# Generalized products and Semiclassical Quantization

Richard Melrose  Department of Mathematics, Massachusetts Institute of
Technology<EMAIL_ADDRESS>
###### Abstract.

The notion of a generalized product, refining that of a (symmetric and smooth)
simplicial space is introduced and shown to imply the existence of an algebra
of pseudodifferential operators. This encompasses many constructions of such
algebras on manifolds with corners. The main examples discussed in detail here
are related to the semiclassical (and adiabatic) calculus as used in the
approach to a twisted form of the Atiyah-Singer index theorem in work with Is
Singer and Mathai Varghese.

###### Key words and phrases:

simplicial space, generalized product, b-fibration, p-embedding,
pseudodifferential operator, semiclassical calculus, manifold with corners

## Introduction

One aim of this paper is to give a full discussion of the (well-known)
semiclassical calculus, as used in the context of index theory in [5, 6, 7].
This includes ‘extensions’ to the semiclassical calculus such as the double
semiclassical calculus for an iterated fibration. The construction of these
operator algebras follows a pattern that has emerged over a long period with
calculi such as the b-calculus, the scattering calculus, the zero calculus and
their many variants. So rather than repeat the (non-automatic) form of these
the notion of a _generalized product_ is introduced and shown to determine an
algebra of operators. This conjecturally includes essentially all the known
algebras obtained this way, although that is not discussed here, except for
the b-calculus itself.

A generalized product is modeled on the fibre products of a fibration. The
notion is a refinement a that of a symmetric simplicial space; for the
topological concept see for instance [2]. Abstractly we consider a
contravariant functor from the integers (or finite sets) to the category of
manifolds, here we are mainly interested in the target category being that of
the compact manifolds with corners. The morphisms of the integers are all the
maps from $J(m)$ to $J(n)$ where $J(m)=\\{1,\dots,m\\}.$ There are important
additional constraints on the functor. Namely the dimensions of the manifolds
associated to the integers, written generically $M[k],$ are required to be of
the form $\dim M[k]=\mu+\kappa(k-1).$ The maps associated to injections are
required to be b-fibrations and those associated to surjections to be
p-embeddings. These are special morphisms in the category of manifolds with
corners where the morphism are b-maps; the notion of a b-fibration is close
to, but crucially weaker than, that of a fibration. The precise definition is
given in §1 which also sets out the stronger notion of a stretched product.

The multidiagonals, just called diagonals below, in a generalized product,
correspond to the surjective maps for $J(m)$ to $J(n)$ ($n<m$) and are of
particular interest as the carriers of the singularities of the kernels of
pseudodifferential operators. In Theorem 1 it is shown that the intersection
properties of these diagonals are essentially the same as the diagonals in a
fibre product. In particular the fact that _the_ diagonal, $D\subset M[2],$ is
a p-embedded manifold allows the definition of pseudodifferential operators in
terms of kernels with conormal structure at the diagonal, to be taken over
directly. These operators are naturally the ‘microlocalization’ of a Lie
algebroid over $M[1]$ although this is by no means a general Lie algebroid.
The precise global structure of the $M[k]$ remains to be elucidated.

Very special examples of generalized products arise from a compact Lie group
with $G[1]$ a point and the algebra resulting being the convolution algebra.
We conjecture that any real Lie group with connected centre arises as the
interior of a generalized product, i.e. has a ‘simplicial compactification’.
Two (essentially familiar) examples of this arise as the compactifcations of
the multiplicative group of the positive reals and the translation group given
by a vector space. These two cases are closely connected to the b-calculus in
the former case and the semiclassical (or scattering) calculus in the latter.

Some results here, especially in §3 are derived from an unfinished project
with Michael Singer on the scattering stretched product; I also thank him for
comments on the manuscript. Helpful conversation with Chris Kottke regarding
generalized products are also happily acknowledged.

##  1\. Generalized and stretched products

An important example of a simplicial space arises from a smooth fibre bundle

(1.1) $\phi:M\longrightarrow Y.$

In case $M$ is a compact manifold without boundary, a submersion induces such
a fibre bundle. In the case of non-compact manifolds and also in the category
of compact manifolds with corners, recalled below, it is necessary to require
more in order that $\phi$ have local trivializations over a suitable cover of
$Y.$ Given this, the fibre products form a simplicial space

(1.2)
$\textstyle{M}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M^{[2]}_{\phi}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M^{[3]}_{\phi}}$$\textstyle{\dots.}$

Here the maps are each a projection off one of the factors.

Going somewhat beyond the standard notion of a simplicial space, observe that
these maps and spaces correspond to part of a contravariant functor from the
category with objects the integers $\mathbf{N},$ identified as the collection
of sets $J(n)=\\{1,\dots,n\\},$ and with morphisms all the maps from $J(n)$ to
$J(m),$ (rather than just the order-preserving maps) to the category of
manifolds. Thus if

(1.3) $\begin{gathered}I:\\{1,\dots,n\\}\longrightarrow\\{1,\dots,m\\}\text{
then }\\\ S_{I}:M^{[m]}_{\phi}\longrightarrow M^{[n]}_{\phi},\
S_{I}(p_{1},\dots,p_{m})=(p_{I(1)},\dots,p_{I(n)}).\end{gathered}$

The maps $S_{I}$ corresponding to injective maps $I$ are fibrations, namely up
to relabelling the various projections off factors in (1.2). The maps
corresponding to surjective maps $I$ are the embeddings of ‘multidiagonals’.
However in the case of fibres which have boundary these do not satisfy the
stronger condition we will require of being p-embeddings, that near boundary
points there is a common local product decomposition of manifold and
submanifold.

For instance for $n=2$ and $m=1$ so $I(1)=I(2)=1,$

(1.4) $S_{I}:M\ni p\longrightarrow(p,p)\in M^{[2]}_{\phi}$

is the embedding of the diagonal in the fibre product. If the fibre has a
boundary then the diagonal passes through the corner and does so in a manner
which precludes a common product decomposition of manifold and submanifold.
This is a particular issue for the kernels of pseudodifferential operators.

So, for the definition, note that in the category of (usually compact)
manifolds with corners the morphisms are b-maps and there are special
surjective maps, the b-fibrations, and special injective maps, the
p-embeddings. This is all discussed in §2 below.

######  Definition 1.

A _generalized product_ of manifolds (with corners) is a contravariant functor
from the category $\mathbf{N},$ with morphisms _all_ the maps from $J(n)$ to
$J(m),$ to the category of manifolds with corners, with objects denoted
$M[k],$ $k\in\mathbf{N},$ $\dim M[k]=\mu+(k-1)\kappa,$ morphisms the interior
b-maps and under which injective maps are taken to b-fibrations and surjective
maps to p-embeddings; all manifolds (including the fibres of b-fibrations) are
assumed to be connected.

We call the images $S_{I}:M[m]\longrightarrow M[n]$ of the maps
$I:J(m)\longrightarrow J(n)$ the _structure maps_.

The main interest lies in the compact case; the assumption of connectedness is
made to simplify the discussion below, it is not really crucial.

######  Definition 2.

A generalized product is said to be _compact_ if all the spaces $M[k]$ are
compact (and connected) and the structural b-fibrations have connected fibres.

The permutations $\sigma:J(n)\longrightarrow J(n)$ for each $n$ must induce
maps which are both b-fibrations and p-embeddings and hence are
diffeomorphisms, the structural diffeomorphisms. Thus the corresponding
$S_{\sigma}$ acting on $M[n]$ induce an action of the permutation group.

######  Lemma 1.

A generalized product is symmetric.

In any generalized product the minimal, or $k$-fold, diagonal in $M[k]$ is the
p-submanifold $D=D_{k}:M[1]\longrightarrow M[k]$ corresponding to the unique
map $J(k)\longrightarrow J(1).$ Consider one of the $k$ b-fibrations
$\Pi:M[k]\longrightarrow M[1]$ arising from a map $J(1)\longrightarrow J(k).$
Since the composite $J(1)\longrightarrow J(k)\longrightarrow J(1)$ is the
identity $\Pi$ restricts to the identity map on the image of $D_{k}.$ If $H$
is a boundary hypersurface of $M[1]$ it follows that the inverse image
$\Pi^{-1}(H)$ is a union of boundary hypersurfaces of $M[k]$ exactly one of
which meets $D_{k}$ (since it is an interior p-submanifold) and does so in
$H.$ We Denote this boundary hypersurface as $H[k].$

######  Proposition 1.

If $M[k]$ is a generalized product and $H$ is a boundary hypersurface of
$M[1]$ the boundary hypersurfaces $H[k]$ have an induced structure as a
generalized product.

###### Proof.

Clearly $\dim H(k)=(\mu-1)+\kappa(k-1).$ The composite of any map
$J(n)\longrightarrow J(m)$ with $J(m)\longrightarrow J(1)$ is the unique map
$J(n)\longrightarrow J(1).$ Taking $m=n=k$ it follows that any of the
structure diffeomorphisms map the maximal diagonal $D_{k}$ into itself as the
identity. As diffeomorphisms they map boundary hypersurfaces to boundary
hypersurfaces so map $H[k]$ into itself giving the permutation action on
$H[k].$

Similarly for any of the structure maps $S_{I}$ for $I:J(n)\longrightarrow
J(m)$ the composite with $J(m)\longrightarrow J(1)$ is the map
$J(n)\longrightarrow J(1)$ so $S_{I}D_{m}=D_{n}.$ Thus $S_{I}$ maps the image
of $D_{m}$ to the image of $D_{n}$ as the identity. Again it maps boundary
hypersurfaces into boundary faces so must map $H[n]$ to $H[m].$ So each of the
structure maps takes $H[m]$ into $H[n]$ and the functorial properties follow.
∎

Since the spaces $M[1],$ $M[2]$ and $M[3]$ are particularly important we use
special notation for some of the structure maps. Thus

(1.5) $\begin{gathered}\Pi_{L},\ \Pi_{R}:M[2]\longrightarrow M[1]\text{ and
}\\\ \Pi_{S},\ \Pi_{C},\ \Pi_{F}:M[3]\longrightarrow M[2]\end{gathered}$

are the b-fibrations which are the images of the maps $L,$
$R:J(1)\longrightarrow J(2),$ $L(1)=1,$ $R(1)=2$ and $S,$ $C,$
$F:J(2)\longrightarrow J(3)$ where $S(1)=1,$ $S(2)=2,$ $C(1)=1,$ $C(2)=3$ and
$F(1)=2,$ $F(2)=3.$ Similarly the partial diagonals are the p-embeddings

(1.6) $D_{i,j}:M[2]\longrightarrow M[3],\ i\not=j,\\\ \text{corresponding to
the surjective maps taking }\\{i,j\\}\text{ to }1$

and $R_{i,j}$ for $i\not=j$ correspond to the reflections exchanging $i$ and
$j.$ The single reflection on $M[2]$ is denoted $R.$

By assumption all the structure maps of a generalized product are interior
b-maps and so have the property that the inverse image of the boundary of the
range space is contained in the boundary of the domain. It follows that they
map the interior into the interior in all cases.

######  Lemma 2.

The interiors $\operatorname{int}(M[k])$ have an induced structure as a
generalized product.

So generalized products can be thought of as compactifications of their
interiors. Restricted to the interiors consider the maps $\Pi_{k}=S_{I},$
$I:J(k-1)\longrightarrow J(k)$ being the strictly increasing map $I(j)=j$ and
$D_{1,2}=S_{L}$ $L:J(k)\longrightarrow J(k-1)$ being, for general $k\geq 3,$
the surjective increasing map with $L(1)=L(2)=1.$ Then

(1.7)
$\begin{gathered}\Pi_{k}:\operatorname{int}M[k]\longrightarrow\operatorname{int}M[k-1]\text{
is a fibration and}\\\
D_{1,2}:\operatorname{int}M[k-1]\longrightarrow\operatorname{int}M[k]\text{ is
an embedding for each }k.\end{gathered}$

Conversely

######  Lemma 3.

If $M[k]$ are compact manifolds with corners with interiors which form a
generalized product then to show that the $M[k]$ form a generalized product,
and hence a compactification, it suffices to show that the structural
diffeomorphisms extend to diffeomorphisms of each $M[k],$ the fibrations
$\Pi_{k}:M[k]\longrightarrow M[k-1]$ extend to b-fibrations and the embeddings
$D_{1,2}:M[k-1]\longrightarrow M[k]$ extend to p-embeddings.

###### Proof.

For any map $I:J(n)\longrightarrow J(m)$ suppose the range has $k$ elements,
then it can be identified with $J(k)$ and $I$ is the composite of a surjection
to $J(k)$ and an injection from $J(k).$ So the functorial properties imply
that it suffices to show the existence of all the $S_{I}$ for surjections and
injections and the special case of bijections, the others can then be obtained
by composition. So, assuming the diffeomorphisms giving the action of the
permutation group on the interior extend to be smooth b-maps, symmetry follows
for the $M[k].$

After composition with permutations, an injection is reduced to a map
$I:J(n)\longrightarrow J(m)$ where $I(j)=n-m+j.$ These in turn are products of
the maps $I:J(n)\longrightarrow J(n+1),$ $I(j)=j.$ Similarly the surjections
can be decomposed into products of permutations and the maps corresponding to
the $D_{1,2}.$ ∎

As is easily seen

######  Lemma 4.

The product $M[k]\times N[k]$ of two generalized products is a generalized
product.

However the product may not be the most natural compactification of the
products of the interiors.

######  Example 1.

The basic example of a generalized product arises, as noted above, as the
fibre products $M[k;\phi]=M^{[k]}$ of a fibration of compact manifolds (1.1).
This ‘resolves’ (and in consequence ‘quantizes’) the Lie algebra of fibre
vector fields on $M.$ These form the Lie algebroid over $M=M[1]$ formed by the
sections of the null bundle, $\operatorname{null}(\phi_{*}),$ over $M=M[1].$

For later reference we give a ‘simplicial’ construction of this Lie algebra
structure using $M[2]$ and $M[3].$ The two projections $\pi_{L},$
$\pi_{R}:M[2]\longrightarrow M[1]$ are fibrations so the null bundles
$(\pi_{L})_{*}$ and $(\pi_{R})_{*}$ are subbundles of $TM[2],$ namely the
right and left fibre tangents. We identify the bundle
$\operatorname{null}(\phi_{*})$ over $M[1]$ more abstractly as a vector bundle
over $M[1]$

(1.8) $E=D^{*}\operatorname{null}((\pi_{R})_{*}).$

Clearly $(\pi_{L})_{*}$ identifes $\operatorname{null}((\pi_{R})_{*})$ over
the (image of) the diagonal with $\operatorname{null}(\phi_{*}).$

Consider the following indirect, but more simplicial construction of the Lie
algebra structure on ${\mathcal{C}}^{\infty}(M[1];E).$ Starting from a section
$V\in{\mathcal{C}}^{\infty}(M[1];E)$ we first identifying it as a section of
$\operatorname{null}((\pi)_{R})_{*})$ over $D.$

  1. (1)

The projection $\Pi_{S}$ maps $D_{1,2}$ as a fibration onto $D.$ The null
space of the differential $\operatorname{null}((\Pi_{F})_{*})$ over $D_{1,2}$
is mapped by $(\Pi_{S})_{*}$ isomorphically onto
$\operatorname{null}((\pi_{L})_{*}$ at each point of $D_{1,2}$ so $V$ lifts
pointwise to a smooth section
$W\in{\mathcal{C}}^{\infty}(D_{1,2};\operatorname{null}((\pi_{F})_{*})).$

  2. (2)

The fibration $\Pi_{C}$ restricts to a diffeomorphism $D_{1,2}\longrightarrow
M[2]$ and $(\Pi_{C})_{*}$ maps $\operatorname{null}((\pi_{F})_{*}))$
isomorphically onto $\operatorname{null}((\Pi)_{R})_{*})$ over $M[2].$ This
defibes
$\tilde{V}=(\Pi_{C})_{*}W\in{\mathcal{C}}^{\infty}(M[2];\operatorname{null}((\Pi)_{R})_{*}).$
The restriction $\tilde{V}\big{|}_{D}=V.$

These claims are easily verified in a local trivialization of $\phi$ over $M$
with base coordinates $y$ and fibre coordinates written
$(z,z^{\prime},z^{\prime\prime})$ in $M[3]$ and
$(z^{\prime},z^{\prime\prime})$ in $M[2]$ induced by fibre coordinates $z$ in
$M.$ Then $V=\sum\limits_{i}v_{i}(y,z^{\prime})\cdot\partial_{z^{\prime}}$ as
a section of $\operatorname{null}((\pi_{R})_{*}$ over
$D=\\{z^{\prime}=z^{\prime\prime}\\}.$ The the pull-back to $D_{1,2}$ as a
section of $\operatorname{null}((\pi_{F})_{*})$ is

$W=\sum\limits_{i}v_{i}(y,z)\cdot\partial_{z}.$

This pushes forward under $(\pi_{C})_{*}$ to the vector field

$\tilde{V}=\sum\limits_{i}v_{i}(y,z^{\prime})\cdot\partial_{z^{\prime}}$

on $M[2]$ which extends the original vector field $V$ to be fibre constant on
the ‘right’ fibres of $M[2].$ These form a Lie subalgebra of the sections of
$\operatorname{null}((\pi_{R})_{*})$ over $M[2]$ (which can depend on the
‘right’ variable $z^{\prime\prime}).$ This defines a Lie algebra structure on
${\mathcal{C}}^{\infty}(M[1];E)$ which is identified with with the obvious
structure of fibre vector fields on $M.$

Whilst trivial in this case, this indirect construction is the basis of the
identification of the Lie algebroid structure in the wider context of a
generalized product in §5 below.

