Compact controlled Whitney spaces admit compatible finite triangulations

Let $W$ be compact with finite compatible tube data and the actual commuting radial collars of [F1]. Retain outer maps while using inner normalized domains. Fix a stratum $X$ and a level $0<a<1$; write $A=\{p\in T_X:{\rho }_X(p)=a\}$ and $L=\bar{A}$ in $W$, which is compact. A limit of points of $A$ leaving $T_X$ must land on the frontier of $X$: in tube parameters either the bases have an interior limit, which retains the tube because the normalized radius is the fixed $a<1$, or the bases approach the frontier, where the imposed displacement bound forces the point and its base to have the same limit. Consequently $L$ meets only upper incident strata, where it is the smooth hypersurface ${\rho }_X=a$, and lower frontier strata of $X$; it misses $X$ itself and incomparable strata.

Membership in $L$ on a lower frontier stratum $Y<X$ is locally constant along $Y$, by transport rather than by an appeal to a Whitney property of $L$. On a small $Y$-base coordinate box construct smooth lifts of each base coordinate field in increasing dimension, with the maximal incident tube controller as in [F1]; the field on $Y$ is the given coordinate field and upper fields satisfy $d{\pi }_{Z}V=V_Z{\pi }_Z$ and $d{\rho }_{Z}V=0$ on the retained tube of every lower $Z$. On the $X$-tube the lift therefore preserves ${\rho }_X=a$. The compact-trapping proof of [F1] gives a jointly continuous short flow even without ambient boundedness on an entire outer tube, and it maps $A$ to itself wherever defined, hence maps its closure $L$ to itself, as does the inverse flow. Ordered coordinate transports along $Y$ prove local constancy of membership, so $L\cap Y$ is an open and closed union of components of $Y$.

The same horizontal-lift construction, with the level constraints imposed in the chain of controllers, shows that the closure-incidence label of an upper level piece is locally constant on each lower level piece. If a disconnected level piece has different incidence labels, split it by the finitely many Boolean labels recording membership in the closures of the finitely many upper pieces; these subsets are open and closed smooth pieces, and since there are at most finitely many labels this makes a finite frontier stratification without assuming finitely many connected components. The maps and collars below restrict to these pieces, because the transports preserve each closure and hence its incidence labels.

For a lower frontier $Y<X$ the restricted $Y$-tube retracts $L$ into $L\cap Y$: radial $Y$-transport preserves ${\rho }_X$ on $A$ by the commuting collar identities, and its limit as ${\rho }_Y\to 0$ is ${\pi }_Y$, so ${\pi }_Y(A\cap T_Y)\subseteq L$ and continuity gives the same inclusion for $L\cap T_Y$. For an upper $Y>X$ the identity ${\rho }_X{\pi }_Y={\rho }_X$ shows directly that ${\pi }_Y$ preserves the level hypersurface. Therefore all inherited projections have their actual target in the corresponding stratum of $L$.

The inherited radius/projection maps are submersive on the smooth strata of $L$. If $Y<X<Z$, then on $Z$ the map $({\pi }_X,{\rho }_X)$ is submersive and $({\pi }_Y,{\rho }_Y)$ factors through ${\pi }_X$, while the latter pair is submersive on $X$; hence the joint map $({\pi }_Y,{\rho }_Y,{\rho }_X)$ is submersive on $Z$, and restricting to ${\rho }_X=a$ gives the full rank of $({\pi }_Y,{\rho }_Y)$ there. If $X<Y<Z$, then $({\pi }_Y,{\rho }_Y)$ is submersive on $Z$ and ${\rho }_X$ factors through ${\pi }_Y$; its restriction to the kernel of $d{\rho }_X$ surjects onto the tangent space of $(Y\cap \{{\rho }_X=a\})\times (0,1)$, because lifting a target vector with $d{\rho }_X=0$ through the original pair and using the factorization forces the lift to satisfy the same equation. This is precisely the needed level-pair submersion.

All control equations and incomparable-domain disjointness restrict, and local properness restricts because $L$ is closed in compact $W$. The radial families restrict too: they preserve $A$ where defined, preserve its closure by continuity, and preserve the new incidence labels; their semigroup, commutation and projection equations are the original equations on actual retained domains, and zero-end convergence still follows from the compact tube estimate. Thus $L$ is a compact finite smooth controlled stratified space with commuting collars. Its maximal stratum dimension is at most $\dim (W)-1$, since upper level strata lose one dimension and lower frontier strata have dimension strictly less than $\dim X\le \dim W$; this is the dimension drop needed by the interior triangulation induction. Nothing here identifies $L$ with an algebraic or semialgebraic set.

