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 \(S_ID_m=D_n.\) 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. ◻

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 on both sides, an injection is reduced to a map \(I:J(n)\longrightarrow J(m)\) where \(I(j)=j.\) 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 \(D_{1,2}.\) ◻

Proof. This follows from the local structure of manifold and the map. ◻

Proof. Assume first that \(F_1\subset F_2.\) 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. ◻

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. ◻

Proof. The composite ?? is certainly a b-map. Suppose \(C\subset H'\) where \(H'\) 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 \(q+1,\) \(q\ge0,\) 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 \(H'.\) 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. ◻

Proof. In case \(Y\) is an interior p-submanifold then \(\phi^{-1}(Y)\) is automatically a p-submanifold. Locally any near point \(q\in\phi^{-1}(Y)\) and its image \(p=\phi(q)\) there are local product neighbourhoods \(B\times B'\) in \(N\) and \(B\times B''\) in \(M\) where \(\{b\}\times B'\) is a neighbourhood of \(p\in Y,\) \(\{b\}\times B''\) is a neighbourhood of \(q\in \phi^{-1}(Y)\) and \(\phi\) is a product map which is the identity on \(B.\) From this the result follows.

In case \(Y\) is a boundary p-submanifold, the assumption that \(\phi^{-1}(Y)\) is a (connected) p-submanifold is non-trivial. If \(Q\) is the smallest boundary face of \(N\) containing \(Y\) it follows that \(\phi^{-1}(Q)\) is a boundary face of \(M,\) rather than a union of such. In particular each of the boundary hypersurfaces of \(N\) containing \(Y\) has preimage just one boundary hypersurface of \(M.\) Then there is a similar product decomposition to the interior case and the result follows. ◻

Proof. All the blow-ups are near the boundary hypersurface \(H\in\mathcal{M}_1(N)\) containing \(Y\) or its preimage in \(M.\) We may therefore localize, replacing \(N\) by \[H\times[0,\epsilon )_x \label{GP46333}\tag{25}\] where \(x\) is a boundary defining function for \(H.\) This gives a local extension of \(Y\) to \[\tilde{Y}=Y\times[0,\epsilon ),\;\epsilon >0, \label{GP46334}\tag{26}\] which is an interior p-submanifold with \(Y=H\cap\tilde{Y}\) one of its boundary hypersurfaces. The preimage of \(Y\) is therefore the part of the preimage \(f^{-1}(\tilde{Y})\) which meets one of the boundary hypersurfaces \(H_i'\in\mathcal{M}_1(M)\) mapping to \(H.\)

Thus the preimage of \(Y\) under \(f\) is a collection of the boundary hypersurfaces of \(f^{-1}(\tilde{Y}).\) It follows from Lemma 8 that the blow-up of the \(X_*\) is permissible in any order and once an order is chosen, \([M,X_*]\) is well-defined.

To show that \(f\) lifts to b-fibration as in ?? we first show that it lifts to a smooth map. This is a local statement near the preimage in \([M,X_*]\) of an arbitrary point \(m'\in f^{-1}(Y)\) with image \(m\in Y.\) Now, near \(m\) we may choose adapted local coordinates \(y_j,\) \(j=1,\dots,n,\) \(x'_l,\) \(y'_r\) in \(H\) such that \(Y\) is defined by \(y=0,\) and the \(x'_l\) define the boundaries hypersurfaces of \(H.\) Since \(f\) is a b-fibration it decomposes as a product near \(m'\) in terms of local coordinates \(\xi_i,\) \(y,\) \(y'\) and boundary defining functions \(\xi_p'\) so locally \[f(\xi,y,\xi',y',z)=(\xi_1\dots\xi_k,y,g(\xi',y')) \label{GP46335}\tag{27}\] where \(g\) is a b-fibration in the remaining variables (which can also be brought to normal form) and the \(z\) are interior fibre variables. All the blow-ups are in the variables \(\xi_i\) and \(y,\) defining the various \(X_i\) locally, so we can ignore the variables \(\xi',\) \(y'\) and \(z\) and suppose that \(Y\) is a point in \(H\) which is simply a ball in \(\mathbb{B}^n_y\) and we are considering the map \[\begin{gather} f:\{(\xi,y)\in[0,1)^k\times\mathbb{B}^n;\xi_1\dots\xi_k<1\}\\ f(\xi,y)=(x,y),\;x=\xi_1\dots\xi_k. \end{gather} \label{GP46336}\tag{28}\]

Consider the lift of a quadratic defining function for \(Y,\) \[f^*(x^2+|y|^2)=\xi_1^2\dots\xi_k^2+|y|^2. \label{GP46337}\tag{29}\] The blow-up of \(X_1=\{\xi_1=0,y=0\}\) introduces the smooth functions \[\eta_1=(\xi_1^2+|y|^2)^{\frac{1}{2}},\; \omega _i=\frac{y_i}{\eta_1},\;\Xi_1=\frac{\xi_1}{\eta_1} \label{GP46338}\tag{30}\] which generate the \({\mathcal{C}}^\infty\)structure of the blow-up. The lift of \(\tilde{Y}\) is now \(\omega =0\) and \[\xi_1=\eta_1\Xi_1,\;f^*(x^2+|y|^2)=\eta_1^2(\Xi_1^2\xi_2\dots\xi_k^2+|\omega|^2). \label{GP46339}\tag{31}\] Here \(\eta_1\) is a defining function for the front face and \(\Xi_1\) is a defining function for the lift of the boundary hypersurface \(\xi_1=0\) containing \(X_1.\) It follows that, away from the lift of \(\tilde{Y},\) \(f\) lifts to a smooth map to the blow-up of \(Y\) since \(|\omega |^2>0\) and hence \[f^*(r)=\eta_1(\Xi_1^2\xi_2\dots\xi_k^2+|\omega|^2)^\frac{1}{2},\;f^*(y/r)=\frac{\eta_1}{r} \omega. \label{GP46340}\tag{32}\]