######  Example 2.

Let $G$ be a connected Lie group. Then the products $G^{k}$ for $k=1,2,\dots,$
form a generalized product (not using the group structure) which is
equivariant under the right action of $G$ via

$G^{k}\times
G\ni((g_{1},\dots,g_{k}),g)\longrightarrow(g_{1}g^{-1},\dots,g_{k}g^{-1})\in
G^{k},$

so $S_{I}(\gamma g^{-1})=S_{I}(\gamma)\cdot g^{-1}$ for all $I.$ The quotients

(1.9) $G[k]=G^{k}/G$

therefore form a generalized product.

For instance the three symmetries on $G^{2}$ corresponding to the involutions
exchanging the first two, first and third and second and third elements are

(1.10) $R_{1,2}(a,b)=(b,a),\ R_{1,3}(a,b)=(a^{-1},ba^{-1}),\
R_{2,3}(a,b)=(ab^{-1},b^{-1})\\\ \text{ on }G[3]=G^{2}.$

The reflection, $R,$ on $G=G[2]$ is $g\longmapsto g^{-1},$ the three
projections from $G[3]=G^{2}$ to $G[2]$ are

$\Pi_{F}(a,b)=b,\ \Pi_{C}(a,b)=a,\ \Pi_{S}(a,b)=ab^{-1},$

the three diagonals, in $G[3],$ which are diffeomorphic to $G,$ are

$D_{1,2}=\\{(g,g)\\},\ D_{1,3}=\\{(\operatorname{Id},g)\\},\
D_{2,3}=\\{(g,\operatorname{Id})\\}$

and the triple diagonal is the point
$D=(\operatorname{Id},\operatorname{Id}).$

The projections $\Pi_{L}=\Pi_{R}$ both map to a point so
$\operatorname{null}((\Pi)_{R})_{*})=T_{\operatorname{Id}}G$ is a ‘bundle’
over $G[1].$ Consider the analogues of the operations carried out in the case
of a fibre bundle above.

  1. (1)

The null space of $(\Pi_{F})_{*}$ at a point $(g,g)\in D_{1,2}$ is
$T_{g}G\times\\{0\\}.$ Under $\Pi_{S},$ $(e^{tv}g,g)$ is mapped to $e^{tv}$ so
$W=(R_{v},0)$ on $D_{1,2}.$

  2. (2)

Thus the image $(\Pi_{C})_{*}W$ is $\tilde{v}=R_{v},$ the right-invariant
vector field on $G$ evaluating to $v$ at $\operatorname{Id}.$

Thus the simplicial construction, formally the same as that for a fibration,
yields the (right) Lie algebra structure on $E=T_{\operatorname{Id}}G.$

######  Definition 3.

A _simplicial compactification_ of a Lie group, $G,$ is a generalized product
$\overline{G}[*],$ in the category of compact manifolds with corners, reducing
to (1.9) in the interior.

Of course for a compact Lie group (1.9) is a simplicial compactification.

######  Conjecture 1.

Any real reductive Lie group with connected centre has a simplicial
compactification.

It may be that this can be seen using results such as in [1]. Two
(commutative) examples of simplicial compactification arise below. For a
vector space, as an abelian Lie group, a simplicial compactification arises
from the construction of the semiclassical stretched product below, through
restriction to the (one) boundary hypersurface as in Theorem 1. For the
multiplicative group of the positive real numbers a simplicial
compactification is produced in §9, it is not the same as the compactification
of the additive group on $\mathbb{R},$ even though the two groups are
diffeomorphic.

There is frequently a larger group of diffeomorphisms preserving the structure
maps of a generalized product beyond the permutation action. If a group $K$
acts in the sense that there is a homomorphism
$K\longrightarrow\operatorname{Dif}(M[k])$ into the diffeomorphism group for
all $k$ and these are intertwined by the structural maps then one can form
bundles with fibres modelled on the generalized product and cocycle in $K.$

One special case of the notion of generalized product is particularly common,
that of a ‘stretched product’. For this we require that the interiors be the
fibre products of a fibration. In typical cases these interiors are actually
the interiors of a fibre bundle of compact manifolds with corners, so that the
stretched products typically arise by ‘resolution’ but we do not demand this
explicitly.

######  Definition 4.

A _stretched product_ is a generalized product $M[k]$ in the category of
compact manifolds with corners such that there is a fibration
$\phi:\operatorname{int}(M[1])\longrightarrow Y^{\prime}$ and the interiors of
the $M[k]$ are identified with their structural maps with the fibre products
of $\phi.$

It turns out that there may be different generalized products compactifying a
given open generalized product, and in particular a particular open fibre
product. So identifiers are used to distinguish them and we write $M[k;T]$ for
the generalized or stretched product with identifier ‘$T$’. As already
suggested, in typical exampes of a stretched product the interior fibration
$\phi$ extends to a fibration of compact manifolds with corners
$\phi:M[1;T]\longrightarrow Y$ and the notation is intended to suggests that
$M[k;T]$ is defined through a resolution process, such as iterated blow-up,
from the fibre products of the $M=M[1;T].$ In the examples of stretched
products below the $M[k;T]$ are constructed by blow-up of boundary
p-submanifolds of these $M^{[k]}_{\phi}.$ This leads directly to symmetry and
then the maps $\Pi_{k}^{T}$ and $D_{1,2}^{T}$ in Lemma 3 are shown to exist
through commutation and lifting properties of blow-up.

Proposition 1 applies to stretched products but (as seen in the adiabatic
example below) the resulting generalized product for a boundary hypersurface
is not a stretched product. Whilst in principle ‘simpler’ than the stretched
product from which they come these boundary generalized products typically
have a factor related to the group example above, i.e. the associated groupoid
(discussed in §6 below) has a non-trivial null space to its anchor map.

##  2\. Manifolds with corners

We briefly recall the concepts involved in Definition 1 and used freely in the
remainder of this paper.

First, to give a direct (if not particularly natural) definition of manifolds
with corners, consider a compact manifold, $\hat{M},$ without boundary and a
_hypersurface involution_. This is a smooth involution, $R,$ of the manifold
which has fixed points and at each of them the differential of $R$ has exactly
one negative eigenvalue. It follows that

$\hat{H}_{R}=\\{m\in\hat{M};Rm=m\\}$

is a closed embedded separating hypersurface (possibly with several
components). In particular there exists
$\xi\in{\mathcal{C}}^{\infty}(\hat{M})$ such that $R^{*}\xi=-\xi$ and
$\hat{H}=\\{\xi=0\\}$ with $d\xi\not=0$ at $\hat{H}.$

Suppose $\hat{M}$ is equipped with $N$ such commuting involutions, $R_{j},$
$j=1,\dots,N$ and corresponding $\xi_{j}$ such that for any subset
$I\subset\\{1,\dots,N\\}$ and point $m$ at which $\xi_{j}(m)=0$ for all $j\in
I$ the $d\xi_{j}(m)$ are linearly independent. Then the space

(2.1) $M=\\{m\in\hat{M};\xi_{j}\geq 0\ \forall\ j\\},\
{\mathcal{C}}^{\infty}(M)={\mathcal{C}}^{\infty}(\hat{M})\big{|}_{M}$

is a compact manifold with corners. Provided it is required as part of the
definition that the boundary hypersurfaces of a manifold with corners be
embedded (as we do), this is equivalent to more standard definitions.

Taking this as the definition, the boundary hypersurfaces of $M$ are the
components of the fixed hypersurfaces $\hat{H}_{i}\cap M;$ the set of them
denoted $\mathcal{M}_{1}(M).$ Each $H_{j}\in\mathcal{M}_{1}(M)$ has a defining
function $0\leq x_{j}\in{\mathcal{C}}^{\infty}(M)$ such that
${\mathcal{C}}^{\infty}(M)x_{j}$ is the ideal of functions vanishing on
$H_{j}.$ These $x_{j}$ are boundary defining functions of $M.$ Note that since
the $\hat{H}_{i}\cap M$ may have several components they are in general
disjoint unions of boundary hypersurfaces and the $\xi_{i}$ restricted to $M$
become ‘collective’ boundary defining functions, i.e. products of the
corresponding $x_{j}.$ The dimension, $n,$ of $M$ is the dimension of
$\hat{M}.$ At any point of $M$ there are ‘adapted’ coordinates consisting of
$k\geq 0$ of the $x_{j}$ which vanish there and $n-k$ locally smooth functions
$y_{l}$ together giving a diffeomorphism of an open neighbourhood of the point
to $[0,\epsilon)^{k}\times(-\epsilon,\epsilon)^{n-k}$ for some $\epsilon>0.$
We call these simply ‘coordinates’ on $M.$

The boundary faces of codimension $k,$ forming $\mathcal{M}_{k}(M),$ are the
components of non-trivial intersections of $k$ boundary hypersurfaces.

A map $F:M_{1}\longrightarrow M_{2}$ between compact manifolds with corners is
smooth if
$F^{*}:{\mathcal{C}}^{\infty}(M_{2})\longrightarrow{\mathcal{C}}^{\infty}(M_{1}).$
Writing complete sets of boundary defining functions $x_{j}^{\prime}$ for
$M_{2}$ and $x_{k}$ for $M_{1},$ a smooth map is a _b-map_ if for each $j$ one
of three possibilities holds

(2.2) $F^{*}x_{j}^{\prime}\equiv 0,\
F^{*}x_{j}^{\prime}=\alpha_{j}\prod\limits_{k}x_{k}^{a(j,k)},\
\alpha_{j}>0\text{ or }F^{*}x_{j}^{\prime}>0.$

Here the $a(j,k)$ are necessarily non-negative integers but may vanish and the
final case is the special form of the second when all the powers vanish. If
the first case does not occur, so the image of $F$ does _not_ lie in a
boundary hypersurface, $F$ is called an interior b-map. Clearly the composite
of b-maps is a b-map and the compact manifolds with corners form a category
with the b-maps as the morphisms (or with interior b-maps as morphisms).

The smooth vector fields on $M$ which are tangent to all boundaries form a Lie
algebra $\mathcal{V}_{\operatorname{b}}(M)$ which is a module over
${\mathcal{C}}^{\infty}(M)$ and locally has a basis, i.e.
$\mathcal{V}_{\operatorname{b}}(M)={\mathcal{C}}^{\infty}(M;{}^{\operatorname{b}}TM)$
for a smooth vector bundle over $M$ with a natural bundle map
${}^{\operatorname{b}}TM\longrightarrow TM$ of corank $k$ at a boundary point
of codimension $k.$ In local coordinates ${}^{\operatorname{b}}TM$ has the
local basis $x_{j}\partial_{x_{j}},$ $\partial_{y_{l}}.$ Thus
$\mathcal{V}_{\operatorname{b}}(M)$ is a _Lie algebroid_. An interior b-map
has a b-differential
$F_{*}:{}^{\operatorname{b}}T_{m}M_{1}\longrightarrow{}^{\operatorname{b}}T_{F(m)}M_{2}$
consistent with the usual differential. For a boundary b-map one must replace
${}^{\operatorname{b}}TM_{2}$ by ${}^{\operatorname{b}}TB$ where $B$ is the
smallest boundary face containing the image of $F.$

To complete the explanation of the Definition 1 we need to define p-embeddings
and b-fibrations which are the analogues of embeddings and fibrations in this
category.

A _p-submanifold_ of $M$ is a closed (and generally assumed connected here)
subset everywhere locally of the form $x_{i}=0,\ 1=1,\dots,d^{\prime},$
$y_{l}=0,\ l=1,\dots,n-p^{\prime}$ in terms of some coordinates. Thus it has a
_p_ roduct decomposition consistent with that of the manifold. It is always a
manifold with corners. A _p-embedding_ is a diffeomorphism of a manifold with
corners onto a p-submanifold. An interior p-submanifold is one with no
component contained in a boundary hypersurface, equivalently it is locally
defined by the vanishing of interior coordinates.

A _b-fibration_ $F:M_{1}\longrightarrow M_{2}$ between compact manifolds with
corners is a surjective b-map with the property that near each point $m\in
M_{1}$ there are local coordinates $x,y$ and also local coordinates
$x^{\prime},y^{\prime}$ near $F(p)\in M_{2}$ such that the
$F^{*}y^{\prime}_{l^{\prime}}$ are amongst the $y_{l}$ and

(2.3) $F^{*}x^{\prime}_{j^{\prime}}=\prod_{j\in
J(j^{\prime})}x_{j}^{\alpha(j^{\prime},j)}\text{ where }J(j^{\prime}_{1})\cap
J(j^{\prime}_{2})=\emptyset\text{ if }j^{\prime}_{1}\not=j^{\prime}_{2}.$

Thus each local defining function for the boundary of $M_{1}$ near $p$ can
occur in at most one of the product decompositions. The only case which arises
below is of a _simple b-fibration_ where the $\alpha(j,j^{\prime})$ are either
$0$ or $1;$ we therefore subsume this in the notion of a b-fibration. As for a
fibration (which corresponds to the case that at most one
$\alpha(j^{\prime},j)=1$ for each $j^{\prime})$ surjectivity is automatic if
$M_{2}$ is connected.

######  Proposition 2.

The inverse image of an interior p-submanifold under a b-fibration is an
interior p-submanifold.

###### Proof.

This follows from the local structure of manifold and the map. ∎

The inverse image of a boundary p-submanifold under a b-fibration is a union
of boundary p-submanifolds.