For each stratum $Z$ define its core by removing the small open tubes of lower-dimensional strata; compactness is literal. A sequence in this core tending to the frontier of $Z$ would eventually lie in the open tube of its limiting lower stratum, contradicting the definition, so the core is a closed subset of $W$ contained in $Z$, hence compact; the same argument applies to a level link after lower-stratum tubes are removed.

At a core point the active lower controllers form a chain and the differentials of their radius functions are independent on the stratum. For a chain $Y_1<\cdots <Y_k<Z$, the submersion $({\pi }_{Y_k},{\rho }_{Y_k})$ on $Z$ and the factorizations of the earlier radius functions through ${\pi }_{Y_k}$ let one lift first through the earlier joint radius map and then through the maximal pair, proving surjectivity of the joint radius map on $Z$. In constant-rank coordinates the inequalities ${\rho }_{Y_j}\ge d_j$ are independent half-space coordinates, so the core is a smooth compact manifold with corners and every designated face or face intersection is a genuine corner face, which checks the corner-face compatibility premise of [F3] and [F4].

The normal level-core map to a fixed base core, ${\pi }_X:Z\cap S_X(a)\to X$, is a proper corner submersion with compatible faces. For $Y<X$ the boundary coordinate ${\rho }_Y-d_Y$ is the pullback of the same base boundary coordinate under ${\pi }_X$, and for $X<Y<Z$ it is an extra source face coordinate; the pair-map factorization proves surjectivity of the joint differential $(d{\pi }_X,d{\rho }_{Y_1},\dots ,d{\rho }_{Y_k})$ for the extra source faces, and the preceding kernel argument also applies to the fixed equation ${\rho }_X=a$. Thus local smooth constant-rank charts put the map and all active faces in the form $(b,u)\mapsto b$, with base inequalities in the coordinates $b$, the corresponding source inequalities equal to those base inequalities, and extra source inequalities in the coordinates $u$; properness follows from the proved compactness of the truncated cores.

To obtain bundle charts choose a sufficiently small corner box in the base. For each base coordinate $b_i$ choose local smooth horizontal fields satisfying $d{\pi }_X{V}_i=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{b}_i$ and $d{h}_j{V}_i=0$ for every active extra source face-defining coordinate $h_j$; the joint differential just proved has a smooth right inverse giving these fields. Choose patches avoiding every inactive closed face and retaining every active one; a smooth subordinate partition preserves all these linear equations, and no derivative of a partition coefficient enters. A source face pulled back from the base automatically has the prescribed derivative of that base coordinate, so it must not incorrectly be required tangent to an outward base field.

On a compact inverse image finitely many extended smooth corner charts give a common short flow time. Flow each coordinate in a fixed order from a selected base point, possibly on a boundary face; the coordinate segment remains in the base orthant, extra source faces are invariant, and corresponding source faces follow the nonnegative base coordinates, so the source path remains in the corner manifold. Extend the smooth field temporarily across the coordinate hyperplanes in its chart to define the one-sided boundary flow and retain only the part with nonnegative base and fibre face coordinates. Compactness and the face constraints exclude finite-time exit, and reversing the coordinate flows returns to the selected fibre and supplies the smooth inverse. These products preserve every named face and the projection exactly, and a finite base cover gives genuine finite smooth bundle product trivializations, which are the additional hypothesis required by the relative smooth bundle-fitting author [F4].

For a finite controlled space $W$ let $W^{(p)}$ be its union of strata of real dimension at most $p$, closed by the frontier axiom. Put $T_p(d)={\bigcup }_{\dim Y\le p}\{{\rho }_Y<d\}$, and for a stratum $X$ of dimension $i$ put $X_d^0=X\setminus T_{i-1}(d)$. Every such core is a compact manifold with corners by step 8.1. For a top stratum $Z$ of dimension $n$ put $S_d^p(Z)=Z_d^0\cap {\bigcup }_{\dim Y\le p}\{{\rho }_Y=d\}$; intersections of these sets are genuine corner faces because incomparable tube domains are disjoint and the joint rank of the radii in every active chain has been proved.

An interior $d$-triangulation triangulates all cores $X_d^0$, retains every intersection with a lower normal level as a subpolyhedron, and makes the level-to-base projections PL. A partial $(d,p)$-triangulation does the same on all lower-dimensional cores and only on $S_d^p(Z)$ for each top stratum. At every stage the prescribed faces and their intersections are full subcomplexes after one common barycentric subdivision. Smooth-core pieces and positive-width collar pieces carry the nondegenerate local inverse and tangent-star contract of [F3] and [F4]; this is not asserted at a singular collapsed end, and singular ends arise only in the final topological gluing.

The induction retains buffers: if $d^{\mathrm{\prime }}<d$, choose $d^{\mathrm{\prime }}<d^{\mathrm{\prime }\mathrm{\prime }}<d$, retain the prescribed pieces on a closed collar, and fit within the larger open collar. All objects are compact and there are finitely many strata and flags, so each supplier's local fine-approximation bounds have positive minima on each of the finitely many compact pieces and a finite subdivision satisfies them. There is no infinite process, and no claim that an arbitrary preassigned incompatible smooth boundary triangulation can be extended.

