CHAPTER 4 Resolved spaces

I will now discuss the notion of an iterated fibration of the boundary hypersurfaces of a compact manifold with corners. This will turn out to include the ‘resolved’ version of a stratified pseudomanifold. Before discussing this resolution I shall motivate the notion of an iterated fibration by discussing the total boundary blow up of a manifold with corners.

1. Manifolds with corners

There is no time to go into a complete discussion of manifolds with corners here. Let me remind you of some of the elementary properties by adopting the following (perfectly correct but somewhat awkward) definition.

Definition 2. A manifold with corners is a topological manifold with boundary, X, for which there is a homeomorphism e : X ⟶ X̃ into a C^{∞} manifold without boundary, X̃, of the same dimension and such that the image

e(X) = {x ∈ X̃; ρ_{j}(x) ≥ 0, j = 1, …, K}

where the functions ρ_{j} ∈ C^{∞}(X̃) have the following independence property:

If x̄ ∈ X̃ and I ⊂ {1, …, K} is such that ρ_{j}(x̄) = 0
∀ j ∈ I then d_{j}(x̄), j ∈ I, are independent in T ∗ ; x̄X̃.

(4.1)

In fact the space of functions

C^{∞}(X) = e^{∗}C^{∞}(X̃),

which is the C^{∞} structure on X, can be determined by the usual local coordinate chart definition, except for the fact that the boundary hypersurfaces

H′_{j} = e^{−1} ({ρ_{j} = 0}) (4.2)

are embedded, as follows automatically from (4.1).

We are mainly interested in compact manifolds with corners, in which case X̃ can be taken to be compact. Let M_{1}(X) be the set of boundary hypersurfaces, given as the components of the H′_{j} in (4.2). More generally, let M_{k}(X) be the set of boundary faces of codimension k. Each F ∈ M_{k}(X) is necessarily a component of the intersection of k of the elements of M_{1}(X). Put

M_{0}(X) = {X} , M′(X) = ⋃_{k≥1} M_{k}(X), M(X) = ⋃_{k≥0} M_{k}(X).

48 4. RESOLVED SPACES

A non-degenerate Riemann metric on X is simply the pull-back of a Riemann metric on X̃. There is a very important difference between the case of a compact manifold with boundary and that of the more general case of a compact manifold with corners. Namely, for a general non-degenerate metric, there is no Hodge decomposition of C^{∞}(X; Λ^{∗}X). This can be seen most directly in the appearance of harmonic forms which are not smooth up to the corner, despite the fact that they satisfy appropriate boundary conditions.

Rather than give any sort of complicated example just consider a sector in ℝ^{2} with angle α < π. Of course this is not compact, but the local behavior of harmonic forms will persist in the compact case. So, simply consider the Laplacian with Dirichlet boundary conditions. In polar coordinates the Laplacian is

Δ = −\frac{∂^{2}}{∂r^{2}} − \frac{1}{r} \frac{∂}{∂r} − \frac{1}{r^{2}} \frac{∂^{2}}{∂θ^{2}} .

It follows that the functions

f_{k}(r, θ) = r^{± \frac{kπ}{α}} sin (\frac{kπθ}{α} ) k ∈ N (4.3)

are harmonic in the sector 0 ≤ θ ≤ α and vanish on the boundaries, θ = 0, α.

[figure: α]

In general, even for large k, these functions are not smooth down to r = 0. An exception is the right angle, α = π/2, for which the positive signs in (4.3) do give smooth functions.

Thus, in general it is not reasonable to try to find an abstract Hodge decomposition with target C^{∞}(X; Λ^{∗}X), even in the case of a non-degenerate metric. In fact we can already suspect from (4.3) that even to get a decomposition with target the space of polyhomogeneous conormal forms it will be necessary to introduce polar coordinates around the corners.

2. Blow-up

I will try not to give a detailed treatment of blow-up! Reasonably complete discussions of real blow up can be found in [7] and [6]. For present purposes it suffices to consider the blow-up of a boundary face of a compact manifold with corners. This is always possible, giving a new manifold with corners and natural blown-down map

[X; F] ⟶^{β} X F ∈ M′(X).