##  3\. Blow up

We also recall the notion of blow-up, which is well-defined for any
p-submanifold $X\subset M$ of a manifold with corners. The construction of $M$
blown up along the centre $X$ is closely related to the existence of a collar
neighbourhood, or normal fibration, for $X$ in $M.$ The result is a manifold
with corners (functorially associated to the pair $(M,X))$ with associated
blow-down (b-)map

(3.1) $\beta:[M;X]\longrightarrow M.$

We will usually consider the case where the centre of blow-up, $X,$ is a
boundary p-submanifold. As a set

(3.2) $[M;X]=S^{+}X\sqcup(M\setminus X),$

the blown-up space is the disjoint union of the complement of $X$ in $M$ and
the inward-pointing part of the spherical normal bundle to $X.$ Thus if $p\in
X$ and $x_{j}$ are a maximal set of $k$ local boundary defining functions for
$M$ near $p$ which vanish identically on $X$ (so $X$ is contained in a
boundary face of codimension $k)$ then

$S^{+}_{p}X=N^{+}_{p}X/T_{p}X,\ T^{+}_{p}X=\\{v\in T_{p}M;vx_{j}(p)\geq 0\\}.$

These form a bundle of the $k$ positive parts of spheres,
$\mathbb{S}^{k}_{+}=\mathbb{S}^{k}\cap[0,\infty)^{k+1},$ over $X.$ The well-
defined ${\mathcal{C}}^{\infty}$ structure on $[M;X]$ makes $S^{+}X$ into a
boundary hypersurface (the ‘front face’ of the blow-up). The
${\mathcal{C}}^{\infty}$ functions near a point of the front face are those
which are smooth in polar coordinates in the normal variables to $X$ and in
tangential variables. The blow-down map is the identity away from the front
face and a fibration of the front face onto the centre.

######  Lemma 5.

A diffeomorphism, $F,$ of $M$ lifts to a diffeomorphism of $[M;X]$ to
$[M;F(X)];$ so $F$ lifts to a diffeomorphism of $[M;X]$ if and only if it
restricts to a diffeomorphism of $X.$

In the sequel we encounter iterated blow-ups. As noted in the Introduction
several of the results of thus section ard clisely related go unpublished work
with Michael Singer.

######  Definition 5.

A finite collection of p-submanifolds is _p-clean_ (generalizing the notion
due to Bott) if near each point of the union there are local coordinates
$x_{j},$ $y_{l}$ in which each submanifold in the set, containing that point,
is defined by a subset of the $x_{j}$ and some linear functions of the
$y_{l}.$

For any two manifolds in such a p-clean collection there is, near each point
of intersection, a joint product decomposition with the manifold. In general
there is not a simultaneous product decomposition of all the elements passing
through a given point.

Under the blow-up of one p-submanifold of a p-clean collection the lift, or
proper transform, of any other element is well-defined as either the inverse
image, if the manifold to be lifted is contained in the centre of blow-up, or
as the closure of the inverse image of the complement of the centre if not. In
either case it is a p-submanifold of the blown up manifold. To proceed with an
iterated blow up it needs to be checked that successive centres are
p-submanifolds when the preceding ones have been blown up.

######  Definition 6.

An order of a p-clean collection of submanifolds is _permissible for blow up_
if at each step the lift of the next element is a p-submanifold.

See the work of Kottke and Rochon, [4], for related ideas.

Note that for collections of boundary faces any order of blow-up is
permissible, since the lift of a boundary face under blow up of another is
always a boundary face. However the final result may depend on the order.

A _dimension order_ for a p-clean collection is an order in which the
dimension is (weakly) increasing.

######  Proposition 3.

If $\mathcal{F}$ is a p-clean collection of p-submanifolds of a compact
manifold with corners, $M,$ which is closed under intersection then any
dimension order is permissible; the lifts of the remaining elements at each
stage form a p-clean collection closed under intersection, and the result,
denoted $[M;\mathcal{F}],$ is independent of choice.

###### Proof.

Certainly the elements are partially order by dimension. The elements of
minimal dimension are necessarily disjoint and hence can be blown up in any
order. Since no other element can be contained in an element of minimal
dimension, $F,$ the lifts of the other elements are the closures of the
inverse images of the complements of $F$ under the blow-down map. The
condition that these form a p-clean collection is everywhere local and
certainly satisfied away from the front face of the blow-up. Consider a point
$p$ in the front face and coordinates $x^{\prime}_{i},$
$x^{\prime\prime}_{j},$ $y^{\prime}_{l},$ $y^{\prime\prime}_{p},$ based at the
image $\beta(p)$ under the blow-down map where $F$ is locally defined by
$x^{\prime}=0,$ $y^{\prime}=0.$ If an element of the p-clean collection
contains $\beta(p)$ then it must contain $F$ locally since otherwise the
intersection with $F$ would be smaller. Thus we can suppose that the other
elements through $\beta(p)$ are each defined by some subset of the
$x^{\prime}_{i}$ and linear functions of the $y^{\prime}_{l}.$ One of the
$x^{\prime}_{i}$ or $y^{\prime}_{l},$ denoted $r,$ dominates the others near
$p$ and it is then a defining function for the front face and the remaining
$x^{\prime}_{i}/r$ and $y^{\prime}_{l}/r$ are coordinates on the front face
near $p.$ Consider the elements which lift to contain $p.$ They are
transversal to the front face so $dr\not=0$ on them and they are each defined
locally by some of the $x^{\prime}_{i}/r$ and some linear functions of the
$y^{\prime}_{l}/r.$ By subtracting the values at $p$ we may renormalize the
(non-trivial) $y^{\prime}_{l}/r$ to give $Y^{\prime}_{l}$ which vanish at $p.$
Then the lifted manifolds are defined by the appropriate $x^{\prime}_{r}/r$
and affine functions of the $Y^{\prime}_{l}.$ To vanish at $p$ the affine
functions must be linear. This shows that the lifted collection is p-clean.

To see that the lifted collection is closed under intersection it suffices to
note that the lift of the intersection is the intersection of the lifts except
in the case two manifolds intersect in $F;$ then their lifts are disjoint.

Thus the iterative blow-up is well-defined and the only choice at each level
is irrelevant one of the order of blow-up of the disjoint elements of minimal
dimension. ∎

######  Lemma 6.

In any permissible order of blow-up for a p-clean family, elements which are
disjoint or transversal at any stage lift to be disjoint or transversal at all
subsequent stages.

###### Proof.

Blow up is essentially local so disjoint elements lift to be disjoint. If two
elements are transversal, the centre cannot contain both, since then it would
have maximal dimension. If it contains one it is transversal to the other and
then the lift of the latter is also equal to its preimage from which the
transversality of the lifts follows. ∎

######  Lemma 7.

If $F_{1}\subset F_{2}$ are two p-submanifolds then

(3.3) $[M;F_{1};F_{2}]=[M;F_{2};F_{1}]$

are naturally diffeomorphic; the same is true if $F_{1}$ and $F_{2}$ are
p-clean and transversal.

###### Proof.

Away from the preimage of $F_{1}$ both blow-ups reduce to the blow-up of
$F_{2}\setminus F_{1}.$ Near $F_{1}$ both are bundles over $F_{1}$ with the
blow-ups being in the fibres. Thus we are reduced to the case that $F_{1}$ is
the origin in $(y,z)\in\mathbb{R}^{p}\times\mathbb{R}^{q}$ and
$F_{2}=\\{y=0\\}.$ Then both blow-ups represent completions of $y\not=0,$ to a
manifold with corners.

Blowing up $F_{1}$ first introduces $R=(|y|^{2}+|z|^{2})^{\frac{1}{2}},$
$y/R,$ $z/R$ as smooth functions generating the ${\mathcal{C}}^{\infty}$
structure. Then blowing $y/R=0$ the functions $r=|y|/R$ and $y/|y|$ become
smooth in $|y|<\epsilon R.$ Note however that in $|y|>\frac{1}{2}\epsilon R,$
$R/|y|$ is smooth so in fact $y/|y|$ and $r$ are globally smooth and induce
the ${\mathcal{C}}^{\infty}$ structure on the blow-up with $r$ and $R$ as
boundary defining functions.

Proceeding in the opposite order the gives first the polar coordinates $s=|y|$
and $y/|y|,$ then blowing up $s=0=|z|$ introduces
$R=(s^{2}+|z|^{2})^{\frac{1}{2}},$ $r=s/R$ and $z/R$ as smooth functions. This
way the manifold looks like the product of the sphere in $y$ and the half-
space in $(s,z)$ blown up at the origin, i.e. the product of a sphere, a half-
sphere and a half-line in $R.$

It follows that the functions on the complement of the larger submanifold
which extend to define the ${\mathcal{C}}^{\infty}$ structures on the two
resolutions are the same, so the two blow-ups are naturally diffeomorphic.
This natural diffeomorphism shows that the blow-up of a sphere along a
coordinate subsphere is the product of a sphere and a half-sphere. ∎

######  Proposition 4.

If three submanifolds $F_{i},$ $i=1,2,3$ meet p-cleanly in a manifold $M$ and
satisfy

(3.4) $F_{1}\cap F_{2}\subset F_{3},\ \text{ neither }F_{1}\text{ nor
}F_{2}\subset F_{3}$

then the lifts of $F_{1}$ and $F_{2}$ to $[M;F_{3}]$ are disjoint.

###### Proof.

The lifts of $F_{1}$ and $F_{2}$ are the closures of their preimages, so are
disjoint away from the front face, since it contains the preimage of the
intersection. So it suffices to show that these lifts do not have a common
point in the front face above any chosen point of $F_{1}\cap F_{2}.$ We may
introduce coordinates near this in which $M=V$ is linear and the manifolds are
linear subspaces. The front face is the (partial) sphere in the normal space
$V/F_{3}.$ Since $F_{1}\cap F_{2}\subset F_{3}$ the intersections of the lifts
of $F_{1}$ and $F_{2}$ with the front face are spheres in non-trivial
subspaces which only meet at the origin. Thus they are disjoint. ∎

######  Proposition 5.

If $F_{i}\subset M,$ $i=1,2,3$ are a p-clean collection satisfying

(3.5) $F_{3}\subset F_{1},\ F_{2}\setminus F_{1}\not=\emptyset,\ F_{3}\text{
transversal to }F_{2}\cap F_{1}\text{ within }F_{1}$

then $F_{2}$ and $F_{3}$ lift to be transversal in $[M;F_{1}].$

###### Proof.

After the blow-up of $F_{1},$ the lift of $F_{2}$ is transversal to the front
face. So it suffices to show that the intersection of $F_{2}$ with the front
face is transversal to the lift of $F_{1}.$ The latter is the union of fibres
of the front face, so this reduces to the transversality of the quotients by
the blow up within $F_{1}$ which holds by hypothesis. ∎

######  Proposition 6.

Suppose $\mathcal{F}$ is a p-clean collection which can be written

(3.6) $\mathcal{F}=\mathcal{G}\sqcup\mathcal{R}$

where $\mathcal{G}$ and $\mathcal{R}$ are separately closed under intersection
and the intersection of an element of $\mathcal{G}$ with an element of
$\mathcal{R}$ lies in $\mathcal{R},$ then an order in which the $\mathcal{G}$
appear first and then the $\mathcal{R},$ both in dimension order, is
permissible and results in the same space

(3.7) $[M;\mathcal{F}]=[[M;\mathcal{G}];\mathcal{R}].$

The lifts of the colllection $\mathcal{R}$ in (3.7) will not in general be a
p-clean collection but they are individually p-submanifolds and the blow up is
permissible.

###### Proof.

We prove the iterative statement that choosing any element $R_{j}$ of
$\mathcal{R}$ and taking a dimension order on
$\mathcal{G}\cup\\{R\in\mathcal{R};R<R_{j}\\}$ followed by a dimension order
on $\\{R\geq R_{j}\\}$ is permissible and gives the same space. To prove this
in turn we proceed iteratively with a dimension order on
$\mathcal{G}\cup\\{R\in\mathcal{R};R<R_{j}\\}$ except that $R_{j}$ appears $k$
steps above its dimension order, followed by $\\{R>R_{j}\\}$ in dimension
order. For $k=0$ this is the statement for $j+1$ and for maximal $k$ for $j$
so we iterate down in $j.$

Consider the situation at $k=0$ where
$\mathcal{F}\cup\\{R\in\mathcal{R};R<R_{j}\\}$ has been blown up in dimension
order and we can suppose that $R_{j}$ is last of its dimension to be blown up.
The manifolds of a given dimension are all disjoint at the time they are to be
blown up since all their intersections have been blown up earlier. Thus it is
enough to consider the remaining manifolds which are not disjoint from $R_{j}$
at this juncture. Again, these are the manifolds $G\in\mathcal{G}$ with
intersection $G\cap R_{j}$ not of smaller dimension than $R_{j}$ and hence
must be equal to it. Thus we only need consider all the manifolds in
$\mathcal{G}$ which contain $R_{j}.$ Denote the first of these, $G_{j}.$ The
remaining elements of $\mathcal{G}$ which contain $R_{j}$ must contain $G_{j}$
since the intersection would otherwise have been blown up earlier and they
would be disjoint.

By Lemma 7 we can indeed reverse the order of blow up of $R_{j}$ and $G_{j}.$
With $G_{j}$ blown up first, $R_{j}$ lifts to its inverse image, contained in
the front face so produced. It follows from Lemma 5 that the remaining
relevant elements of $\mathcal{G}$ are transversal to the lift of $R_{j}$
after the blow up of $G_{j}$ so the commutation can be continued from $k=0$ to
maximal $k$ with the final space always the same. ∎

######  Lemma 8.

If $\phi:M\longrightarrow N$ is a b-fibration and $D\subset M$ is a
p-submanifold such that $F=\phi^{-1}(D)$ is a p-submanifold then $\phi$ lifts
by continuity to a b-fibration

(3.8) $\phi:[M;F]\longrightarrow[N;D].$

###### Proof.

Taking defining functions $x,$ $y$ for $D$ near a point it follows that their
inverse images must be defining functions for $F$ near any point in that fibre
of $\phi.$ This brings $\phi$ to the local form

(3.9) $\operatorname{Id}\times f:U\times F^{\prime}\longrightarrow U\times
D^{\prime},\ f:F^{\prime}\longrightarrow D^{\prime}\text{ a b-fibration}$

where $U$ is a ball in $x,y$ and $D^{\prime}$ is an open set in $D$ with
$F^{\prime}$ its preimage. The blow up is now of the origin in $U$ on both
sides and the result follows. ∎

######  Proposition 7.

If $\phi:M\longrightarrow N$ is a b-fibration of compact manifolds with
corners and $C\subset M$ is a boundary p-submanfold not contained in a
boundary face of codimension greater than one and such that
$\phi:C\longrightarrow H$ is a b-fibration with image a boundary hypersurface
of $N$ then the composite

(3.10) $\beta\circ\phi:[M;C]\longrightarrow N\text{ is a b-fibration.}$

###### Proof.

The composite (3.10) is certainly a b-map. Suppose $C\subset H^{\prime}$ where
$H^{\prime}$ is the hypersurface containing $C$ – by assumption it is then the
minimal boundary face doing so, i.e. interior points of $C$ have codimension
$1$ in $M.$ Consider the preimage in $C$ of a general point $p\in H$ of
codimension $q+1,$ $q\geq 0,$ i.e. having codimension $q$ as a point in $H.$
Let $m\in C$ be in the preimage under $\phi$ of $p.$ Near $m$ there are local
coordinates in $N$ near $p,$ $x,$ $y$ and $\xi$ $y,z$ in $M$ near $m$ such
that $H=\\{x_{1}=0\\},$ $C=\\{\xi_{1}=0,\ z=0\\}$ with $\phi^{*}x_{1}=\xi_{1}$
and the $\phi^{*}x_{j}$ $j>1$ products (without common factors) of the
$\xi_{p},$ $p>1.$ Thus $R=(|z|^{2}+\xi_{1}^{2})^{\frac{1}{2}}$ is a defining
function for the front face of $[M;C]$ and $r=\xi_{1}/R$ is a defining
function for the lift of $H^{\prime}.$ Thus the lift of $x_{1}$ becomes $rR$
and the other lifts remain unchanged. It follows that $\phi\circ\beta$ is
everywhere locally, and hence globally, a b-fibration. ∎