Thus we can proceed iteratively blowing up the lift of the next \(X_j\) \[\{\xi_j=0,\;\omega ^{(j-1)}=0\},\;\text{ where } f^*(r^2)=\eta_1^2\dots\eta_{j-1}^2(\Xi_1^2\dots\Xi_{j-1}^2\xi_j^2\dots\xi_k^2+ |\omega^{(j-1)}|^2) \label{GP46341}\tag{33}\] with essentially the same argument justifying the smoothness away from the remaining \(X_*.\) After the final step \[f^*(r^2)=\eta_1^2\dots\eta_{j-1}^2(\Xi_1^2\dots\Xi_{k}^2+|\omega^{(k)}|^2). \label{GP46344}\tag{34}\] Here the final lift of \(\tilde{Y}\) is \(\{\omega ^k=0\}\) but this p-submanifold is disjoint from all the lifted hypersurfaces \(\{\Xi_i=0\}\) so the last factor is strictly positive.

By induction it follows that \(f\) lifts to a smooth map ?? near the arbitrary point \(m'\) and \[\begin{gather} \tilde{f}^*(r)=a\eta_1\dots\eta_k,\;a>0\text{ and smooth,}\\ \tilde{f}^*(x/r)=a'\Xi_1\dots\Xi_k,\;a'>0\text{ and smooth.} \end{gather} \label{GP46343}\tag{35}\] Thus \(\tilde{f}\) is a b-normal b-map since these are the defining functions for the boundary faces of \([N;Y].\)

To see that \(\tilde{f}\) is a b-fibration it suffices to show that it is a b-submersion, i.e.any element of \(\mathcal{V}_{\operatorname{b}}([N;Y])\) has a lift to an element of \(\mathcal{V}_{\operatorname{b}}([M;X_*]).\) These b-vector fields on \([N;Y]\) are generated by the lift from \(N\) of \[\mathcal{W}=\{W\in\mathcal{V}_{\operatorname{b}}(N);W\text{ is tangent to }\tilde{Y}\}+x\mathcal{V}_{\operatorname{b}}(N). \label{GP46346}\tag{36}\] This can be seen, after the simplifications discussed above, from the fact that these reduce in local coordinates to \[x\partial_x,\;y_j\partial_{y_l},\;x\partial_{y_j}\text{ and }x^2\partial_x \label{GP46347}\tag{37}\] for all \(j\) and \(l.\) Under the iterated blow-up of the \(X_i\) these lift to be smooth, so it follows that \(\tilde{f}\) is a b-fibration. ◻

Proof. The boundary hypersurfaces of an interior p-submanifold certainly form a p-clean collection since locally, near any point, \(Y\) is defined by the vanishing of some tangential coordinates, \(y_i,\) \(i=1,\dots,q.\) Under the blow up of one boundary hypersurface, \(Y\) lifts to again be an interior p-submanifold and the remaining boundary hypersurfaces lift to be boundary hypersurfaces of the lift. Thus the result follows by induction. ◻

Proof. 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 where they are unchanged. Consider a point \(p\) in the front face and coordinates \(x'_i,\) \(x''_j,\) \(y'_l,\) \(y''_p,\) based at the image \(\beta (p)\) under the blow-down map where \(F\) is locally defined by \(x'=0,\) \(y'=0.\) 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 \(x'_i\) and linear functions of the \(y'_l.\) One of the \(x'_i\) or \(y'_l,\) denoted \(r,\) dominates the others near \(p\) and it is then a defining function for the front face and the remaining \(x'_i/r\) and \(y'_l/r\) 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 \(x'_i/r\) and some linear functions of the \(y'_l/r.\) By subtracting the values at \(p\) we may renormalize the (non-trivial) \(y'_l/r\) to give \(Y'_l\) which vanish at \(p.\) Then the lifted manifolds are defined by the appropriate \(x'_r/r\) and affine functions of the \(Y'_l.\) 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 of any two elements is the intersection of the lifts except in case two manifolds intersect in \(F;\) then their lifts are disjoint. ◻

Proof. Certainly the elements are partially ordered by dimension, so a size order is simply a total order compatible with this partial order. The elements of minimal dimension are necessarily disjoint and hence can be blown up in any order.

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

Proof. Applying Lemma 7 repeatedly gives the first part of the result. The description of \(\operatorname{ff}_C,\) for \(C\in\mathfrak{C},\) follows similarly. The blow-up of all the elements of \(\mathfrak{C}\) which are smaller than \(C\) replaces it by \([C;\mathfrak{C}\cap C]\) and giving the blow-down map \(\beta_C.\) Blowing up the lift of \(C\) up gives an initial fibration \(\phi'_C.\) Subsequent (larger) elements of \(\mathfrak{C}\) either lift to be disjoint from the lift of \(C\) or originally contained it. The latter form a pic consisting of sections of the fibration \(\phi'_C\) so these blow-ups replace \(\phi'_C\) by a final fibration \(\phi_C:\operatorname{ff}_C\longrightarrow [C;\mathfrak{C}\cap C]\) with resolved fibres. ◻

Proof. Transversal intersection includes the case that \(C_1\) and \(C_2\) are disjoint, in which case the result is obvious.