If F ∈ M_{1}(X) is a boundary hypersurface then [X; F] = X and β is the identity.

In general if F ∈ M_{k}(X), k > 1, then [X; F] has one more boundary hypersurface H_{F} ∈ M_{1} ([X, F]) representing the blow-up of F, β(F) = H_{F}. The map β : H_{F} → F is a fibration with fibre diffeomorphic to

𝕊^{k−1,k} = {t ∈ ℝ^{k}; t_{j} ≥ 0 for 1 ≤ j ≤ k, |t| = 1} .

3. TOTAL BOUNDARY BLOW-UP 49

The other boundary hypersurfaces of [X; F] can be identified via β with the lifts of the boundary hypersurfaces of X. In all cases the closure of the inverse image

clos (β^{−1}(H\F)) ≅ [H; H ∩ F] , H ∈ M (X) .

3. Total boundary blow-up

If X is a compact manifold with corners we define a new manifold X_{tb} by blowing up all the boundary faces of X. There is no point in blowing up the boundary hypersurfaces; it is important to specify the order of the blow-up. We shall always proceed in order of decreasing codimension. Thus if X has maximum boundary codimension k, we first define

X_{k} = [X, M_{k}(X)] , β_{k} : X_{k} → X.

The boundary faces of maximum codimension are manifolds without boundary. There may be several disjoint faces but these lead to independent blow ups, so M_{k} is well defined. Furthermore all the boundary faces of X of codimension at most k − 1 lift, in the sense of

clos β_{k}^{−1}(G\ ⋃ M_{k}(X)) = G_{k} ∈ M′(X_{k}),

to boundary faces, of the same codimension of X_{k}. Indeed

M_{j}(X_{k}) ≡ M_{j}(X) 1 < j < k.

We now proceed iteratively, defining

X_{l} = [X_{l+1}, M_{l}(X)] , k < l < 1,

where M_{l}(X) ≡ M_{l}(X_{l+1}) is the set of boundary faces of codimension l.

The end result of this process of iterated blow-up is the total boundary blow-up, X_{tb} = X_{2}, which can be written as a repeated blow-up

X_{tb} = [X; M_{k}(X); M_{k−1}(X); …; M_{2}(X)] .

Proposition 21. The boundary hypersurfaces of X_{tb} are in 1-1 correspondence with the boundary faces of X, with the relationship the total blow-down map β. Two boundary hypersurfaces H, H′ ∈ M_{1}(X_{tb}) intersect if and only if either β(H) ⊂ β(H′) or β(H) ∪ β(H′).

Proof. The correspondence between M_{1}(X_{tb}) and M′(X), i.e. between the boundary hypersurfaces of X_{tb} and all non-trivial boundary faces of M(X), is essentially by construction. To see the intersection relation, suppose that F_{1}, F_{2} ∈ M′(X) are such that neither F_{1} ⊂ F_{2} nor is F_{2} ⊂ F_{1}. Then F_{1} ∩ F_{2} is a collection of boundary faces (or is empty). All the faces in the intersection have larger codimension than either F, or F_{2}. Thus is the construction of X_{tb} the faces in F_{1} ∩ F_{2}, if any, will be blown-up before either F_{1} or F_{2}. Since F_{1} and F_{2} are closed, F_{1} ∩ F_{2} consists of maximal boundary faces (disjoint if they have different dimensions G_{1}, …, G_{p}, together with all F ∈ M′(X) such that F ⊂ G_{i} for some i. As the boundary faces of the G_{i} are blown-up in order of decreasing codimension the structure of the intersection does not change essentially – F_{1}, F_{2} and G_{i} are replaced by blown up versions but the intersection of the lifts of F_{1} and F_{2} remains the lift of G_{i}. When one of the G_{i}, locally of maximal dimension in the intersection, is blown up the lifts of F_{1} and F_{2} become disjoint in a neighborhood of the new boundary hypersurface, since they are disjoint in the spherical conormal bundle of G_{i}. Thus, after the G_{i} are blown up the lifts of the F_{1} and F_{2} are disjoint.