##  4\. Diagonals and multidiagonals

Consider a compact generalized product, where the manifolds $M[k]$ are all
supposed connected. The (multi)diagonals in $M[k]$ are labelled by partitions,
$\mathfrak{p},$ of $J(k)$ corresponding to disjoint subsets $\mathcal{P}_{l},$
$l=1,\dots,L(\mathfrak{p}),$ each with at least two elements, the remainder
being singletons. The corresponding diagonal

(4.1) $D_{\mathfrak{p}}=D_{\mathfrak{p}}^{(k)}\subset M[k]$

is the image of a p-embedding. In general there are several choices of
surjective map $I:J(k)\longrightarrow J(k_{\mathfrak{p}}),$ where
$k_{\mathfrak{p}}$ is $L(\mathfrak{p})$ plus the number of singletons such
that $D_{\mathfrak{p}}$ is the image of
$S_{I}:M[k_{\mathfrak{p}}]\longrightarrow M[k].$ It is convenient to make a
specific choice. Namely we take

(4.2) $I_{\mathfrak{p}}:J(k)\longrightarrow J(k_{\mathfrak{p}})$

to be the map which sends each of the sets $\mathcal{P}_{j},$ constituting
$\mathfrak{p},$ to $j$ and is strictly increasing on the singletons. The
associated p-embedding will also denoted $D_{\mathfrak{p}}.$

The functorial properties of the generalized product show that the image
$D_{\mathfrak{p}}$ under any of the structural diffeomorphisms of $M[k],$
forming a representation of the permutation group, is another
$D_{\mathfrak{p}^{\prime}}.$

Under the partial order of partitions, where $\mathfrak{q}<\mathfrak{p}$ if
every $\mathcal{Q}_{j}$ is contained in some $\mathcal{P}_{l},$

(4.3) $\mathfrak{q}<\mathfrak{p}\Longrightarrow D_{\mathfrak{p}}\subset
D_{\mathfrak{q}}$

since the surjection defining $D_{\mathfrak{p}}$ can be factored through that
defining $D_{\mathfrak{q}}.$

Consider the diagonals, $D_{i,j},$ corresponding to a partition with one non-
trivial subset containing just two distinct element, $i$ and $j.$ For any
partition $\mathfrak{p}$ there is a structural diffeomorphism mapping
$D_{\mathfrak{p}}$ to $D_{\mathfrak{q}}$ where the elements of each
$\mathfrak{q}_{j}$ are successive integers in $J(k)$

(4.4) $\mathcal{Q}_{j}=\\{r_{j},r_{j}+1,\dots,r_{j}+k_{j}\\}.$

It follows by functoriality that

(4.5)
$D_{\mathfrak{q}}\subset\bigcap_{\\{i;i,i+1\in\mathcal{Q}_{j}\\}}D_{i,i+1}$

where subsets of $J(k)$ with at least two elements can be identifed with
partitions. Note that the corresponding $D_{\mathcal{P}}$ are the true
diagonals for a fibration and we think of $D_{i,j}$ corresponding to
$\mathcal{P}=\\{i,j\\}$ as a ‘simple diagonal’. Indeed, the map
$J(k)\longrightarrow J(k^{\prime})$ defining the diagonal can be factored
through the $J(k)\longrightarrow J(k-1)$ defining any one of the simple
diagonals on the right in (4.5).

We proceed to show that there is equality in (4.5) (which of course is trivial
unless $k\geq 3)$ but first consider a single simple diagonal. Let
$\Pi_{i,j}:M[k]\longrightarrow M[2],$ for $i<j\leq k,$ be the structural
b-fibrations corresponding to the injective map $J(2)\longrightarrow J(k)$
taking $1$ to $i$ and $2$ to $j.$

######  Lemma 9.

The diagonal $D_{i,j}$ in $M[k]$ for $i<j$ and $k\geq 3$ satisfies

(4.6) $D_{i,j}=\Pi_{i,j}^{-1}(D).$

###### Proof.

The basic diagonal $D\subset M[2]$ is the image of the embedding corresponding
to the unique map $J(2)\longrightarrow J(1).$ For any $i<j$ the functorial
properties show that

(4.7) $\Pi_{i,j}D_{i,j}=DS_{I},\ I:J(1)\longrightarrow J(k-1),\ I(1)=i.$

Thus it follows that as submanifolds $\Pi_{i,j}(D_{i,j})\subset D.$ The
inverse image, $\Pi_{i,j}^{-1}(D),$ of an interior p-submanifold under a
b-fibration is an interior p-submanifold and it follows that the dimension is
$m+(k-2)\kappa.$ Since this is the same as that of $M[k-1]$ and hence of the
submanifold $D_{i,j},$ $D_{i,j}$ must be a component of $\Pi_{i,j}^{-1}(D).$
Since we are requiring the connectivity of the fibres of b-fibrations (4.6)
holds. ∎

######  Lemma 10.

The p-submanifolds $D_{1,2}$ and $D_{1,3}$ in $M[k]$ for $k\geq 3$ intersect
in the diagonal $D_{\mathcal{P}}$ corresponding to the single subset
$\mathcal{P}=\\{1,2,3\\}$ in $J(k)$ and the intersection is transversal.

The conormal directions here can be thought of as in the dual to the b-tangent
space but since all the manifolds are interior p-submanifolds, this is the
same as the ordinary conormals.

###### Proof.

Consider the b-fibration $\Pi^{k}_{3}:M[k]\longrightarrow M[k-1]$
corresponding to the increasing injection $J(k-1)\longrightarrow J(k)$ with
image omitting $3.$ By functoriality the image $\Pi^{k}_{3}(D_{1,2})$ is the
corresponding simple diagonal $D_{1,2}\subset M[k-1].$ However $\Pi^{k}_{3}$
is a diffeomorphism of $D_{1,3}$ to $M[k-1].$ It follows that the intersection
is transversal, since a fibre of $\Pi^{k}_{3}$ which is contained in $D_{1,2}$
is mapped into a point in $D_{1,2}$ so must be transversal to $D_{1,3}$ at the
intersection. Moreover a point in $D_{1,3}$ is uniquely of the form
$D_{1,3}(m)$ with $m\in M[k-1]$ and $\Pi^{k}_{3}(D_{1,3}(m))=m.$ Thus if
$D_{1,3}(m)$ is in the intersection, i.e. $D_{1,3}(m)\in D_{1,2},$ it follows
that $\Pi^{k}_{3}(m)\in D_{1,2}\subset M[k-1].$ The intersection is therefore
the image of $D_{1,2}\subset M[k-1]$ under $D_{1,3}$ and this is functorially
identified with the diagonal $D_{\mathcal{P}}\subset M[k]$ for
$\mathcal{P}=\\{1,2,3\\}.$ ∎

######  Lemma 11.

For any $3\leq l\leq k$ the intersection

(4.8) $D_{1,2}\cap D_{2,3}\cap\dots\cap D_{l-1,l}\text{ in }M[k]$

is transversal and equal to the diagonal corresponding to the single subset
$\mathcal{P}=\\{1,\dots,l\\}\subset J(k).$

###### Proof.

Lemma 10 is this statement for $l=3.$ We proceed iteratively in $l$ assuming
the result for $l<L$ and all $k.$ Since the structural diffeomorphisms can be
used to shift the indices we can assume that transversality holds for

$D_{1,2}\cap\dots\cap D_{L-2,L-1}\text{ in }M[k-1]$

with the intersection identified with $D_{\mathcal{P}},$
$\mathcal{P}=\\{1,\dots,L-1\\}.$ The b-fibration
$\Pi_{1}^{k}:M[k]\longrightarrow M[k-1],$ corresponding to the increasing
injective map from $J(k-1)$ to $J(k)$ omitting $1$ maps $D_{1,2}$ onto
$M[k-1]$ and each $D_{j,j+1}$ for $j>1,$ onto $D_{j-1,j}\subset M[k-1].$ From
this we deduce the inclusion of the left in the right in

(4.9) $D_{2,3}\cap\dots\cap D_{L-1,L}=\Pi^{k}_{L}\left(D_{1,2}\cap\dots\cap
D_{L-2,L-1}\right)\text{ in }M[k].$

Equality follows from the equality of dimensions (4.6). Since the fibres of
$\Pi^{k}_{L}$ at a point of the intersection (4.8) contained in $D_{1,2}$
cannot be tangent to $D_{2,3}\cap\dots\cap D_{l-1,l}$ transversality follows,
as does the identification of the intersection. ∎

######  Theorem 1.

The diagonals in a compact (connected) generalized product form a p-clean
collection closed under intersection. The ‘true diagonals’ $D_{\mathcal{P}}$
for single subsets $\mathcal{P}\subset J(k)$ form a p-clean collection closed
under non-transversal intersection.

###### Proof.

Consider a diagonal $D_{\mathcal{P}}$ in $M[k].$ Using the structural
diffeomorphisms it suffices to assume that $\mathcal{P}=\\{1,2,\dots,l\\},$
$2\leq l\leq k.$ Then $D_{\mathcal{P}}\subset D_{i,j}$ if
$\\{i,j\\}\subset\mathcal{P}$ and conversely the diagonals containing
$D_{\mathcal{P}}$ are those corresponding to partitions contained in
$\mathcal{P}.$

We know that the b-fibration $\Pi_{1,j}:M[k]\longrightarrow M[2],$ $2\leq
j\leq l,$ restricts to a b-fibration of $D_{1,j}$ over $D$ and that

(4.10) $D_{\mathcal{P}}=D_{1,2}\cap D_{1,3}\dots\cap D_{1,l}\text{ is
transversal.}$

Moreover the image of a point in $D_{\mathcal{P}}$ under each of the
$\Pi_{1,j},$ $2\leq j\leq l,$ is the same. Since, by functoriality,
$\Pi_{1,j}R_{1,j}=R\Pi_{1,j}$ and (hence) $D$ is a component of the fixed
point set of $R$ near a point in $D.$ So we may choose $\kappa$ independent
functions, denoted collectively $z,$ each odd under $R$ and such that
$D=\\{z=0\\}$ locally. Then the functions

(4.11) $z_{j}=(\Pi_{1,j})^{*}z,\ j=2,\dots,l\text{ satisfy
}R_{1,j}^{*}z_{j}=-z_{j},\ D_{1,j}=\\{z_{j}=0\\}\text{ locally.}$

By Lemma 11 their differentials must (all) be independent.

Now consider the other $D_{i,r}$ where $1,i<r\leq l.$ The structural identity

(4.12)
$\Pi_{1,i}R_{i,r}=\Pi_{1,r}\Longrightarrow(R_{ir})^{*}(z_{i}-z_{r})=z_{r}-z_{i}.$

Since the components of $z_{r}-z_{i}$ are linearly independent it follows that

(4.13) $D_{i,r}=\\{z_{i}-z_{r}=0\\}\text{ locally.}$

Thus all the diagonals containing one $D_{\mathcal{P}}$ for
$\mathcal{P}\subset J(k)$ form a p-clean collection. The general case follows
from this since if $\mathfrak{p}$ consists of several disjoint sets
$\mathcal{P}_{j}$ then

(4.14) $D_{\mathfrak{p}}=\bigcap_{i}D_{\mathcal{P}_{i}}$

is transversal and the local linear defining functions (4.13) can be
constructed independently. ∎

##  5\. The Lie algebroid

As noted above a generalized product is associated to a Lie algebroid, which
it resolves. Since it is a b-fibration the null spaces of the differentials
$(\Pi_{R})_{*}\subset{}^{\operatorname{b}}TM[2]$ form a vector bundle.

######  Definition 7.

The vector bundle $E$ over $M[1]$ is defined as the (abstract) pull-back

(5.1) $E=D^{*}\operatorname{null}((\Pi_{R})_{*})).$

######  Theorem 2.

The space $\mathcal{E}$ of smooth sections of $E$ over $M[1]$ is a Lie
algebroid with anchor map

(5.2)
$(\Pi_{L})_{*}:\mathcal{E}\longrightarrow{\mathcal{C}}^{\infty}(M[1];{}^{\operatorname{b}}T).$

###### Proof.

That (5.2) is a smooth bundle map follows from the fact that $\Pi_{L}$ is a
b-fibration.

The main part of the proof is to see that $\mathcal{E}$ has a natural Lie
algebra structure. To do so we follow the discussion in Examples 1 and 2
above. Thus, take an element $V\in\mathcal{E},$ i.e. a smooth section of $E$
over $M[1]$ or equivalently a smooth section of
$\operatorname{null}((\Pi_{R})_{*})$ over the diagonl $D.$

The projection $\Pi_{S}$ restricts to a b-fibration of $D_{1,2}$ onto $D.$
Moreover the identity

(5.3)
$\Pi_{R}\Pi_{S}=\Pi_{L}\Pi_{F}\Longrightarrow(\Pi_{S})_{*}:\operatorname{null}((\Pi_{C})_{*})\longrightarrow\operatorname{null}((\Pi_{R})_{*}).$

At $D_{1,2}$ the null space of $(\Pi_{S})_{*}$ is tangent to $D_{1,2}$ whereas
$\operatorname{null}((\Pi_{C})_{*})$ is transversal to $D_{1,2}.$ Since the
dimensions are the same the push-forward map in (5.3) is therefore an
isomorphism at each point. It follows that $V$ lifts to a unique smooth
section

(5.4)
$W\in{\mathcal{C}}^{\infty}(D_{1,2};\operatorname{null}((\Pi_{C})_{*}))\text{
satisfying }(\Pi_{S})_{*}W(p)=V(\Pi_{S}(p))$

at each point.

Now, the projection $\Pi_{F}$ restricts to a diffeomorphism of $D_{1,2}$ onto
$M[2].$ Moreover, the identity

$\Pi_{R}\Pi_{F}=\Pi_{R}\Pi_{C}$

shows that the differential of $\Pi_{F}$ maps
$\operatorname{null}((\Pi_{C})_{*})$ into $\operatorname{null}((\Pi_{R})_{*})$
everywhere. Thus $W$ defined by (5.4) pushes forward to a smooth global
section

(5.5)
$\tilde{V}=(\Pi_{F})_{*}W\in{\mathcal{C}}^{\infty}(M[2];\operatorname{null}((\Pi_{R})_{*})).$

This global section restricts to $D$ to be the original section $V.$ Indeed,
on the triple diagonal, $\Pi_{S}=\Pi_{F}=\Pi_{C}$ are all the identity in
terms of the embeddings. Also _at_ the triple diagonal,
$\operatorname{null}((\Pi_{F})_{*})=TD_{2,3}.$ Now $\Pi_{C}R_{2,3}=\Pi_{S}$
and $R_{2,3}$ acts as the identity on $D_{2,3}$ so it follows that
$(R_{2,3})_{*}W=W$ and hence $(\Pi_{F})_{*}W=V.$

The extension of $V$ to $\tilde{V}$ also gives an extension of $W$ to a
section
$\tilde{W}\in{\mathcal{C}}^{\infty}(\Omega;\operatorname{null}((\Pi_{C})_{*})$
at least for a small open neighbourhood of the triple diagonal, $T.$ Indeed,
in such a neighourhood $(\Pi_{F})_{*}$ maps
$\operatorname{null}((\Pi)_{C})_{*})$ isomorphically onto
$\operatorname{null}((\Pi_{R})_{*})$ so $\tilde{W}$ is well-defined pointwise
by

(5.6) $\tilde{V}=(\Pi_{F})_{*}\tilde{W}\text{ near }T.$

Then $\tilde{V}$ and $\tilde{W}$ are $\Pi_{F}$-related smooth vector fields.
The extended vector fields $\tilde{V}$ are tangent to the fibres of $\Pi_{R}$
so given two vector fields $V_{1},$ $V_{2}\in\mathcal{E}$ the commutators of
the extensions $[\tilde{V}_{1},\tilde{V}_{2}]$ and
$[\tilde{W}_{1},\tilde{W}_{2}]$ are $\Pi_{F}$-related as well. From this it
follows that the lift of the commutator defined on $\mathcal{E}$ by
restriction is the commutator of the lifts. In particular, the Jacobi identity
holds.