If the intersection is non-empty and transversal it suffice to consider a small neighbourhood of point of \(Q.\) If this is not in \(C_1\cap C_2\) then again this commutativity result holds trivially. Take a neighbourhood of a point in the intersection which is the product of three balls \(B_1\times B_2\times B\) where there are points \(p_i\in B_i\) such that locally \[C_1=\{p_1\}\times B_2\times B,\;C_2=B_1\times \{p_2\}\times B. \label{GP46363}\tag{44}\] Then the blow-up of \(C_1\) is \([B_1,\{p_1\}]\times B_2\times B\) and \(l_1(C_2)=[B_1,\{p_1\}]\times\{p_2\}\times B.\)

In case \(Q\subset C_1\) it is of the form \(\{p_1\}\times F\) with lift \(l_1(Q)=\operatorname{ff}([B_1,\{p_1\}])\times F.\) This is liftable for the blow-up of \(l_1(C_2)\) if \(l_1(Q)\subset l_1(C_2),\) which is equivalent to \(F=\{p_2\}\times F'\) so \(Q\subset C_1\cap C_2\) and \[l_2(l_1(Q))=\operatorname{ff}([B_1,\{p_1\}])\times\operatorname{ff}([B_2,\{p_2\}])\times F', \label{GP46364}\tag{45}\] which is symmetric so the result follows. The other possibility, given that \(Q\subset C_1,\) is that \((l_1(Q)\setminus\operatorname{ff}([B_1,\{p_1\}])\times \{p_2\}\times B\) is dense in \(l_1(Q)\) which is equivalent to the density of \(F'=F\setminus(\{p_2\}\times B)\) in \(F\) and hence of \(Q\setminus C_2\) in \(Q.\) Thus \(Q\) is liftable for the blow-up of \(C_2\) with lift \(\{p_1\}\times \tilde{F}\) where \(\tilde{F}\) is the closure of \(F'\) in \([B_2\times \{p_2\}]\times B.\) This is contained in \(l_2(C_1)\) and the double lifts are the same.

Next consider the case, as in 44 , that \(Q\setminus C_1\) is dense in \(Q.\) Then \(l_1(Q)\subset l_1(C_2)\) if and only if \(Q\subset C_2\) and the result follows readily. In the remaining case \(Q\setminus(C_1\cup C_2)\) is dense in \(Q\) and again the result follows. This completes the case that \(C_1\) and \(C_2\) meet transversally.

Now we may suppose \(C_1\subset C_2.\) If \(Q\setminus C_1\) is dense in \(Q\) then \(l_1(Q)\subset l_1(C_2)\) if and only if \(Q\subset C_2.\) Thus \(l_1(Q)\) is liftable under blow up of \(C_2\) with lift the union of the fibres over \(Q.\) The \(l_2(Q)\setminus l_2(C_1)\) is the union of the fibres over \(Q\setminus C_1\) so is dense in \(l_2(Q)\) and the result follows. If \(l_1(Q)\setminus l_1(C_2)\) is dense in \(l_1(Q)\) then \(Q\setminus C_2\) is dense in \(Q\) and both double lifts are defined and equal to the closure of the double lift of \(Q\setminus C_2.\) In case \(Q\subset C_1\subset C_2\) all lifts are preimages and commutativity follows. ◻

Proof. Clearly this is true when \(G\) is the first element so we can proceed by induction over the order, assuming that \(\mathfrak{C}_K\) is closed under intersection for all \(K<G.\) Then it suffices to show that \(F\cap G\in\mathfrak{C}_G\) if \(F<G\) which is Definition 9. ◻

Proof. We proceed by induction over the number of elements of the collection. To prove permissibility of blow-up we may assume the result for the collection consisting of the elements preceding the last using Lemma 10. We start by considering the special intersection given by a size order on all the elements preceing the last element, \(G.\) Thus \(G\) has some position in a size order on the whole collection. The elements of strictly smaller dimension than \(G\) may then be blown up in this size order and the lift of \(G\) is one of the minimal remaining elements. The initial intersection of each of the remaining elements is no larger than \(G.\) If this intersection is strictly smaller than \(G\) then it has already been blown up and the lift of the element is disjoint from \(G.\) Thus the remaining lifted elements which meet \(G\) are those which initially contained it. In fact they, without \(G,\) form a pic since the intersection of any two must, in the initial intersection order, precede one of them and hence strictly precede \(G.\) Thus at this point in the resolution the remaining elements which intersect \(G\) form a pic with a unique minimal element \(H.\) Proceeding from the complete blow up in size order, \(G\) and \(H\) may be interchanged, by commutativity. When \(H\) is blown up all the other (relevant) lifted centres meet the front face it produces in sections of the fibration over \(H\) and the lift of \(G\) is its inverse image under this blow-up. It follows that the lift of \(G\) and each of the remaining elements is transversal. Thus \(G\) can be commuted back to the last position showing that this special intersection order is permissible for resolution and equivalent to a size order.