Conversely if F_{1} ⊂ F_{2} then the boundary hypersurface produced by blow-up of F_{1} and F_{2} are disjoint.

The space obtained by iterated boundary blow-up of X has more structure than X itself.

Proposition 22. Each of the boundary hypersurfaces of X_{tb} is a product; if H_{F} ∈ M_{1}(X_{tb}) arises by the blow-up of F ∈ M′(X) then

H_{F} ≅ F_{tb} × 𝕊_{tb}^{k−1,k}, F ∈ M_{k}(X) (4.4)

where 𝕊^{k−1,k} is the intersection of the (k − 1)-sphere with the totally positive space:

𝕊^{k−1,k} = {t ∈ [0, ∞)^{k}; |t|^{2} = 1} .

The product decomposition is realized by the continuous extension of the blow-down map from the interior of H_{F}.

Proof. In a neighborhood of the given boundary face F the manifold X is a product, namely

X_{F} ≅ F × [0, ∞)^{k} near F ∈ M_{k}(X). (4.5)

Thus it suffices to work with this local model, the process of blow-up being itself local.^{1} The boundary faces of X_{F} are simply products of boundary faces, so

M′(X_{F}) = ⋃_{k} ⋃_{l} M_{l}(F) × M_{k−l} ([0, ∞)^{k}) .

If F has maximum codimension p the faces of X_{F} of maximal codimension, p + k, are the products of these faces with {0}. Thus the first blow-up gives

(X_{F})_{p+k} = [F × [0, ∞)^{k}; M_{p}(F) × {0}] .

We are interested in a neighborhood of the lift of F × {0} to this new manifold. This is of the form

(X_{F})_{p+k} ⊃ X_{F}^{(p+k)} ≅ F_{p} × [0, ∞)^{k} ,

i.e. is just the product of F with its faces of maximal codimension blown up and the model normal space [0, ∞)^{k}.

At the next stage of the iterated blow-up occur the lifts of products M_{p}(F) × M_{1} ([0, ∞)^{k}) and M_{p−1}(X) × {0}. The former are all disjoint from the lift of F ×{0}, i.e. F, and so only the latter need to be considered. By the same argument as before this shows that there is a neighborhood of the lift of F in

(X_{F})_{p+k−1} ⊃ X_{F}^{(p+k−1)} ≃ F_{p−1} × [0, ∞)^{k} ,

just consisting of F with faces of codimension p, and of codimension p − 1, blown up.

This discussion can be repeated until the codimension of blow-up reaches k. At this point a neighborhood of the lift of F is

F_{tb} × [0, ∞)^{k}

1Local in the manifold which is blown up, not in the blown up manifold!

and the manifolds to be blown up are the lifts of

F × {0} and M_{l}(F) × M_{k−l} ([0, ∞)^{k}) , for l ≥ 1.

The latter lifts are disjoint from the lift of F, so F lifts to a hypersurface and nearby the manifold becomes

F_{tb} × [0, ∞) × 𝕊^{k−1,k}

with F lifting to F_{tb} × {0} × 𝕊^{k−1,k}.

For codimensions less than k the only lifts of submanifolds not disjoint from the lift of F are the products

F_{tb} × [0, ∞) × M′(𝕊^{k−1,k}).

Thus at the end of the sequence of blow-ups the lift of F has a neighborhood

F_{tb} × [0, ∞) × 𝕊_{tb}^{k−1,k}

in which the middle variable is the boundary defining function from the lift of F. This proves the proposition.

Thus we see that the boundary hypersurface H_{F} ∈ M_{1}(X_{tb}) arising from F ∈ M_{k}(X) has a product decomposition, which we write as a fibration:

[figure: 𝕊_{tb}^{k−1,k}; H_{F}; φ_{F}; F_{tb}.] (4.6)

It remains to check how these fibrations are related to each other at points of intersection of the boundary hypersurfaces.