It also follows directly that the anchor map (5.2) maps commutators to
commutators so $\mathcal{E}$ does indeed have the structure of a Lie
algebroid. ∎

It is straightforward to check that the Lie algebroid structure on
$\mathcal{E}$ is the same as that induced from the associative algebra of
differential operators formed by the pseudodifferential operators, discussed
in the next section, with kernels supported on the diagonal.

######  Remark 1.

The interior of $M[2]$ is a Lie groupoid quantizing the Lie algebroid
$\mathcal{E}.$ However in most interesting cases $M[2]$ itself is not a Lie
groupoid. Rather, the intention is the notion of generalized product should
correspond to ‘good’ compactifications of certain Lie groupoids.

Note that, as for a Lie group, it is totally unreasonable to expect that the
compactification of a Lie groupoid should be a Lie groupoid.

##  6\. Smoothing and pseudodifferential algebras

Each generalized product gives rise to ‘small’ algebra of pseudodifferential
‘operators’ determined by kernels on $M[2]$ conormal with respect to the
diagonal. There are representations of these algebras, acting on smooth
functions on (or more generally corresponding algebras action on sections of a
vector bundle over) $M[1].$ In the case of a stretched product the kernels are
determined by these actions as operators. However, as is clear from Example 2,
where the action is trivial, this is not in general the case and there are
‘hidden variables’ corresponding to the fact that these pseudodifferential
operators always give a ‘microlocalization’ of the Lie algebroid discussed in
§5.

The algebra of fibrewise smoothing operators for a fibration (1.1), with
compact boundaryless fibres, arises directly from (1.2) and the basic fact
that for any fibration of compact manifolds, and any smooth complex vector
bundle $W\longrightarrow Y$ over the base the push-forward map, or fibre
integration, gives a continuous (and surjective) linear map

(6.1)
$\phi_{*}:{\mathcal{C}}^{\infty}(M;\Omega\otimes\phi^{*}W)\longrightarrow{\mathcal{C}}^{\infty}(Y;\Omega\otimes
W)$

where $\Omega$ is the density bundle on each respective space.

For any fibration there is an induced short exact sequence for the
differential

(6.2) $T_{\phi}M\longrightarrow TM\overset{\phi_{*}}{\longrightarrow}TY,$

where $T_{\phi}M=\operatorname{null}(\phi_{*})$ is the fibre tangent bundle.
This induces a natural isomorphism for the associated density bundles

(6.3) $\Omega(M)=\phi^{*}\Omega(Y)\otimes\Omega_{\phi}$

where $\Omega_{\phi}$ is the bundle of densities on the fibres of $\phi.$

The algebra of smoothing operators, on sections of a complex vector bundle
$V\longrightarrow M,$ is then identified as a space with

(6.4)
$\Psi^{-\infty}_{\phi}(M;V)={\mathcal{C}}^{\infty}(M^{[2]}_{\phi};\Pi_{L}^{*}V\otimes\Pi_{R}^{*}V^{\prime}\otimes\Pi_{R}^{*}\Omega_{\phi}).$

The action on ${\mathcal{C}}^{\infty}(M;V)$

(6.5)
$\Psi^{-\infty}_{\phi}(M;V)\times{\mathcal{C}}^{\infty}(M;V)\longrightarrow{\mathcal{C}}^{\infty}(M;V)$

comes from pushforward as in (6.1) and (6.3). Namely for
$A\in\Psi^{-\infty}_{\phi}(M;V)$ and $u\in{\mathcal{C}}^{\infty}(M;V)$ to
define $Au$ we may choose a positive global section
$\nu\in{\mathcal{C}}^{\infty}(M;\Omega)$ and set

(6.6) $Au=(\Pi_{L})_{*}(\Pi_{L}^{*}\nu\cdot A\cdot\Pi_{R}^{*}u)/\nu.$

The product in (6.6) includes the pairing of $\Pi_{R}^{*}V^{\prime}$ and
$\Pi_{R}^{*}V$ and the tensor product identification of densities gives
$\Pi_{R}^{*}\nu\cdot
A\cdot\Pi_{R}^{*}u\in{\mathcal{C}}^{\infty}(M^{[2]}_{\phi};\Pi_{L}^{*}V\otimes\Omega)$
which pushes forward to define $(Au)\nu.$ That this is independent of the
choice of $\nu$ follows from the fact that for (6.1) in general
$\phi_{*}(v\phi^{*}f)=f\phi_{*}(v)$ for all $f\in{\mathcal{C}}^{\infty}(Y)$
and $v\in{\mathcal{C}}^{\infty}(M;\Omega\otimes\phi^{*}W).$

The triple product corresponds to the composition of operators

$\textstyle{M^{2}_{\phi}}$$\textstyle{M^{3}_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{S}}$$\scriptstyle{\Pi_{F}}$$\scriptstyle{\Pi_{C}}$$\textstyle{M^{2}_{\phi}}$$\textstyle{M^{2}_{\phi}}$

Then the kernel of the product is given by push-forward of densities

(6.7)
$K_{AB}=(\Pi_{C})_{*}\left(\Pi_{S}^{*}K_{A}\Pi_{1}^{*}\nu\times\Pi_{F}^{*}K_{B}\right)/\Pi_{1}^{*}\nu$

where $\nu$ is again a positive smooth density on $M$ and the tensor product
pushed forward is naturally a density on $M^{[3]}_{\phi}.$

We proceed to generalize these constructions, especially the product, to a
compact generalized product $M[k]$ in place of the fibre products
$M^{[k]}_{\phi}.$

For a b-fibration of compact manifolds with corners, still denoted as in
(1.1), there is a short exact sequence generalizing (6.2) for the
b-differential

(6.8)
${}^{\operatorname{b}}T_{\phi}M\longrightarrow{}^{\operatorname{b}}TM\overset{\phi_{*}}{\longrightarrow}{}^{\operatorname{b}}TY,$

so (6.3) is replaced by

(6.9)
$\Omega_{\operatorname{b}}(M)=\phi^{*}\Omega_{\operatorname{b}}(Y)\otimes\Omega_{\operatorname{b},\phi}.$

The fibrations of a generalized product satisfy

$\Pi_{R}\Pi_{S}=\Pi_{L}\Pi_{F}$

so

(6.10)
$(\Pi_{F})_{*}:\operatorname{null}((\Pi_{S}))_{*})\longrightarrow\operatorname{null}((\Pi_{L}))_{*}).$

This is an isomorphism near the triple diagonal.

######  Definition 8.

A generalized product is said to be _regular_ if the determinant of (6.10) is
non-vanishing as a section

(6.11)
$\delta\in\rho^{\zeta}{\mathcal{C}}^{\infty}(M[3];\lambda^{d}\operatorname{null}((\Pi_{F})_{*})\otimes\lambda^{d}\operatorname{null}((\Pi_{L}))_{*}))$

for some integer multiindex $\zeta.$

In general the push-forward of a smooth b-density under a b-fibration is _not_
smooth, in fact it may not exist with the obstruction to existence being the
possibility of ‘fixed hypersurfaces’ of $M,$ $H\in\mathcal{M}_{1}(M)$ such
that $\phi(H)=M.$ Apart from this, logarithmic terms in general arise when two
(intersecting) boundary hypersurfaces of $M$ are mapped to one boundary
hypersurface of $Y.$

######  Definition 9.

A smooth function (or section of a vector bundle) on a compact manifold with
corners $Z_{1}$ is said to have _simple support_ with respect to a b-fibration
$F:Z_{1}\longrightarrow Z_{2}$ if it vanishes to infinite order at the
intersection of any two boundary hypersurfaces mapped to the same boundary
hypersurface of $Z_{2}$ by $F$ and also to infinite order at fixed
hypersurfaces.

Really only simple vanishing at the fixed hypersurfaces is relevant for
existence and smoothness of push-forwards.

######  Proposition 8.

If $F:Z_{1}\longrightarrow Z_{2}$ is a simple b-fibration then fibre
integration defines

(6.12) $F_{*}:\\{u\in{\mathcal{C}}^{\infty}(Z_{1};F^{*}W\otimes\Omega);u\text{
has simple
support}\\}\longrightarrow{\mathcal{C}}^{\infty}(Z_{2};W\otimes\Omega).$

###### Proof.

This is a special case of the push-forward theorem from [9]. ∎

The usual way in which this support property is seen to be satisfied is for
$u$ to vanish to infinite order at all but one of the boundary hypersurfaces
of $Z_{1}$ in the preimage of each boundary hypersurface of $Z_{2}.$ If $u$
vanishes to infinite order on all the hypersurfaces in the preimage of a
hypersurface of $Z_{2}$ then the push-forward also vanishes to infinite order
at that hypersurface.

The kernels of the smoothing ‘operators’ associated to a compact generalized
product $M[k]$ are smooth sections of a bundle over $M[2].$ As noted in
Proposition 3 each boundary hypersurface of $M[1]$ determines a diagonal
boundary hypersurface of $M[2]$ which maps into it under either of the
projections $\Pi_{L},$ $\Pi_{R}:M[2]\longrightarrow M[1].$

######  Definition 10.

The space of (small) smoothing kernels for a vector bundle $V$ over $M[1]$ for
a compact generalized product $M[*]$ is

(6.13)
$\Psi^{-\infty}(M[*];V)=\\{A\in{\mathcal{C}}^{\infty}(M[2];\Pi_{L}^{*}V\otimes\Pi_{R}^{*}V^{\prime}\otimes\Omega_{L});\\\
A\equiv 0\text{ at }H\in\mathcal{M}_{1}(M[2])\text{ if }H\cap D=\emptyset\\}$

where $\Omega_{L}$ is the space of fibre b-densities for
$\Pi_{L}:M[2]\longrightarrow M[1],$ that is the densities for the bundle of
null spaces of $\Pi_{L}.$ The small space(s) of pseudodifferential kernels are
similarly the space(s) of conormal densities with singularities on $D$

(6.14) $\Psi^{m}_{\sigma}(M[*];V)=\\{B\in
I^{m+\frac{1}{4}(m-\kappa)}_{\sigma}(M[2],D;\Pi_{L}^{*}V\otimes\Pi_{R}^{*}V^{\prime}\otimes\Omega_{L});\\\
A\equiv 0\text{ at }H\in\mathcal{M}_{1}(M[2])\text{ if }H\cap D=\emptyset\\}.$

Here $\sigma$ refers to the class of conormal distributions admitted, for
‘classical operators’ it is omitted, for the class of type $1,0$ with bounds
we write $\sigma=\infty.$

In the case of a concrete generalized product written $M[k;T]$ where $T$
denotes a ‘structure’ on $M$ we use the simpler notation

(6.15) $\Psi^{m}_{T}(M;V)=\Psi^{m}(M[*;T];V).$

Although the main concern here is with smoothing operators, recall that the
kernels of pseudodifferential operators are particular classes of conormal
distributions the basic properties of which we briefly describe. We interpret
the conormal distributions at a point as the quantization of a simplicial
compactification of the translation group.

Let $V$ be a real finite-dimensional vector space. As a translation group the
Lie algebra is also identified with $V$ and its enveloping algebra is
naturally identified with the polynomials on $V^{*}$ or the distributional
densities on $V$ supported at the origin. These views are intertwined by
Fourier transform

(6.16)
$\mathcal{F}:{\mathcal{S}}^{\prime}(V;\Omega)\longrightarrow{\mathcal{S}}^{\prime}(V^{*}).$

Let $\overline{V}^{*}$ be the radial compactification of $V^{*}$ which is a
compact manifold with boundary (a closed ball), let $\rho$ be a boundary
defining function, for instance $\rho=(1+|w|^{2})^{-\frac{1}{2}}$ in terms of
a metric on $V^{*}.$ Then the ‘space of classical symbols of order $m$’ is by
definition

(6.17)
$\rho^{-m}{\mathcal{C}}^{\infty}(\overline{V}^{*})\subset{\mathcal{S}}^{\prime}(V^{*}),\
m\in\mathbb{C}.$

Since these are tempered distibution on $V^{*}$ the distributional inverse
Fourier transforms may be applied and

(6.18)
$\mathcal{F}^{-1}(\rho^{-m}{\mathcal{C}}^{\infty}(\overline{V}^{*}))=I^{m+\frac{1}{4}\dim
V}_{\mathcal{S}}(V,\\{0\\};\Omega)\subset{\mathcal{S}}^{\prime}(V;\Omega)$

is identified as the space of the classical conormal distributions of order
$m+\frac{1}{4}\dim V.$ These distributions are singular only at the origin and
decay rapidly with all derivatives at infinity. The normalization of the order
(due to Hörmander) is more natural than it might appear to be!

These space form a filtered convolution algebra

(6.19)
$I^{m}_{\mathcal{S}}(V,\\{0\\};\Omega)*I^{m^{\prime}}_{\mathcal{S}}(V,\\{0\\};\Omega)=I^{m+m^{\prime}}_{\mathcal{S}}(V,\\{0\\};\Omega).$

The distributions supported at the origin, the derivatives of the Dirac delta,
are included in these spaces with a derivative of order $\alpha$ lying in
$I^{|\alpha|+\frac{1}{4}\dim V}_{\mathcal{S}}(V,\\{0\\};\Omega),$ so the
constant-coefficent pseudodifferential operators satisfy

(6.20) $\Psi^{m}_{\operatorname{ti}}(V)=I^{m-\frac{1}{4}\dim
V}_{\mathcal{S}}(V,\\{0\\};\Omega)\supset\operatorname{Dff}^{m}_{\operatorname{ti}}(V)\text{
act on }\mathcal{S}(V).$

Even though (6.19) follows immediately from the obvious multiplicative
property

(6.21)
$\rho^{-m}{\mathcal{C}}^{\infty}(\overline{V}^{*})\cdot\rho^{-m^{\prime}}{\mathcal{C}}^{\infty}(\overline{V}^{*})=\rho^{-m-m^{\prime}}{\mathcal{C}}^{\infty}(\overline{V}^{*})$

it is convenient to realize it in terms of functorial properties of conormal
functions and the mapping diagram

(6.22)
$\textstyle{V}$$\textstyle{V^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{C}}$$\scriptstyle{\pi_{S}}$$\scriptstyle{\pi_{F}}$$\textstyle{V}$$\textstyle{V}$
$\textstyle{v}$$\textstyle{(v,w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v-w}$$\textstyle{w.}$

Namely

(6.23)
$\kappa_{1}*\kappa_{2}=(\pi_{C})_{*}(\pi_{S}^{*}\kappa_{1}\cdot\pi_{F}^{*}\kappa_{2}).$

In this case the three (face) maps in (6.22) are projections with fibre and
base $V.$ Under the lower two maps the inverse images of zero in $V$ are
respectively

(6.24) $\pi_{S}^{-1}(\\{0\\})=\\{(v,v)\in V^{2};v\in
V\\}=\operatorname{Diag}(V^{2}),\ \pi_{F}^{-1}(\\{0\\})=V\times\\{0\\}.$

In general distributions cannot be pulled back under smooth maps but in this
case this is admissible by wavefrontset considerations and the spaces
$I^{m+\frac{1}{4}\dim V}(V,\\{0\\};\Omega)$ pull back into the more general
spaces $I^{m}(V^{2};\pi_{S,F}^{*}(\\{0\\});\pi_{S,F}^{*}\Omega)$ dicussed
below for an embedded submanifold of a manifold. The transversality of the two
manifolds $\pi_{S}^{*}(\\{0\\})$ and $\pi_{F}^{*}(\\{0\\})$ means that the
product on the right in (6.23) is well-defined. It is a ‘conormal distribution
of product type’; the tensor product of the two pulled-back density lines is
canonically the density line of $V^{2}.$ The ‘composite’ projection $\pi_{C}$
is transversal to the two lifts of $\\{0\\}$ and as a result the push-forward
is smooth outside the image
$\pi_{C}(\pi_{S}^{*}(\\{0\\})\cap\pi_{F}^{*}(\\{0\\}))=\\{0\\}\in V$ and
(6.23) follows from an appropriate push-forward theorem for conormal
distributions.

