102, Springer-Verlag, New York, 1984.

34. S. Weinberger, The topological classification of stratified spaces, Chicago Lectures in Math., Univ. Chicago Press, Chicago, 1994.

35. H. Whitney, Local properties of analytic varieties, Differentiable and combinatorial topology (S. Cairns, ed.), Princeton Univ. Press, Princeton, 1965, pp. 205–244.

Department of Mathematics, Vanderbilt University, Nashville, TN 37240

   E-mail address: hughes@math.vanderbilt.edu