Proposition 23. If p ∈ X_{tb} is a boundary point of codimension q then the boundary hypersurfaces of X_{tb} through p are strictly ordered by the codimension of the boundary faces from which they arise and there is a local product decomposition

X_{tb} ≅ [0, ∞)_{x}^{q} × ℝ_{y}^{n−q} near p

where the boundary hypersurface {x_{i} = 0}, i = 1, …, q has fibration (4.6) locally having fibres

(4.7) {(x, y); x_{i} = 0, x_{s} = const, s > i, y_{l} = const, l ≥ R_{i}}

where R_{i} is a non-decreasing function of i.

Proof. From Proposition 21 the q hypersurfaces intersecting at p must arise by the blow-up of a chain, under inclusion, of boundary faces of X.. We order the hypersurfaces by increasing codimension, i.e. H_{i} corresponds to F_{i} where

F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{q}.

Notice that the dimensions are strictly decreasing but may decrease by more than one, i.e. F_{l} need not be a boundary hypersurface of F_{l−1}, but is necessarily a boundary face of it.

Of course there is a complete chain of boundary faces containing β(p) ∈ X which we can write as

G_{1} ⊂ G_{2} ⊂ ⋯ ⊂ G_{k} ∋ β(p), G_{l} ∈ M_{l}(X).

Then F_{i} = G_{s(i)} for some strictly increasing function s(i).

Consider local coordinates x̃_{1}, …, x̃_{k}, ỹ_{1}, …, ỹ_{n−k} around β(p), where, as usual, the x̃_{l}’s are local boundary defining functions so that

G_{i} = {x̃_{l} = 0, l ≤ i} near β(p).

The lifts of these coordinate functions are smooth near p. Moreover the blow-down map β is a b-map so each β^{∗}x̃_{i} is the product of a positive function and boundary defining functions near p. For each l, x̃_{s(l)} vanishes on the F_{i} for i ≥ l. Thus if ρ_{j} ∈ C^{∞}(X_{tb}) define the hypersurfaces H_{j} near p,

β^{∗}x̃_{s(l)} = a_{s(l)} ∏_{j≥l} ρ_{j}, 0 < a_{s(l)} ∈ C^{∞}.

This shows that the lifts of the quotients x̃_{s(j)}/x̃_{s(j+1)}, for j < q, and x̃_{s(q)} itself, can be introduced in place of the ρ_{j} as defining functions for the boundary hypersurfaces. Having done this, a_{s(l)} ≡ 1 (near p) and

ρ_{j} = β^{∗} (x̃_{s(j)}/x̃_{s(j+1)}) , j < q, ρ_{q} = β^{∗}x̃_{s(q)}.

The other local boundary defining functions x̃_{j}, j ≠ s(i), have similar decompositions. Thus if we consider the quotients

β^{∗} (x̃_{l}/x̃_{s(j)}) s(j) < l < s(j + 1) (4.8)

we obtain positive smooth functions. Let us denote by y_{r} these functions, with a constant subtracted to make them vanish at p and relabeled so that 1 ≤ r ≤ k − q. For r > k − q set

y_{r} = β^{∗}(ỹ_{r−k}) − ỹ_{r−k}(q) k − q < r ≤ n − q.

The fact that β is a b-submersion shows that these functions, x_{1}, …, x_{q}, y_{1}, …, y_{n−q} give local coordinates near p ∈ X_{tb}. By constriction β is then the map

β(x, y) = (x̃, ỹ)
x̃_{s(i)} = ∏_{l=i}^{i} x_{l} ,  i < q
x̃_{s(q)} = x_{q}
x̃_{j} = y_{T (j)} ∏_{l=i}^{q} x_{l} s(i) < j < s(i + 1)(or s(q) < j < k)
ỹ_{j} = y_{k−q+j}, j = 1, …, n − k.