To proceed further we alter the size order on the elements preceding \(G\) step by step, using 42 (in reverse) to successively exchange neighbours. Thus we can again assume inductively that up to some point these resolutions are all permissible and the elements preceeding the pair exchanged are in size order. Then again it follows that the lifts through the blow-up of these earlier elements leave the lifts of \(F_i\) and \(F_{i+1}\) either transversal (including disjoint) or with the larger included in the smaller. Then Lemma 9, applied with \(Q\) the lift of \(G,\) proves the inductive step showing that the new order is also permissible. ◻

Proof. This is an intersection order. ◻

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 \[\Pi_{i,j}D_{i,j}=DS_I,\;I:J(1)\longrightarrow J(k-1),\;I(1)=i. \label{GP46203}\tag{55}\] 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 \(\mu +(k-2)\kappa.\) 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 54 holds. ◻

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 \(D_{P}\subset M[k]\) for \(P=\{1,2,3\}.\) ◻

Proof. Lemma 12 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 \[D_{1,2}\cap\dots\cap D_{L-2,L-1}\text{ in }M[k-1] \label{GP46215}\tag{57}\] with the intersection identified with \(D_{P},\) \(P=\{1,\dots,L-1\}.\) 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 \[D_{2,3}\cap\dots\cap D_{L-1,L}=\Pi^{k}_{L}\left (D_{1,2}\cap\dots\cap D_{L-2,L-1}\right)\text{ in }M[k]. \label{GP46216}\tag{58}\] Equality follows from the equality of dimensions 54 . Since the fibres of \(\Pi^{k}_{L}\) at a point of the intersection 56 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. ◻

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

We know that the b-fibration \(\Pi_{1,j}:M[k]\longrightarrow M[2],\) \(2\le j\le l,\) restricts to a b-fibration of \(D_{1,j}\) over \(D\) and that \[D_{P}=D_{1,2}\cap D_{1,3}\dots\cap D_{1,l}\text{ is transversal.} \label{GP46220}\tag{59}\] Moreover the image of a point in \(D_{P}\) under each of the \(\Pi_{1,j},\) \(2\le j\le l,\) 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 \(R,\) such that \(D=\{z=0\}\) locally. Then the functions \[z_j=(\Pi_{1,j})^*z,\;j=2,\dots,l\text{ satisfy } R_{1,j}^*z_j=-z_j,\;D_{1,j}=\{z_j=0\}\text{ locally.} \label{GP46218}\tag{60}\] By Lemma 13 their differentials must (all) be independent.

Now consider the other \(D_{i,r}\) where \(1, i<r\le l.\) The structural identity \[\Pi _{1,i}R_{i,r}= \Pi _{1,r}\Longrightarrow (R_{ir})^*(z_i-z_r)=z_r-z_i. \label{GP46219}\tag{61}\] Since the components of \(z_r-z_i\) are linearly independent it follows that \[D_{i,r}=\{z_i-z_r=0\}\text{ locally.} \label{GP46221}\tag{62}\]

Thus all the diagonals containing one \(D_{P}\) for \(P\subset J(k)\) form a p-clean collection. The general case follows from this since if \(\mathfrak{P}\) consists of several disjoint sets \(P_j\) then \[D_{\mathfrak{P}}=\bigcap_{i}D_{P_i} \label{GP46222}\tag{63}\] is transversal and the local linear defining functions 62 can be constructed independently. ◻

Proof. This follows from the corresponding properties for diagonals discussed above. ◻

Proof. That 67 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, more abstractly, the discussion in Examples 1 and 2 above. 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 diagonal \(D\subset M[2].\)

The projection \(\Pi_S\) restricts to a b-fibration of \(D_{1,2}\) onto \(D.\) Moreover the identity \[\Pi_R\Pi_S=\Pi_L\Pi_F\Longrightarrow (\Pi_S)_*:\operatorname{null}((\Pi_F)_*)\longrightarrow \operatorname{null}((\Pi_R)_*). \label{GP46305}\tag{68}\] At \(D_{1,2}\) the null space of \((\Pi_S)_*\) is tangent to \(D_{1,2}\) whereas \(\operatorname{null}((\Pi_F)_*)\) is transversal to \(D_{1,2}.\) Since the dimensions are the same the push-forward map in 68 is therefore an isomorphism at each point. It follows that \(V\) lifts to a unique smooth section \[W\in{\mathcal{C}}^{\infty}(D_{1,2};\operatorname{null}((\Pi_F)_*))\text{ satisfying }(\Pi_S)_*W(p)=V(\Pi_S(p)) \label{GP46306}\tag{69}\] at each point.

Now, the projection \(\Pi_C\) restricts to a diffeomorphism of \(D_{1,2}\) onto \(M[2].\) Moreover, the identity \[\Pi_R\Pi_F=\Pi_R\Pi_C \label{GP46316}\tag{70}\] shows that the differential of \(\Pi_C\) maps \(\operatorname{null}((\Pi_F)_*)\) into \(\operatorname{null}((\Pi_R)_*)\) at each point of \(D_{1,2}.\) Thus \(W\) defined by 69 pushes forward to a smooth global section \[\tilde{V}=(\Pi_C)_*W\in{\mathcal{C}}^{\infty}(M[2];\operatorname{null}((\Pi_R)_*)). \label{GP46307}\tag{71}\]

This global section restricts to \(D\) to be the original section \(V.\) Indeed, on the triple diagonal, \(T,\) \(\Pi_S=\Pi_F=\Pi_C\) are all the identity in terms of the embeddings of \(M[1]\) as \(T\) and \(D.\) Also at the triple diagonal, \(\operatorname{null}((\Pi_F)_*)\subset TD_{2,3}.\) Now \(\Pi_CR_{2,3}=\Pi_S\) and \(R_{2,3}\) acts as the identity on \(D_{2,3}\) so it follows that \((R_{2,3})_*W=W\) and hence \((\Pi_C)_*W=(\Pi_S)_*W=V\) at \(D.\)