So, although the discussion of (6.23) is far more involved than the use of the
Fourier transform to prove (6.19) which becomes

(6.25)
$\Psi^{m}_{\operatorname{ti}}(V)\circ\Psi^{m^{\prime}}_{\operatorname{ti}}(V)=\Psi^{m+m^{\prime}}_{\operatorname{ti}}(V)$

it is this latter approach which is generalized below.

The main interest in the space $I^{M}_{\infty}$ and corresponding kernels is
the fact that include the classical space $I^{M}$ and ${\mathcal{C}}^{\infty}$
is dense in $I^{M}_{\infty}$ in the topology of $I^{M^{\prime}}_{\infty}$ for
any $M^{\prime}>M.$

######  Theorem 3.

The small pseudodifferential kernels for a compact connected generalized
product define an order-filtered associative algebra with identity with a
surjective and multiplicative symbol map

(6.26) $\sigma:\Psi^{m}(M[*];V)\longrightarrow S^{m}(E^{*}M[1];\hom(V))$

which gives a short exact sequence for each $m$

(6.27)
$\textstyle{\Psi^{m-1}(M[*];V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Psi^{m}(M[*];V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{m}(E^{*}M[1];\hom(V)).}$

Here $EM[1]$ structure bundle of the associated Lie groupoid.

###### Proof.

We concentrate on the smoothing operators, since the extension to
pseudodifferential operators only involves the behaviour of conormal
singularities at the diagonals and this turns out to be essentially the same
as in the case of a fibration.

The product is defined by direct generalization of (6.7). Thus if $A$ and
$B\in\Psi^{-\infty}(M[*];V)$ then the kernel of the composite is

(6.28)
$K_{AB}=(\Pi_{C})_{*}\left(\Pi_{S}^{*}A\cdot\Pi_{1}^{*}\nu\times\Pi_{F}^{*}B\right)/\Pi_{1}^{*}\nu$

To see that this is well-defined first note that the densities behave in the
expected manner:

(6.29)
$(\Pi_{L}\Pi_{S})^{*}\Omega_{\operatorname{b}}(M[1])\otimes\Pi_{S}^{*}\Omega(\operatorname{null}((\Pi_{L})_{*})\otimes\Pi_{F}^{*}\Omega(\operatorname{null}((\Pi_{L})_{*})\longrightarrow\Omega_{\operatorname{b}}(M[3])$

is a smooth map of line bundles which is non-vanishing near the triple
diagonal. The product of the kernels has simple support relative to $\Pi_{C}$
so (6.28) defines a bilinear map

(6.30)
$\Psi^{-\infty}(M[*];V)\times\Psi^{-\infty}(M[*];V)\longrightarrow\Psi^{-\infty}(M[*];V).$

The space $M[4]$ can be used to see that this product is associative. Consider
the structural b-fibrations

(6.31) $\Pi_{[i,j]}:M[4]\longrightarrow M[2],\ 1\leq i<j\leq 4$

associated to the strictly increasing map $I_{i,j}:J(2)\longrightarrow J(4)$
which omits $i$ and $j.$ The densities behave as in (6.29) so giving a
trilinear map

(6.32)
$\begin{gathered}\Psi^{-\infty}(M[*];V)\times\Psi^{-\infty}(M[*];V)\times\Psi^{-\infty}(M[*];V)\longrightarrow\Psi^{-\infty}(M[*];V),\\\
(A,B,C)\longmapsto(\Pi_{[2,3})_{*}\left(\Pi_{[3,4}^{*}A\cdot\Pi_{L}^{*}\nu\times\Pi_{[1,4]}^{*}B\times\Pi_{[1,2]}\right)/\Pi_{1}^{*}\nu.\end{gathered}$

This factors through the products on the left and right to give associativity.

The pseudodifferential extension folliws from the structure of the diagonals,
especially (4.13). ∎

######  Proposition 9.

For a stretched product, $M[k;T],$ the kernels in (6.13) of the (small)
algebra of smoothing operators, act as linear operators on
${\mathcal{C}}^{\infty}([M[1];V)$ by analogy with (and extension of) (6.6):

(6.33)
$Au=(\Pi_{L}^{T})_{*}\left(A\cdot(\Pi_{2}^{T})^{*}u\cdot(\Pi_{L})^{*}\nu_{\operatorname{b}}\right)/\nu_{\operatorname{b}},\
0<\nu_{\operatorname{b}}\in{\mathcal{C}}^{\infty}(M[1;T];\Omega_{\operatorname{b}}).$

###### Proof.

As shown above the diagonal boundary hypersurfaces of $M[2],$ those meeting
$D,$ map to the correspoing hypersurfaces of $M[1]$ under both projections.
Thus the kernels in (6.13) are simply supported relative to $\Pi_{L}.$ The
same is therefore true for the argument of the push-forward in (6.33) which by
(6.9) is a b-density on $M[2;T].$ With the independence of choice of
$\nu_{\operatorname{b}}$ following as before that this gives the bilinear map

(6.34)
$\Psi^{-\infty}_{T}(M;V)\times{\mathcal{C}}^{\infty}(M[1];V)\longrightarrow{\mathcal{C}}^{\infty}[M[1];V).$

Thus it only remains to prove that composites lie in the same space, that

(6.35)
$\Psi^{-\infty}_{T}(M[*];V)\circ\Psi^{-\infty}_{T}(M[*];V)\subset\Psi^{-\infty}_{T}(M[*];V).$

To do so we use properties of the triple product and the three stretched
projections

(6.36)
---
$\textstyle{M[2]}$$\textstyle{M[3]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi_{C}}$$\scriptstyle{\Pi_{S}}$$\scriptstyle{\Pi_{F}}$$\textstyle{M[2]}$$\textstyle{M[2].}$

The kernel of the product of two elements, $A_{1}$ and $A_{2}$ is given as the
push-forward

(6.37)
$(\Pi_{C}^{T})_{*}\left((\Pi_{S}^{T})^{*}(A_{1})\cdot(\Pi_{F}^{T})^{*}(A_{2})\cdot(\Pi_{C}^{T})^{*}(\nu)\right)/\nu$

Similarly the first factor vanishes to infinite order at the front faces other
than those corresponding to $T$ and $D_{1,2}.$ Thus the product vanishes to
infinite order except at the triple front face correponding to $T(\phi).$ Thus
before push-forward the numerator in (6.37) is a smooth b-density. Then (6.12)
applies to show that the push-foward is a smooth b-density on $M[2;T]$ to get
the product (6.35).

The multiplicative property (and extension to bundle coefficients)

(6.38) $\sigma_{T}(A_{1}A_{2})=\sigma_{T}(A_{1})\cdot\sigma_{T}(A_{2})$

then follows either by use of an appropriate oscillatory testing argument or
by identifying the product of the kernels, restricted to the front face in
(6.37) with the kernel of the fibrewise convolution of the restricted kernels.
∎

##  7\. Semiclassical generalized product

The construction of the Atiyah-Singer map in [5] uses the semiclassical, and
more generally the adiabatic, calculus associated to a fibration or iterated
fibration. This is discussed explicitly in the next section, but Theorem 1
allows a semiclassical generalized product to be constructed from a given
generalized product, which as usual we assume to be compact and connected.

Let $M[k]$ be such a generalized product. Then for each $k$ the products

(7.1) $[0,1]\times M[k]$

also form a generalized product with all structure maps being extended by the
identity on $[0,1].$ Then Theorem 1 shows that the (multi)diagonals
$\mathcal{D}\subset M[k]$ form, for each $k,$ a p-clean collection of
submanifolds closed under intersection.

It follows that the collections

(7.2) $\mathcal{D}_{\operatorname{sl}}=\\{\\{0\\}\times
D;D\in\mathcal{D}\\}\subset[0,1]\times M[k]$

also form a p-clean collection closed under intersection.

######  Definition 11.

The semiclassical spaces corresponding to a compact, connected generalized
product are the compact manifolds with corners

(7.3) $M[k;\operatorname{sl}]=[[0,1]\times
M[k];\mathcal{D}_{\operatorname{sl}}]$

where as usual the blow-up is to be performed in a dimension order using
Proposition 3.

When a particular generalized product is denoted $M[k;T]$ with ‘$T$’ being
some label to indicate the structure then the semiclassical spaces can be
denoted $M[k;\operatorname{sl}T].$

######  Theorem 4.

The semiclassical spaces associated to a compact connected generalized product
$M[k]$ form a compact connected generalized product.

###### Proof.

The collection $\mathcal{D}$ is invariant under the structural diffeomorphisms
of $M[k]$ so $\mathcal{D}_{\operatorname{sl}}$ is invariant under the
structural diffeomorphism of $[0,1]\times M[k]$ and it follows that these lift
to diffeomorphisms of $M[k;\operatorname{sl}].$ Thus Lemma 3 shows that it
suffices to check that the b-fibrations and p-embeddings

(7.4) $\begin{gathered}\Pi_{k}:(0,1]\times M[k]\longrightarrow(0,1]\times
M[k-1],\\\ D_{1,2}:(0,1]\times M[k-1]\longrightarrow(0,1]\times M[k],\
k>1,\end{gathered}$

extend to b-fibrations and p-embeddings of the $M[*;\operatorname{sl}].$

The multidiagonals in $\mathcal{D},$ and hence
$\mathcal{D}_{\operatorname{sl}},$ are each associated with a partition
$\mathfrak{p}$ of $J(k).$ We divide

(7.5) $\mathcal{D}_{\operatorname{sl}}=\mathcal{G}\sqcup\mathcal{R}$

where $\mathcal{G}$ consists of the diagonals ‘which do not involve $k$’, i.e.
$k$ is a singleton, so the $\mathcal{P}_{j}\subset\\{1,\dots,k-1\\}$ may be
identified as subsets of $J(k-1).$ The remaining elements, in $\mathcal{R},$
‘do involve $k$’ in that $k\in\mathcal{P}_{j}$ for some $j.$ Clearly these two
collections satisfy the hypotheses of Proposition 6 so

(7.6) $M[k;\operatorname{sl}]=[[[0,1]\times M[k];\mathcal{G}];\mathcal{R}].$

As noted above, each element of $D_{\mathfrak{p}}\in\mathcal{G}$ at level $k$
is associated with an element of
$D_{\mathfrak{q}}\in\mathcal{D}_{\operatorname{sl}}$ at level $k-1$ and

$\mathfrak{p}=I_{k}(\mathfrak{q}),\ I_{k}:J(k-1)\longrightarrow J(k)$

being the increasing injection missing $k.$ Thus under the associated
b-fibration

(7.7) $D_{\mathfrak{q}}\subset\Pi_{k}(D_{\mathfrak{p}})\Longrightarrow
D_{\mathfrak{p}}=\Pi_{k}^{-1}(D_{\mathfrak{q}})$

by the assumption of connectedness, since the dimensions are the same.

Now

(7.8) $\Pi_{k}:[[0,1]\times M[k];\mathcal{G}]\longrightarrow
M[k-1;\operatorname{sl}]=[[0,1]\times
M[k-1]];\mathcal{D}_{\operatorname{sl}}]$

is a b-fibration. This is true at each step as elements of $\mathcal{G}$ on
the left and of $\mathcal{D}_{\operatorname{sl}}$ on the right, related by
(7.7), are blown up, as follows by the repeated application of Lemma 8.

Returning to (7.6) it remains to show that $\Pi_{k}$ lifts to a b-fibration
through the blow up of the successive centres in $\mathcal{R}.$ From the proof
of Proposition 6 it follows that these centres, before they are to be blown
up, map under $\Pi_{k}$ onto the boundary hypersurface face of
$M[k-1;\operatorname{sl}]$ corresponding to the diagonal onto which they
project. Thus Proposition 7 applies to give the successive lifts.

It follows that all the projections in $M[*;\operatorname{sl}]$ defined by
continuous extension from $\epsilon>0$ are b-fibrations.

So consider the p-embedding in (7.4). Appending $D_{1,2}$ to the
$\mathcal{D}_{\operatorname{sl}}$ gives a p-clean collection closed under
intersection with $D_{1,2}$ as the largest element so it certainly lifts to a
p-submanifold of $M[k;\operatorname{sl}].$ Consider the decomposition of the
collection of the centres as

(7.9) $\mathcal{D}_{\operatorname{sl}}=\mathcal{F}\sqcup\mathcal{F}_{12}$

where the diagonals defining $\mathcal{F}$ have $1$ and $2$ appearing in
different sets of the partition; so if either is a singleton or if they appear
in disjoint $\mathcal{P}_{j}.$ Those partitions defining $\mathcal{F}_{12}$
have $1$ and $2$ in the same $\mathcal{P}_{j}.$ Then

(7.10) $C\in\mathcal{F}\Longrightarrow C\cap D_{1,2}\in\mathcal{F}_{12}$

with the intersection of strictly smaller dimension. Thus in the successive
blow-ups defining $M[k;\operatorname{sl}]$ the lift of $D_{1,2}$ is always
disjoint from elements of $\mathcal{F}$ since the intersection has been blown
up earlier. Now $D_{1,2}$ is the image of $M[k-1]$ under a p-embedding and
these successive blow-ups are of all the images under this embedding of the
elements of $\mathcal{D}_{\operatorname{sl}}\subset\\{0\\}\times M[k-1].$ The
diffeomorphism invariance of blow-up shows that the p-embedding lifts to a
p-embedding of $M[k-1;\operatorname{sl}]$ defining $D_{1,2}\subset
M[k;\operatorname{sl}].$

Noting that connectedness is preserved throughout, $M[k;\operatorname{sl}]$ is
a connected generalized product. ∎

######  Remark 2.

If a group $K$ acts as diffeomorphisms on the $M[k]$ with the actions
intertwined by the strucural maps then it lifts to such an action on
$M[k;\operatorname{sl}].$

Next we introduce a simplicial compactification of the translation group which
we identify with a vector space $V,$ its Lie algebra. It is very closely
related to ‘many-body’ spaces in scattering theory. See in particular the
survey by Vasy [10]. In $V^{k-1}$ consider the ‘diagonal’ subspaces defined in
terms of the same variable denoted $v_{j}$ in the $j$th factor

(7.11) $D_{j}=\\{v_{j}=0\\},\ D_{jl}=\\{v_{j}=v_{l}\\},\ j,\ l=1,\dots,k-1,\
j<l.$

These are the images of $V^{k-2}$ under the obvious inclusion maps.

######  Proposition 10.

The p-submanifolds formed from the intersections of subcollections of the
$D_{*}$ with the sphere forming the boundary of the radial compactification of
$V^{k-1},$ form a p-clean collection $\mathcal{I},$ closed under intersection,
and

(7.12) $\overline{E}[k]=[\overline{V^{k-1}};\mathcal{I}]$

is a simplicial compactification of the translation group $E.$

###### Proof.

Consider the semiclassical product constructed from the $V^{k}$ as a
‘generalized’ product. The first centre blown up in the construction of the
space at level $k$ is the $k$-fold diagonal at $\epsilon=0.$ It is naturally
$V\times\overline{V^{k-1}}$ with the boundary of the $k$-fold diagonal being
$V\times\\{0\\}.$ Subsequent blow-ups are transversal to the boundary and meet
it at precisely the $\mathcal{I}$ in (7.12). It follows that these manifolds

(7.13) $V\times\overline{E}[k]$

form a generalized product with all structure maps being invariant under the
translation action through the action on the first factor. Thus (7.12) is
indeed a generalized product and satisfies the conditions of Definition 3 for
the translation group. ∎

Note that the product action of the group of linear transformation of $L$ on
each $V^{k-1}$ lifts to an action on $\overline{E}[k].$

######  Proposition 11.

The boundary generalized product associated to
$H_{\operatorname{sl}}=\\{0\\}\times M[1]$ of $M[1,\operatorname{sl}],$ of a
semiclassical generalized product, fibres over $M[1]$

(7.14)
$\textstyle{\overline{E}[k]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\operatorname{sl}}[k]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M[1]}$

with fibres at each point forming the simplicial compactification of the fibre
of the Lie algebroid $EM[1]\longrightarrow M[1]$ as a vector space.