Here j ↦ T (j) is the map defined by the reordering of the y_{r}’s in the discussion following (4.8). The fibres of β over the interior of H_{j} are then given by the constancy of the x_{l}, for l > j, the y_{r} arising from (4.8) with l > s(j) and the y_{r} for r > k − q. This defines R_{i} and gives 4.7.

4. Iterated fibrations

The properties of the space X_{tb}, with its boundary hypersurfaces each having a fibration, are typical of the resolved spaces we shall need below, except that the fibrations in that case are all products, with fibres being parts of spheres. Dropping these restrictions we arrive at:

5. ASSOCIATED METRICS 53

Definition 3. An iterated fibration of the boundary hypersurfaces of a compact manifold with corners X is a fibration (in the sense of manifolds with corners) of each boundary hypersurface

(4.9)  [figure: Z_{H}; H; φ_{H}; B_{H}.]

such that each boundary point p ∈ X has a coordinate neighborhood with coordinates x_{1}, …, x_{q}, y_{1}, …, y_{n−q} in terms of which the fibres of φ_{{x_{i}=0}} are locally of the form

(4.10) {(x, y); x_{i} = 0, x_{s} = const, s > i, y_{l} = const, l ≤ R(i)}

where R(i) is a non-decreasing function of i ∈ {1, …, q}.

The iterative properties of this notion are important, since this is the approach we need to take in most proofs.

Lemma 11. If X is a manifold with corners with an iterated fibration then each boundary hypersurface of X with fibres of (locally) maximal dimension has a naturally induced structure as a manifold with corners with iterated fibration; the dimensions of the fibres are strictly less than those of H.

Proof. The conditions in (4.10) show that if H is a boundary hypersurface with fibres of maximal dimension then in the local coordinate form of the fibrations around a point p ∈ H, H itself must always be the defined by x_{q} = 0, since this corresponds to the largest fibre. It follows from (4.10) that the fibrations of the hypersurfaces meeting H restrict to give fibrations of the boundary hypersurfaces of H, since x_{q} = 0 is necessarily a union of fibres. Furthermore the local structure condition (4.10) then restricts to give the same condition for the induced fibrations of the boundary hypersurfaces of H. The fibre dimensions are less than those of H itself.

In fact we can improve on this proposition a little by noting that the product decomposition near boundary faces, which is the main property of manifolds with corners, extends to include the fibration. That is, if H has maximal fibre dimension then H has an open neighbourhood O_{H} ⊂ X which can be decomposed smoothly as a product, O_{H} ≡ [0, ϵ) × H in such a way that the various fibrations remain consistent.

Proposition 24. If H ∈ M_{1}(X) is a boundary hypersurface with fibres of maximal dimension for an iterated fibration of X then ϕ_{H} can be extended to a fibration of an open neighbourhood O_{H} of H, ϕ̃_{H} : O_{H} ⟶ Y_{H} × [0, ϵ), ϵ > 0 such that

5. Associated metrics

Reverting again to the case of a manifold with corners, X, and its total boundary blow-up, X_{tb}, consider what happens to a smooth, non-degenerate, metric on X under this ‘resolution’. To express this we first introduce the notion of an iteratively conic metric associated to an iterated fibration of a manifold with corners.

As usual we can introduce this notion inductively, over the maximal dimension of the fibres. In case the fibrations are all diffeomorphism, so have zero dimensional fibrations, we take an iteratively conic metric to be a non-degenerate metric, one that extends smoothly as a metric ‘across the boundaries’. Now, suppose we have defined the notion when the maximal fibre dimension is at most l. If X is a manifold with corners having an iterated fibration with fibres having dimension up to, but not exceeding, l +1 we define an iteratively conic metric on X to be a metric on the interior X^{∘}, which restricts to an iterately conic metric in a neighbourhood of all boundary hypersurfaces with fibrations of fibre dimension l or less. Since they are disjoint it suffices to consider the behaviour of the metric near one boundary face, H with fibres of dimension l + 1. Near H we demand that there is a decomposition as in Proposition 24

g = d