The extension of \(V\) to \(\tilde{V}\) also gives an extension of \(W\) to a section \(\tilde{W}\in{\mathcal{C}}^{\infty}(\Omega ;\operatorname{null}((\Pi_F)_*)\) at least for a small open neighbourhood of the triple diagonal, \(T.\) Indeed, in such a neighourhood \((\Pi_C)_*\) maps \(\operatorname{null}((\Pi)_F)_*)\) isomorphically onto \(\operatorname{null}((\Pi_R)_*)\) so \(\tilde{W}\) is well-defined pointwise by \[\tilde{V}=(\Pi_C)_*\tilde{W}\text{ near }T. \label{GP46308}\tag{72}\] Then \(\tilde{V}\) and \(\tilde{W}\) are \(\Pi_C\)-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 \(\Pi_C\)-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 67 maps commutators to commutators so \(\mathcal{E}\) does indeed have the structure of a Lie algebroid. ◻

Proof. This is a special case of the push-forward theorem from [10]. ◻

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 80 . Thus if \(A\) and \(B\in\Psi^{-\infty}(M[*];V)\) then the kernel of the composite is \[K_{AB}=(\Pi_C)_*\left(\Pi_S^*A\cdot\Pi_{1}^*\nu\times\Pi_F^*B\right)/ \Pi_{1}^*\nu \label{GP46310}\tag{101}\]

To see that this is well-defined first note that the densities behave in the expected manner: \[(\Pi_L\Pi_S)^*\Omega _{\operatorname{b}}(M[1])\otimes \Pi_S^*\Omega (\operatorname{null}((\Pi_L)_*)\otimes \Pi_F^*\Omega (\operatorname{null}((\Pi_L)_*)\longrightarrow\Omega _{\operatorname{b}}(M[3]) \label{GP46312}\tag{102}\] 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 101 defines a bilinear map \[\Psi^{-\infty}(M[*];V)\times \Psi^{-\infty}(M[*];V)\longrightarrow \Psi^{-\infty}(M[*];V). \label{GP46313}\tag{103}\]

The space \(M[4]\) can be used to see that this product is associative. Consider the structural b-fibrations \[\Pi_{[i,j]}:M[4]\longrightarrow M[2],\;1\le i< j\le 4 \label{GP46314}\tag{104}\] associated to the strictly increasing map \(I_{i,j}:J(2)\longrightarrow J (4)\) which omits \(i\) and \(j.\) The densities behave as in 102 so giving a trilinear map \[\begin{gather} \Psi^{-\infty}(M[*];V)\times \Psi^{-\infty}(M[*];V)\times \Psi^{-\infty}(M[*];V)\longrightarrow \Psi^{-\infty}(M[*];V),\\ (A,B,C)\longmapsto (\Pi_{2,3})_*\left(\Pi_{3,4}^*A\cdot\Pi_{L}^*\nu\times\Pi_{1,4}^*B\times \Pi_{1,2}\right)/ \Pi_{1}^*\nu. \end{gather} \label{GP46315}\tag{105}\] This factors through the products on the left and right to give associativity.

The pseudodifferential extension follows from the structure of the diagonals, especially 62 . ◻

Proof. As shown above the diagonal boundary hypersurfaces of \(M[2],\) those meeting \(D,\) map to the corresponding hypersurfaces of \(M[1]\) under both projections. Thus the kernels in 86 are simply supported relative to \(\Pi_L.\) The same is therefore true for the argument of the push-forward in ?? which by 82 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 \[\Psi^{-\infty}_T(M;V)\times {\mathcal{C}}^{\infty}(M[1];V)\longrightarrow {\mathcal{C}}^{\infty}[M[1];V). \label{GP46134}\tag{106}\] Thus it only remains to prove that composites lie in the same space, that \[\Psi^{-\infty}_T(M[*];V)\circ\Psi^{-\infty}_T(M[*];V)\subset\Psi^{-\infty}_T(M[*];V) . \label{SCL461}\tag{107}\]

To do so we use properties of the triple product and the three stretched projections \[\xymatrix{ &M[2]&\\ &M[3]\ar[u]_{\Pi_C}\ar[dl]_{\Pi_S}\ar[dr]^{\Pi_F}\\ M[2]&&M[2]. } \label{SCL462}\tag{108}\]

The kernel of the product of two elements, \(A_1\) and \(A_2\) is given as the push-forward \[(\Pi_C^{T})_*\left((\Pi_S^{T})^*(A_1)\cdot(\Pi_F^{T})^*(A_2)\cdot(\Pi_C^{T})^*(\nu)\right)/\nu \label{SCL465}\tag{109}\]

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 109 is a smooth b-density. Then ?? applies to show that the push-foward is a smooth b-density on \(M[2;T]\) to get the product 107 .