###### Proof.

This follows by repeating the argument used above to prove Proposition 10. ∎

##  8\. Adiabatic stretched product

The primary use of a generalized product in [5] is for the construction of the
associated algebra of smoothing operators for the semiclassical product
associated by Theorem 4 to the fibre products of a fibre bundle of compact
manifolds without boundary. In fact to allow ‘operators with values in
smoothing operators’ we proceed to discuss the more general adiabatic case,
which can be thought of as ‘partially semiclassical’. For a single manifold
$M$ this calculus was introduced in work with Mazzeo, [8], corresponding to a
fibration of a compact manifold.

Since we wish to include smooth parameters, consider an iterated fibration

(8.1)
$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\phi}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{Y}$

Here we work, at least initially, in the context of connected compact
manifolds without boundary so the maps $\gamma$ and $\psi$ are simply
submersions and $\phi=\psi\circ\gamma.$ The adiabatic algebra corresponds to
degeneration in a parameter which we introduce as for the semiclassical case
by taking the product with a closed interval $[0,1]_{\epsilon}.$ Then, as for
the semiclassical product, consider the fibre products

(8.2) $[0,1]\times M^{[k]}_{\phi}.$

The adiabatic Lie algebroid is the ${\mathcal{C}}^{\infty}$ module over this
space generated by smooth vector fields on $M$

(8.3) $V+\epsilon W$

where $W$ is tangent to the fibres of $\psi$ and $V$ is tangent to the fibres
of the finer fibration $\phi.$

There are ‘diagonal’ submanifolds of $M^{[k]}_{\phi}$ but with respect to the
map $\gamma.$ Thus $\gamma^{k}:M^{k}\longrightarrow F^{k}$ restricts to a
fibration which we denote

(8.4) $\gamma^{[k]}:M^{[k]}_{\phi}\longrightarrow F^{[k]}_{\psi}.$

If $\mathfrak{p}$ is a partition of $J(k)$ as discussed in §4 then
$D_{\mathfrak{p}}F\subset F^{[k]}_{\psi}$ and we consider the p-clean
collection of submanifolds

(8.5)
$\mathcal{D}_{\operatorname{ad}}=\left\\{\\{0\\}\times(\gamma^{[k]})^{-1}(D_{\mathfrak{p}}F)\right\\}.$

######  Definition 12.

The adiabatic manifolds associated to an iterated fibration (8.1) are the
spaces, defined by blow-up in dimension order using Proposition 3,

(8.6) $M[k;\operatorname{ad}]=[[0,1]\times
M^{[k]}_{\phi};\mathcal{D}_{\operatorname{ad}}].$

######  Theorem 5.

The adiabatic spaces, $M[k;\operatorname{ad}],$ for any iterated fibration of
connected compact manifolds (8.1), form a regular stretched product with
interior $(0,1)\times M^{[k]}_{\phi}.$

###### Proof.

We follow the structure of the proof of Theorem 4 closely. The factor exchange
diffeomorphisms of $M^{[k]}_{\phi},$ extended to $[0,1]\times M^{[k]}_{\phi}$
as the identity on the parameter space, interchange the elements of
$\mathcal{D}_{\operatorname{ad}}$ within a given dimension. Thus they lift to
give the symmetry diffeomorphims of $M[k;\operatorname{ad}]$ and Lemma 3
applies. The projections $\Pi_{k}$ and diagonal embeddings $D_{1,2}$ are
defined on the unresolved products $[0,1]\times M^{[k]}_{\phi}.$ Thus it
remains to show that they extend from the interior to b-fibrations and
p-embeddings of the $M[k;\operatorname{ad}].$

We proceed to examine $\Pi_{k}$ starting as in (7.5) but now

(8.7) $\mathcal{D}_{\operatorname{ad}}=\mathcal{G}\sqcup\mathcal{R}$

where the $\mathcal{G}$ are the diagonals in which $k$ is a singleton. The
argument can be followed with only minor notation change to show that
$\Pi_{k}$ does indeed lift to a b-fibration.

Similarly the lift of $D_{1,2}$ can be seen using the argument starting at
(7.9) writing

(8.8) $\mathcal{D}_{\operatorname{ad}}=\mathcal{F}\sqcup\mathcal{F}_{12}$

where $\mathcal{F}_{12}$ corresponds to the diagonals in which $1$ and $2$
appear in the same set $\mathcal{P}_{j}$ of the corresponding partition and
adding $D_{1,2}$ to the p-clean collection. Note that this enlarged collection
is not closed under intersection, since $D_{1,2}$ is the $\phi$ diagonal.
However in examining the lift of $D_{1,2}$ under the blow-ups defining
$M[k;\operatorname{ad}]$ it is still the case that the blow-up of the elements
of $\mathcal{F}$ are disjoint from $D_{1,2}.$ Indeed, each such centre is
associated with a lower-dimensional centre in $\mathcal{F}_{12}$ by adding $1$
and $2$ as a subset if they are both absent or by adding $2$ to the set
containing $1$ or $1$ to the set containing $2$ if one of them is not a
singleton. This element of $\mathcal{F}_{12}$ has the property of $F_{3}$ in
Proposition 4. It contains the intersection but neither of the two
submanifolds nor conversely. Thus again only the elements of
$\mathcal{F}_{12}$ affect the lift of $D_{1,2}$ and these correspond to the
blow-up of $\mathcal{D}_{\operatorname{ad}}$ for $D_{1,2}$ as a p-submanifold.
Then Lemma 8 shows that the p-embedding lifts through the blow-ups. ∎

It is now straightforward to check that the boundary products for the
adiabatic space associated to (8.1) are bundles over $F$ with fibre at each
point the product $E[k]\times M^{[k]}_{\gamma}$ where $E[k]$ is the simplicial
compactification of the fibre of the $T_{\psi}F.$

##  9\. The b-stretched product

Next we consider perhaps the most basic stretched product corresponding to a
compact manifold with corners. Since it involves little more effort consider a
fibration again denoted $\phi:M\longrightarrow Y$ but where now the fibres are
compact manifolds with corners. Alternatively the bundle case can be deduced
from the diffeomorphism invariance of the construction for a single manifold.

As already noted above, the fibre diagonal in $M^{[2]}_{\phi}$ is not a
p-submanifold if the boundary is non-trivial. To resolve this we consider the
‘fixed’ boundary hypersurface of $M,$ those that are mapped to $Y$ by $\phi.$
For each such boundary hypersurface $H$ consider the corresponding boundary
faces of $M^{[k]}_{\phi}$ given by the fibre products with factors either $M$
or $H.$ Thus for each $k$ and each $\mathcal{P}\subset J(k)$ consider

(9.1) $H_{\mathcal{P},k}=\\{(m_{1},m_{2},\dots,m_{k})\in
M^{[k]}_{\phi};m_{j}\in H\text{ if }j\in\mathcal{P}\\}.$

######  Lemma 12.

The boundary faces
$H_{\mathcal{P},k}\in\mathcal{M}_{\\#(\mathcal{P})}(M^{[k]}_{\phi}),$ for
fixed $k,$ all $\mathcal{P}$ and all $\phi$-fixed $H\in\mathcal{M}_{1}(M)$
form a p-clean family,
$\mathcal{F}_{\operatorname{\operatorname{b}\kern-1.5pt\phi},k},$ closed under
non-transversal intersection.

###### Proof.

Elements $H_{\mathcal{P},k}$ and $H^{\prime}_{\mathcal{P}^{\prime},k}$ for
distinct hypersurfaces and any subsets $\mathcal{P}$ and
$\mathcal{P}^{\prime}$ intersect transversally since $H$ and $H^{\prime}$ do
so. For a fixed $H$ the collection is closed under intersection with

(9.2) $H_{\mathcal{P},k}\cap
H_{\mathcal{Q},k}=H_{\mathcal{P}\cup\mathcal{Q},k}.$

∎

######  Proposition 12.

The manifolds

(9.3)
$M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]=[M^{[k]}_{\phi};\mathcal{F}_{\operatorname{\operatorname{b}\kern-1.5pt\phi},k}]$

are well-defined by blowing up all centres in a dimension order and form a
regular stretched product.

The arguments are similar to those in Theorem 4 in this combinatorially
simpler setting where partitions of $J(k)$ are replaced by subsets but
complicated by the fact that $D_{1,2}$ is not initially a p-submanifold.

###### Proof.

At each level, in terms of dimension, of blow up in a dimension order the
intersection of non-transversal elements has already been blown up, so all
elements in (9.3) are transversal (or disjoint) and hence can be reordered
freely. It follows that $M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$
is symmetric, since exchanging factors amounts to reordering with
$\\#(\mathcal{P})$ fixed. Thus Lemma 3 applies and to see that this is a
stretched product we only need consider the existence of the stretched
projection $\Pi_{k}$ from
$M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ to
$M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}],$ for each $k>1,$
corresponding to omission of the last factor and the existence of the
p-embedding of $M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ in
$M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ as the diagonal in the
first two factors.

The transversality of any two centres involving different boundary
hypersurfaces means, using Lemma 6, that we may freely change the order of
blow-up between different $H.$ This effectively reduces the discussion to the
case of a manifold with boundary, proceeding over some ordering of the (fixed)
boundary hypersurfaces of $M.$

We may divide the p-clean collection as

(9.4)
$\mathcal{F}_{\operatorname{\operatorname{b}\kern-1.5pt\phi},k}=\mathcal{G}\sqcup\mathcal{R}$

where the first set corresponds to the $\mathcal{P}$ which do not contain $k$
and the second to those which do. So each of these subsets is labeled by the
$\mathcal{Q}\subset J(k-1)$ either by inclusion or by addition of $k.$ Thus

(9.5) $\mathcal{G}\ni C^{\prime}=H_{\mathcal{Q},k-1}\times_{\phi}M,\
\mathcal{R}\ni C^{\prime\prime}=H_{\mathcal{Q},k-1}\times_{\phi}H$

for the fixed hypersurface depending on the set. Clearly, from (9.2),
$\mathcal{G}$ and $\mathcal{R}$ are separately closed under non-transversal
intersection and the non-transversal intersection of an element of
$\mathcal{G}$ with one in $\mathcal{R}$ lies in $\mathcal{R}.$ Thus
Proposition 9.2 applies separately for each $H$ and

(9.6)
$M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]=[[M^{[k]}_{\phi};\mathcal{G}];\mathcal{R}].$

Since they do not involve the $k$th factor the iterated blow up of
$\mathcal{G}$ maps through the fibre product

$[M^{[k]}_{\phi};\mathcal{G}]=M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}]\times_{\phi}M.$

Moreover Lemma 8 applies to $\Pi_{k}$ at each step so it lifts to the
b-fibration

(9.7) $\Pi_{k}:[M^{[k]}_{\phi};\mathcal{G}]\longrightarrow
M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}].$

Proposition 7 applies to all the successive blow-ups of the centres in
$\mathcal{R}$ so $\Pi_{k},$ and hence all the projections
$M[k,\operatorname{\operatorname{b}\kern-1.5pt\phi}]\longrightarrow
M[k-1,\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ exist and are
b-fibrations.

To see the existence of the p-embedding $D_{1,2}:M[k-1]\longrightarrow M[k]$
consider the collection $\mathcal{B}_{1,2}$ of those boundary faces of the
form

(9.8) $B_{H}=H\times_{\phi}H\times_{\phi}M^{[k-2]}_{\phi}.$

within the elements of $\mathcal{F}_{\operatorname{b},k}$ corresponding to the
same $H.$ Proposition 6 applies to

(9.9)
$\mathcal{F}_{\operatorname{b},k}=\mathcal{B}_{1,2}\sqcup\mathcal{B}^{\prime}$

where $\mathcal{B}^{\prime}$ consists of the other elements. Given the
transversality between the collections for different $H$ it follows that

(9.10)
$M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]=[[M^{[k]}_{\phi};\mathcal{B}_{1,2}];\mathcal{B}^{\prime}].$

Now $D_{1,2}\subset M^{[k]}_{\phi}$ is not a p-submanifold rather it is a
‘b-submanifold’ defined locally near the boundary

(9.11) $D_{1,2}=\\{x_{1}^{H}=x_{2}^{H},\ z=z^{\prime}\\}$

where for each $H$ the $x_{i}^{H}$ are defining functions for $H,$ the same in
each factor. The blow up of $B_{H}$ in (9.8) replaces $x_{1}^{H}$ and
$x_{2}^{H}$ by polar coordinates $r^{H}\omega_{i}^{H}$ and so the lift of
$D_{1,2}$ to the first level of blow up in (9.10) is a p-submanifold defined
by the interior equations $\omega_{1}^{H}=\omega_{2}^{H},$ $z=z^{\prime}.$ So
at this stage the $\mathcal{B}^{\prime}$ and $D_{1,2}$ form a p-clean
collection. Thus the final lift of $D_{1,2}$ is a p-submanifold.

The first lift of $D_{1,2}$ is disjoint from the proper transforms of the
hypersurfaces $x_{1}^{H}=0$ and $x_{2}^{H}=0.$ The lifts of the elements
$\mathcal{B}^{\prime}$ in (9.10) depend on whether the subset $\mathcal{P}$
contains $1,$ $2,$ neither or both. In the first two cases (so the boundary
face is initially contained in $x_{1}^{H}=0$ or $x_{2}^{H}=0$ but not both)
the lift is disjoint from that of $D_{1,2}.$ So the only relevant blow-ups for
$D_{1,2}$ are those corresponding to $\mathcal{P}\cap\\{1,2\\}=\emptyset,$
which are not contained in the faces $B_{H}$ and those which are contained in
$B_{H}.$ The latter lift to their inverse images and the former to the
closures of their complements. These intersect with the lifted $D_{1,2},$
which is still diffeomorphic to $M^{[k-1]}_{\phi},$ repectively to the
boundary faces forming $\mathcal{F}_{\operatorname{b},k-1}$ namely to those
contained in $H\times M^{[k-2]}_{\phi}$ and those which are not. The
$\mathcal{B}^{\prime}\cap D_{1,2}$ can be adjusted to dimension order using
inclusions and transversality so it follows from Lemma 8 that the initial
embedding as $M^{[k-1]}$ lifts to a p-embedding of
$M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ as $D_{1,2}.$

Thus the $M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ form a
stretched product, compactifying the fibration restricted to the union of the
interiors of the fibres of $\phi.$ ∎

The b-calculus, which is the operator algebra arising from the generalized
product, is closely related to the positive real numbers as a multiplicative
group. We compactify $(0,\infty)$ using the map

(9.12) $G_{+}=(0,\infty)_{t}\hookrightarrow I=[-1,1]_{s}\text{ where
}s=(t-1)/(t+1)).$

Then we have constructed the stretched product $I[k;\operatorname{b}].$

######  Proposition 13.

The diagonal boundary hypersurfaces of $I[k;\operatorname{b}]$ associated to
$\\{1\\}$ as a boundary ‘hypersurface’ of $I$ give a simplicial
compactification, $G_{+}[k],$ of the multiplicative group of the positive real
numbers and for a bundle of compact manifold with corners $M$ the generalized
product associated to a fixed boundary $H$ of $M$ in the b-stretched product
is diffeomorphic to

(9.13)
$H[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]\times\overline{G_{+}}[k],\
G_{+}=(0,\infty).$

The freedom in the diffeomorphism to (9.13) arises from a choice of boundary
defining function for $H.$

###### Proof.

The boundary generalized products are easily identified. Namely, if $H$ is a
fixed boundary hypersurface of the total space
$M=M[1,\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ then the diagonal
boundary hypersurfaces of
$M[2;\operatorname{\operatorname{b}\kern-1.5pt\phi}]$ are $H\times[-1,1]_{s}$
where the interval can be identified as the compactification of the
multiplicative group ∎

Theorem 4 immediately yields a semiclassical version of the b-stretched
product; the algebra of smoothing operators associated to this is used (in a
rather minor way) in [5] and the pseudodifferential operators much more
substantially (and independently of this discussion) by Hintz in [3].