The multiplicative property (and extension to bundle coefficients) \[\sigma _{T}(A_1A_2)=\sigma _{T}(A_1)\cdot\sigma _{T}(A_2) \label{SCL4610}\tag{110}\] 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 109 with the kernel of the fibrewise convolution of the restricted kernels. ◻

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 \[\begin{gather} \Pi_k:(0,1]\times M[k]\longrightarrow (0,1]\times M[k-1],\\ D_{k-1,k}:(0,1]\times M[k-1]\longrightarrow (0,1]\times M[k],\;k>1, \end{gather} \label{GP46228}\tag{114}\] extend to b-fibrations and p-embeddings of the \(M[*;\operatorname{sl}].\)

The decomposition 64 of the multidiagonals in \(\mathcal{D}(k)\) induces a corresponding decomposition \(\mathcal{D}_{\operatorname{sl}}\) \[\mathcal{D}_{\operatorname{sl}}(k)=\mathcal{G}\sqcup\mathcal{R} \label{GP46229}\tag{115}\] where \(\mathcal{G}\) consists of the centres ‘which do not involve \(k\)’, i.e.the \(D_{\mathfrak{P}}\times\{0\}\) where \(k\) is a singleton and the remaining elements, in \(\mathcal{R},\) ‘do involve \(k\)’ so are of the form \(D_{\mathfrak{Q}}\times\{0\}\) where \(k\in\mathcal{Q}_j\) for some \(j.\) Clearly these two collections satisfy the hypotheses of the Corollary to Theorem 1 so \[M[k;\operatorname{sl}]=[[[0,1]\times M[k];\mathcal{G}];\mathcal{R}]. \label{GP46230}\tag{116}\]

From Lemma  14 it follows under that the product extension of \(\Pi_k\) to \([0,1]\times M[k]\) each \(\{0\}\times D_{\mathfrak{P}}\) is the preimage of \(\{0\}\times D_{\mathfrak{P}'}\) corresponding to removing the singleton \(k.\) Thus the under successive blow-ups of the elements \(\{0\}\times D_{\mathfrak{P}}\) in \(\mathcal{G}\) and the corresponding elements \(\{0\}\times D_{\mathfrak{P}'}\) forming \(\mathcal{D}_{\operatorname{sl}}(k-1),\) Proposition 5 applies and shows that \(\Pi_k\) lifts to a b-fibration from \([[0,1]\times M[k];\mathcal{G}]\) to \(M[k;\operatorname{sl}].\) After the blow-ups of the elements of \(\mathcal{G}\) each element \(\{0\}\times D_{\mathfrak{Q}}\in\mathcal{R}\) is mapped by the lift of \(\Pi_k\) onto the boundary hypersurface of \(M[k-1,\operatorname{sl}]\) which is the front face for the blow-up of \(\{0\}\times D_{\mathfrak{Q}'}.\) Thus Proposition 5 applies to each blow-up showing that \(\Pi_k\) lifts to a b-fibration of \(M[k,\operatorname{sl}]\) to \(M[k-1,\operatorname{sl}].\)

So consider the p-embedding in 114 . Appending \([0,1]\times D_{k-1,k}\) to \(\mathcal{D}_{\operatorname{sl}}(k)\) gives a p-clean collection closed under intersection with \([0,1]\times D_{k-1,k}\) as the largest element, so it certainly lifts to a p-submanifold of \(M[k;\operatorname{sl}].\) To see that it is a p-embedding of \(M[k-1,\operatorname{sl}]\) consider the decomposition the collection of the centres as \[\mathcal{D}_{\operatorname{sl}}(k)=\mathcal{F}_{k-1,k}\sqcup\mathcal{G}\sqcup\mathcal{F}. \label{GP46261}\tag{117}\] Here the diagonals defining \(\mathcal{F}_{k-1,k}\) have \(k-1\) and \(k\) appearing in the same set in the partition. The elements of \(\mathcal{F}\) are those in which both \(k-1\) and \(k\) are singletons and \(\mathcal{G}\) is the remainder, corresponding to the partitions where \(k-1\) and \(k\) appear in different sets of the partition, but at most one is a singleton. Clearly \(\mathcal{F}_{k-1,k}\) and \(\mathcal{F}\) are closed under intersection and the union of the first two sets in 117 is also closed under intersection. So a size order on each gives an intersection order on the union which is permissible and \[M[k,\operatorname{sl}]=[[[[0,1]\times M[k];\mathcal{F}_{k-1,k}];\mathcal{G}];\mathcal{F}]. \label{GP46355}\tag{118}\]

Under the extension of the reflection \(R_{k-1,k}\) the elements of \(\mathcal{F}_{k-1,k}\) and \(\mathcal{F},\) and \([0,1]\times D_{k-1,k},\) are invariant whereas each element of \(\mathcal{G}\) is mapped to a different element of \(\mathcal{G}.\) It follows that under the blow-up of \(\mathcal{F}_{k-1,k}\) in 118 \(R_{k-1,k}\) lifts to be a diffeomorphism leaving the lift of \([0,1]\times D_{k-1,k}\) invariant and exchanging the lifts of the elements of \(\mathcal{G}.\) Now the intersection an an elment of \(\mathcal{G}\) and its image under \(R_{k-1,k}\) lies in \(\mathcal{F}_{k-1,k},\) so after this first partial resolution elements of \(\mathcal{G}\) are disjoint from their images under \(R_{k-1,k}\) and hence are disjoint from the lift of \([0,1]\times D_{k-1,k}.\) Thus in a neighbourhood of the lift of \([0,1]\times D_{k-1,k}\) the blow-up of the elements of \(\mathcal{G}\) are irrelevant. Now as the image of the embedding of \([0,1]\times M[k-1],\) each element of \(\mathcal{D}_{\operatorname{sl}}(k-1)\) is the intersection of a unique element of either \(\mathcal{F}_{k-1,k}\) or \(\mathcal{F}\) with \([0,1]\times D_{k-1,k}.\) Thus, applying Lemma 7 repeatedly shows that the p-embedding in 114 lifts to a p-embedding.

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

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 ?? . It follows that these manifolds \[V\times \overline{E}[k] \label{GP46268}\tag{120}\] form a generalized product with all structure maps being invariant under the translation action through the action on the first factor. Thus ?? is indeed a generalized product and satisfies the conditions of Definition 3 for the translation group. ◻

Proof. This follows by repeating the argument used above to prove Proposition 11. ◻

Proof. The proof of Theorem 5 can be followed almost verbatim. 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 \(D_{k-1,k}\) are defined on the unresolved products \([0,1]\times M^{[*]}_{\phi}.\) Thus it remains to show that they extend from the interior to b-fibrations and p-embeddings of the adiabatic resolution. For the lift of \(\Pi_k\) 115 applies but now for the fibre diagonals. Similarly that \(D_{k-1,k}\) lifts to a p-embedding can be seen using the argument starting at 117 again applied to the fibre diagonals which have the same intersection properties. ◻

Proof. Elements \(H_{P,k}\) and \(H'_{P',k}\) for distinct hypersurfaces and any subsets \(P\) and \(P'\) intersect transversally since \(H\) and \(H'\) do so. For a fixed \(H\) the collection is closed under intersection with \[H_{P,k}\cap H_{Q,k}=H_{P\cup Q,k}. \label{GP46187}\tag{128}\]  ◻

Proof. At each level, in terms of dimension, of blow up in a size order the intersection of non-transversal elements has already been blown up, so all elements in ?? 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 \(\#(P)\) 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 last 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 \[\mathfrak{C}_{\operatorname{\operatorname{b}\kern-1.5pt\phi}}(k)=\mathcal{G}\sqcup \mathcal{R} \label{GP46176}\tag{129}\] where the first set corresponds to the \(P\) which do not contain \(k\) and the second to those which do. So each of these subsets is labeled by the \(Q\subset J(k-1)\) either by inclusion or by addition of \(k.\) Thus \[\mathcal{G}\ni C'=H_{Q,k-1}\times _\phi M,\; \mathcal{R}\ni C''=H_{Q,k-1}\times _\phi H \label{GP46177}\tag{130}\] for the fixed hypersurface depending on the set. Clearly, from 128 , \(\mathcal{G}\) 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 \(\mathcal{R}\) and of smaller dimension than either. Thus Proposition 128 applies separately for each \(H\) and \[M[k;\operatorname{\operatorname{b}\kern-1.5pt\phi}]=[[M^{[k]}_\phi;\mathcal{G}];\mathcal{R}]. \label{GP46283}\tag{131}\]

Since they do not involve the \(k\)th factor the iterated blow up of \(\mathcal{G}\) maps through the fibre product \[[M^{[k]}_\phi;\mathcal{G}]=M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}]\times _\phi M. \label{GP46284}\tag{132}\] Moreover Proposition 5 applies to \(\Pi_k\) at each step so it lifts to the b-fibration \[\Pi_k:[M^{[k]}_\phi;\mathcal{G}]\longrightarrow M[k-1;\operatorname{\operatorname{b}\kern-1.5pt\phi}]. \label{GP46285}\tag{133}\]

Proposition 4 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 \(D_{k-1,k}:M[k-1,\operatorname{\operatorname{b}\kern-1.5pt\phi}]\longrightarrow M[k,\operatorname{\operatorname{b}\kern-1.5pt\phi}]\) consider the collection \(\mathcal{B}_{k-1,k}\) of those boundary faces of the form \[B\times_\phi H\times _\phi H \label{GP46287}\tag{134}\] with \(B\) corresponding to the same \(H.\) Similarly let \(\mathcal{G}\) be the collection of the form 134 where one of the last two fibre factors is \(M\) and the other is \(H\) and finally let \(\mathcal{B}\) be the collection where the last two fibre factors are \(M.\) Thus \[\mathfrak{C}_{\operatorname{b}}(k)=\mathcal{B}_{k-1,k}\sqcup\mathcal{G}\sqcup\mathcal{B}. \label{GP46288}\tag{135}\] A size order on each collection in order is an intersection order so Proposition 1 applies.

Moreover we can change the order on \(\mathcal{B}_{k-1,k}\) by placing the \(M^{[k-2]}_{\phi}\times_\phi H\times_\phi H\) first and still have an intersection order. An elementary calculation shows that after the blow up of these (transversal) factors the diagonal \(D_{k-1,k}\) becomes a p-submanifold. The remaining elements, which the lift of \(D_{k-1,k}\) form a collection closed under non-transversal intersection and it follows that \(D_{k-1,k}\) lifts to a p-submanifold of \(M[k,\operatorname{\operatorname{b}\kern-1.5pt\phi}].\)

The reflection \(R_{k-1,k}\) is a diffeomorphism leaving elements of the first and last collections in 135 fixed and maping each element of the second collection to a different element. The intersection of each element of \(\mathcal{G}\) with its image under the reflection lie in \(\mathcal{B}_{k-1,k},\) so after these faces are blown up all such pairs are disjoint. Since \(D_{k-1,k}\) is the fixed set of \(R_{k-1,k}\) the blow up of the elements of \(\mathcal{G}\) do not affect the lift of \(D_{k-1,k}.\) As in the proof of Theorem 5 it follows that the the lift of \(D_{k-1,k}\) is naturally diffeomorphic to \(M[k-1,\operatorname{\operatorname{b}\kern-1.5pt\phi}].\)