##  10\. Double semiclassical product

Now we return to an iterated fibration as in (8.1) and proceed to define the
‘doubly semiclassical product’ involving two semiclassical parameters
corresponding to the parameter square

(10.1) $\Delta=[0,1]_{\epsilon}\times[0,1]_{\delta}.$

The idea is to reproduce the adiabatic product for the iterated fibration on
$\epsilon=1$ with $\delta$ as parameter, the semiclassical product for $\phi$
on $\delta=1,$ with $\epsilon$ as parameter and a pulled-back version of the
semiclassical product for $\gamma$ on $\delta=0$ with $\epsilon$ as parameter.
The single space is

(10.2) $M[1;\operatorname{2scl}]=\Delta\times M.$

The Lie algebroid which is resolved by the double semiclassical product
consists of the (${\mathcal{C}}^{\infty}$ module generated by the) smooth
vector fields on $\Delta\times M$

(10.3) $\epsilon V+\epsilon\delta W$

where $V$ and $W$ are smooth vector fields on $M$ respectively tangent to the
fibres of $\phi$ and to the fibres of $\gamma.$

We first consider the semiclassical stretched product for $\phi$ in (8.1) with
$\epsilon$ as parameter, as constructed in §8

(10.4) $M[k;\operatorname{sl}\phi]=[[0,1]\times
M^{[k]}_{\phi};\\{0\\}\times\mathcal{D}_{\phi}]$

where $\mathcal{D}_{\phi}$ is the collection of diagonals with respect to the
composite fibration. The idea then is to add the parameer $\delta\in[0,1]$ and
perform adiabatic resolution over the top of the semiclassical resolution.

Recall that the products of $\gamma:M\longrightarrow F$ lift to fibrations
$\gamma^{[k]}:M^{[k]}_{\phi}\longrightarrow F^{[k]}_{\psi}$ and the adiabatic
resolution arises from the fibre diagonals for $\gamma,$ meaning the inverse
images under these maps of the diagonals in $F^{[k]}_{\psi}.$

######  Lemma 13.

For an iterated fibration, under the blow up in (10.4) the fibre diagonals for
$\gamma$ lift from $[0,1]\times M^{[k]}_{\phi}$ to a p-clean collection in
$M[k,\operatorname{sl}\phi].$

###### Proof.

For simplicity we will generally denote the $\psi$-diagonals
$(\gamma^{[k]})^{-1}(D^{\psi}_{\mathfrak{p}})$ as
$D^{\psi}_{\mathfrak{p}}\in\mathcal{D}_{\psi}.$ To see that these lift to a
p-clean collection in $M[k;\operatorname{sl}\phi]$ we examine the local
structure of the full collection

(10.5) $\\{0\\}\times\mathcal{D}_{\phi}\cup[0,1]\times
D^{\psi}_{\mathfrak{p}}\text{ in }[0,1]\times M^{[k]}_{\phi}$

based on the relationships that for each partition $\mathfrak{p}$

(10.6) $D^{\psi}_{\mathfrak{p}}=(\gamma^{[k]})^{-1}(D_{\mathfrak{p}}),\
D_{\mathfrak{p}}\subset[0,1]\times F^{[k]}_{\psi},\
(\gamma^{[k]})(\\{0\\}\times D^{\phi}_{\mathfrak{p}})=\\{0\\}\times
D^{\psi}_{\mathfrak{p}}.$

Consider first a point in the minimal diagonal $\\{0\\}\times
D^{\phi}_{J(k)}.$ This is a point $(m,\dots,m).$ We may therefore take local
coordinates in $F$ near $\psi(m)$ and local coordinates in the fibre so that
in terms of the differences, $u_{i}$ and $y_{i},$

(10.7) $\\{0\\}\times D^{\phi}_{J(k)}=\\{\delta=0,\
u_{1}=\dots=u_{k-1}=0=y_{1}=\dots=y_{k-1}\\},\\\
D^{\psi}_{J(k)}=\\{u_{1}=\dots=u_{k-1}=0\\}.$

The other diagonals through this point then take a similar form

(10.8) $\\{0\\}\times D^{\phi}_{\mathfrak{p}}=\\{\delta=0,\
L_{\mathfrak{p}}(u_{1},\dots,u_{k-1})=0=L_{\mathfrak{p}}(y_{1},\dots,y_{k-1})\\},\\\
D^{\psi}_{\mathfrak{p}}=\\{L_{\mathfrak{p}}(u_{1},\dots,u_{k-1})=0\\}$

where the collection, $L_{\mathfrak{p}}$ of linear functions of $k-1$
variables only depends on $\mathfrak{p}.$

Now consider the blow up of $\\{0\\}\times D^{\phi}_{J(k)}.$ This introduces a
new boundary hypersurface. None of the other manifolds is contained in this
centre so each lifts to the closure of the complement of the centre. Near the
interior of the front face we may use the projective coordinates $U=u/\delta,$
$Y=y/\delta,$ $\delta\geq 0$ and some more lifted functions. All the other
diagonals $\\{0\\}\times D^{\phi}_{\mathfrak{p}}$ lift into the proper
transform of $H_{\delta}$ so do not meet the interior. The lifts of the
interior p-submanifolds are the

(10.9) $D^{\psi}_{\mathfrak{p}}=\\{L_{\mathfrak{p}}(U_{1},\dots,U_{k-1})=0\\}$

so form a p-clean family. It remains to analyse a neighbourhood of a boundary
point of the front face. Here one of the functions $u$ or $y$ dominates the
others and $\delta.$ Suppose one of the components of one of the $u_{i}$
dominates. We can make a linear change of coordinates, the same in the $u_{i}$
and $y_{i}$ (so the same for all components) and apply a symmetry so that the
first component of $u_{1}$ denoted $v$ is dominant. Then $v,$ $\eta=\delta/v,$
$U_{i}=u_{i}/v$ and $Y_{i}=y_{i}/v$ become coordinates (with the first
component of $U_{1}$ being identically one). In these coordinates (which do
not necessarily vanish at the basepoint), the lifts satisfy the affine
equations

(10.10)
$\beta^{\\#}(D^{\psi}_{\mathfrak{p}})=\\{L_{\mathfrak{p}}(U_{1},\dots,U_{k-1})=0\\},\\\
\beta^{\\#}(\\{0\\}\times D^{\phi}_{\mathfrak{p}})=\\{\eta=0,\
L_{\mathfrak{p}}(U_{1},\dots,U_{k-1})=0=L_{\mathfrak{p}}(U_{1},\dots,U_{k-1})=0\\}.$

The first lift passes through the chosen base point if and only if the
coordinates satisfy all the affine constraints. Then, normalizing the
coordinates by subtracting the values at the basepoint they are all given by
linear equations in the normalized $U_{i}.$ So they form a local p-clean
collection of interior p-submanifolds. Next consider which of the lifts of the
$\beta^{\\#}(\\{0\\}\times D^{\phi}_{\mathfrak{p}})$ pass through the current
basepoint. They must project onto one of the manifolds just discussed which
does pass through the basepoint but in addition the
$L_{\mathfrak{p}}(Y_{1},\dots,Y_{k-1})$ must vanish at the base point. If this
holds these lifts are given by linear equations in terms of the normalzed
$Y_{i}.$ So in this case we see that the lifted submanifolds through a give
point consist of some collection of the $\beta^{\\#}(D^{\psi}_{\mathfrak{p}})$
defined by linear constraints in interior coordinates $U^{\prime}$ (the
collection necessarily closed under intersection) and some generally smaller
collection of the $\beta^{\\#}(\\{0\\}\times D^{\phi}_{\mathfrak{p}})$ given
by the same linear constraints in the $U^{\prime}$ and some extra linear
constraints in the $Y^{\prime}$ variables.

If none of the components of the $u_{i}$ is dominant then one of the
components of the $y_{i}$ is instead and becomes a defining function for the
front face and the same conclusion holds for the defining functions.

We will proceed by induction, over a dimension order on the $\\{0\\}\times
D^{\phi}_{\mathfrak{p}}.$ We make the following inductive hypothesis at a
given stage of the blow-up of the $\\{0\\}\times D^{\phi}_{\mathfrak{p}}.$
Namely we assume that the $D^{\psi}_{\mathfrak{q}}$ lift to a p-clean family
of interior p-submanifolds and near any point in the lift of the boundary
$\delta=0$ the $D^{\psi}_{\mathfrak{q}}$ through that point are given by
linear equations in some interior coordinates $U^{\prime}$ and the
$\beta^{\\#}(\\{0\\}\times D^{\phi}_{\mathfrak{r}})$ which pass through that
point correspond to a subset of the $\mathfrak{q}$ which do so and they are
defined by the same linear equations in the $U^{\prime}$ and some other linear
equations in some independent interior coordinates $Y^{\prime}.$ Both
collections are manifestly closed under intersection.

The discussion above is the first step but the inductive step is very similar.
Thus we are considering a point, in the lift of $H_{\delta=0},$ in the lift of
a minimal (remaining) $\\{0\\}\times D^{\phi}_{\mathfrak{p}}.$ By the
inductive hypothesis above the lift of $D^{\psi}_{\mathfrak{p}}$ must pass
through this point but there may also be smaller $D^{\phi}_{\mathfrak{q}}$
which do so. Since the collection is closed under intersection these must
correspond to $\mathfrak{q}\supset\mathfrak{p}.$ On the other hand there can
be (the lifts of) larger $D^{\psi}_{\mathfrak{q}}$ through this point and
larger $\\{0\\}\times D^{\phi}_{\mathfrak{q}}$ through the point corresponding
to a subcollection of the $D^{\psi}_{\mathfrak{q}},$
$\mathfrak{q}\subset\mathfrak{p}.$ So, by the inductive hypothesis we have
local coordinates $U,$ $Y$ ($\eta,$ defining the lift of $H_{\delta=0})$ in
which

(10.11) $\beta^{\\#}(D^{\psi}_{\mathfrak{p}})=\\{L_{\mathfrak{p}}(U)=0\\},\
\beta^{\\#}(D^{\psi}_{\mathfrak{p}})=\\{L_{\mathfrak{p}}(U)=0,\
l_{\mathfrak{p}}(Y)=0\\}$

for some linear functions $L$ and $l.$ By relabelling the variables $U$ and
using independence we can suppose that

(10.12) $\beta^{\\#}(D^{\psi}_{\mathfrak{p}})=\\{U^{\prime}=0\\},\
\beta^{\\#}(D^{\psi}_{\mathfrak{p}})=\\{U^{\prime}=0,\ Y^{\prime}=0\\}$

Here $U=(U^{\prime},U^{\prime\prime})$ is some division of the variables. It
follows that the smaller $D^{\psi}_{\mathfrak{q}}$ passing through that point
must be of the form

(10.13) $\beta^{\\#}(D^{\psi}_{\mathfrak{q}})=\\{U^{\prime}=0,\
L^{\prime\prime}_{\mathfrak{q}}(U^{\prime\prime})=0\\},$

by removing any dependence of the additional linear functions
$L^{\prime\prime}$ on $U^{\prime}.$ The larger $\psi$ diagonals through the
point, and any $\phi$ diagonal which projects onto it and passes through the
point must then be of the form

(10.14)
$\beta^{\\#}(D^{\psi}_{\mathfrak{q}})=\\{L_{\mathfrak{q}}U^{\prime}=0\\},\
\beta^{\\#}(\\{0\\}\times
D^{\phi}_{\mathfrak{q}})=\\{L_{\mathfrak{q}}U^{\prime};l_{\mathfrak{q}}(Y^{\prime})=0\\}.$

This allows the discussion above of the initial blow-up to be followed
closely. The variables $U^{\prime}$ (and $Y^{\prime\prime}$) are not involved
in the blow-up at all the inductive hypothesis is confirmed after this blow-up
so holds in general and the Lemma is proven. ∎

Thus if $\mathcal{D}_{\psi}$ is the collection of the $\psi$-diagonals pulled
back to $M[k;\operatorname{sl}\phi]$ in this way then they have the same
intersection properties as before and we define the doubly semiclassical space
by

(10.15) $M[k;\operatorname{2scl}]=[[0,1]_{\delta}\times
M[k;\operatorname{sl}\phi];\\{0\\}\times\mathcal{D}_{\psi}]$

as usual in terms of a dimension order in each case.

######  Theorem 6.

The doubly-semiclassical spaces (10.15) form a regular stretched product
resolving $\Delta\times M^{[k]}_{\phi}$ over the interior, its boundary
generalized products are $M[k;\operatorname{sl}\phi]$ over $\delta=1,$
$M[k;\operatorname{ad}]$ over $\epsilon=1$ and the pull-back of
$M[k;\operatorname{sl}\gamma]$ to $\overline{T_{\psi}F}$ over $\delta=0.$

###### Proof.

Symmetry follows by noting that the permutations on
$M[k;\operatorname{sl}\phi]$ map the $\psi$-fibre diagonals functorially and
so lift under the blow-ups to diffeomorphisms.

Thus, by Lemma 3, it remains to show that the semiclassical structural maps

(10.16) $\Pi_{k}^{\operatorname{sl}\phi}:[0,1]_{\delta}\times
M[k;\operatorname{sl}\phi]\longrightarrow[0,1]_{\delta}\times
M[k-1;\operatorname{sl}\phi]$

lift to a b-fibrations and the p-embeddings

(10.17) $D_{1,2}:[0,1]\times
M[k-1;\operatorname{sl}\phi]\longrightarrow[0,1]\times
M[k;\operatorname{sl}\phi]$

lift to a p-embeddings.

As before the first follows by applying Proposition 6 to the decomposition

(10.18) $\mathcal{D}=\mathcal{F}\sqcup\mathcal{G}$

into the $\psi$-diagonals not involving $k,$ i.e. in which it is a singleton,
and those in which it lies in one of the $\mathcal{P}_{l}.$

The former are precisely the inverse images of the corresponding diagonals,
over $\delta=0,$ in $M[k-1;\operatorname{ad}]$ so the b-fibration in (10.16)
lifts to a b-fibration

(10.19) $F:[[0,1]\times M[k;\operatorname{ad}];\mathcal{F}]\longrightarrow
M[k-1;\operatorname{2scl}].$

Each element of $\mathcal{G}$ is associated to a partition
$\mathfrak{q}\subset J(k-1)$ by removing $k-1,$ $k$ or both from
$\mathfrak{p}\subset J(k).$ The image under $\Pi_{k}^{\operatorname{ad}}$ of
the boundary face is then the corresponding
$D_{\mathfrak{q}}\subset\\{0\\}\times M[k-1;\operatorname{ad}].$ It follows
that after the blow-up of $\mathcal{F}$ these map to the corresponding front
face of $[[0,1]\times M[k-1;\operatorname{ad}];\mathcal{F}].$

That $\mathcal{D}_{1,2},$ the $\phi$-diagonal, is a p-submanifold of
$M[k;\operatorname{2scl}]$ follows as in the adiabatic case starting at (8.8).
∎

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