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 \(\phi.\) ◻

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 ◻

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 \[\{0\}\times \mathfrak{C}_{\phi}\cup [0,1]\times D^\psi_{\mathfrak{P}}\text{ in }[0,1]\times M^{[k]}_{\phi} \label{16468462023462}\tag{141}\] based on the relationships that for each partition \(\mathfrak{P}\) \[D^\psi_{\mathfrak{P}}=(\gamma ^{[k]})^{-1}(D_{\mathfrak{P}}),\;D_{\mathfrak{P}}\subset [0,1]\times F^{[k]}_\psi,\;(\gamma^{[k]})(\{0\}\times D^\phi_{\mathfrak{P}})=\{0\}\times D^\psi_{\mathfrak{P}}. \label{16468462023463}\tag{142}\]

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 \(y_i,\) \[\begin{gather} \{0\}\times D^\phi_{J(k)}=\{\delta =0,\;u_1=\dots=u_{k-1}=0=y_1=\dots=y_{k-1}\},\\ D^\psi_{J(k)}=\{u_1=\dots=u_{k-1}=0\}. \label{16468462023464} \end{gather}\tag{143}\] The other diagonals through this point then take a similar form \[\begin{gather} \{0\}\times D^\phi_{\mathfrak{P}}=\{\delta =0,\;L_{\mathfrak{P}}(u_1,\dots,u_{k-1})=0=L_{\mathfrak{P}}(y_1,\dots,y_{k-1})\},\\ D^\psi_{\mathfrak{P}}=\{L_{\mathfrak{P}}(u_1,\dots,u_{k-1})=0\} \label{16468462023465} \end{gather}\tag{144}\] 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 \(U=u/\delta,\) \(Y=y/\delta,\) \(\delta \ge0\) 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 \[D^\psi_{\mathfrak{P}}=\{L_{\mathfrak{P}}(U_1,\dots,U_{k-1})=0\} \label{16468462023466}\tag{145}\] 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 \[\begin{gather} \beta ^{\#}(D^\psi_{\mathfrak{P}})=\{L_{\mathfrak{P}}(U_1,\dots,U_{k-1})=0\},\\ \beta ^{\#}(\{0\}\times D^\phi_{\mathfrak{P}})=\{\eta =0,\;L_{\mathfrak{P}}(U_1,\dots,U_{k-1})=0=L_{\mathfrak{P}}(U_1,\dots,U_{k-1})=0\}. \label{16468462023467} \end{gather}\tag{146}\]

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 \(U'\) (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 \(U'\) and some extra linear constraints in the \(Y'\) 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 size 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 \(U'\) 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 \(U'\) and some other linear equations in some independent interior coordinates \(Y'.\) 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 \[\beta^{\#}(D^\psi_{\mathfrak{P}})=\{L_{\mathfrak{P}}(U)=0\},\; \beta^{\#}(D^\psi_{\mathfrak{P}})=\{L_{\mathfrak{P}}(U)=0,\;l_{\mathfrak{P}}(Y)=0\} \label{GP46249}\tag{147}\] for some linear functions \(L\) and \(l.\) By relabelling the variables \(U\) and using independence we can suppose that \[\beta^{\#}(D^\psi_{\mathfrak{P}})=\{U'=0\},\; \beta^{\#}(D^\psi_{\mathfrak{P}})=\{U'=0,\;Y'=0\} \label{GP46250}\tag{148}\] Here \(U=(U',U'')\) is some division of the variables. It follows that the smaller \(D^{\psi}_{\mathfrak{Q}}\) passing through that point must be of the form \[\beta^{\#}(D^\psi_{\mathfrak{Q}})=\{U'=0,\;L''_{\mathfrak{Q}}(U'')=0\}, \label{GP46251}\tag{149}\] by removing any dependence of the additional linear functions \(L''\) on \(U'.\) 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 \[\beta^{\#}(D^\psi_{\mathfrak{Q}})=\{L_{\mathfrak{Q}}U'=0\},\; \beta^{\#}(\{0\}\times D^\phi_{\mathfrak{Q}})=\{L_{\mathfrak{Q}}U';l_{\mathfrak{Q}}(Y')=0\}. \label{GP46252}\tag{150}\] This allows the discussion above of the initial blow-up to be followed closely. The variables \(U'\) (and \(Y''\)) 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. ◻

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 \[\Pi_k^{\operatorname{sl}\phi}:[0,1]_\delta \times M[k;\operatorname{sl}\phi]\longrightarrow [0,1]_\delta \times M[k-1;\operatorname{sl}\phi] \label{GP46245}\tag{152}\] lift to a b-fibrations and the p-embeddings \[D_{k-1,k}:[0,1]\times M[k-1;\operatorname{sl}\phi]\longrightarrow [0,1]\times M[k;\operatorname{sl}\phi] \label{GP46246}\tag{153}\] lift to a p-embeddings.

As before the first follows by applying Proposition 1 to the decomposition \[\mathfrak{C}=\mathcal{F}\sqcup\mathcal{G} \label{GP46247}\tag{154}\] 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 152 lifts to a b-fibration \[F:[[0,1]\times M[k;\operatorname{ad}];\mathcal{F}]\longrightarrow M[k-1;\operatorname{2scl}]. \label{GP46248}\tag{155}\] 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 \(D_{k-1,k},\) the \(\phi\)-diagonal, is a p-submanifold of \(M[k;\operatorname{2scl}]\) follows as in the adiabatic case. ◻