Given an already triangulated level $J=f^{-1}(S_X(d))$, attach the product $J\times [d^{\mathrm{\prime }},d]$ along $J\times \{d\}$ and map $(q,t)\mapsto R_X(t)f(q)$; this is an embedding onto the annular $X$-collar whose inverse records the outer level point and ${\rho }_X$. The product has a finite triangulation, for instance by ordering simplex vertices and decomposing each simplex prism into finitely many monotone staircase simplices, with the same vertex order on shared faces; restrict this subdivision to every marked face. If several collars are extended, the radial commutation identity makes their two product parameterizations agree on overlap, including their corner parameters, so a common subdivision of the finitely many product faces glues them without changing their maps. Every radius preserved by a transport has the same marked level set before and after extension, and every lower projection transported equivariantly retains its induced PL coordinate expression; this is the precise sense in which extension is independent of the order of incidence of collar operations.

The mutually inductive assertions are $B_n$: extend an interior triangulation of the lower skeleton to an interior triangulation of an $n$-dimensional space after decreasing $d$; and $C_{(n,p)}$: extend a partial top-boundary triangulation through the faces whose lower controlling stratum has dimension $p$, after decreasing $d$. The order is $B_{n-1}$, then $C_{(n,0)},\dots ,C_{(n,n-1)}$, then $B_n$. The base consists of compact zero-dimensional cores, which are finite because each point is open in its zero-manifold and compactness gives a finite subcover by singleton neighbourhoods. The relative strengthening retains the prescribed finite collar faces and their inherited projections, and every application below is to boundary data produced by the preceding collar and core construction, with the star contract and compatible corners; it is not an unrestricted relative triangulation theorem for arbitrary subsets.

For $C_{(n,p)}$ process the $p$-dimensional lower strata $X$ separately; their tubes are disjoint. Work at the original outer level $S_X(d)$, reserving $d^{\mathrm{\prime }}<d^{\mathrm{\prime }\mathrm{\prime }}<d$. The already triangulated portion consists of its intersections with the lower skeleton and the top boundary faces previously handled by $C_{(n,q)}$ for $q<p$. Extend those prescribed portions along every other lower collar to the intermediate parameter $d^{\mathrm{\prime }\mathrm{\prime }}$; their intersections with $S_X(d)$ are still subpolyhedra by radius preservation. Pass to the compact closure of the level link, whose controlled structure and strict dimension drop were proved in steps 6.1 and 7.1, and apply the already completed dimension-$(n-1)$ interior and partial extension induction to that link, retaining the prescribed collar intersections. This triangulates all lower link pieces and every required boundary of the still untriangulated top link core.

The map to the fixed triangulated $X$-core is PL on this prescribed portion. On a portion controlled by a higher stratum $Y>X$ it factors as the inherited link projection to $Y\cap S_X(d)$ followed by the already triangulated map to $X$, and both factors are PL by the induction data. On portions near a lower $Y<X$ it is equivariant with the previously defined collar transports, so the product coordinate maps are the transported old PL maps. All these factorizations hold on actual tube overlaps, and the radius equations keep their source and target in the retained cores.

The relative bundle prerequisite is a diagram on an entire closed collar, not merely on its boundary; it is supplied before fitting. Extend every already triangulated boundary face by a positive-width radial product collar. For a controller $Y>X$ one has ${\pi }_X{R}_Y={\pi }_X$, so the affine diagram map ignores that collar coordinate; for $Y<X$ one has ${\pi }_X{R}_Y=R_Y{\mathrm{\mid }}_X{\pi }_X$, and the fixed target triangulation is extended by the identical $Y$-collar operation, so the diagram map carries that collar coordinate identically to the base. On multiple-face intersections the radial families commute and give the same product coordinate maps in either order. Thus ${\pi }_{X}F=hr$ holds throughout the closed collar and $r$ is PL after a common product subdivision; a smaller collar is protected and the positive-width buffer separates every later fitting support from it. This is the actual collar-neighbourhood hypothesis of [F4], not an assertion that a manifold boundary-extension theorem automatically preserves a bundle projection.

What remains in the top link stratum is a smooth compact manifold-with-corners core whose map to the fixed $X$-core has the actual finite face-compatible bundle charts proved in steps 9.1-11.1. The relative smooth bundle theorem [F4] triangulates it over that fixed base, making the projection PL; the projection-preserving fitting supplier fits it to the retained positive-width boundary collar, fixes the prescribed subcomplex and retains the projection exactly. The nondegenerate local star premise holds on both pieces, being supplied for the smooth core by [F3] and for the collar by the product construction. Full-subcomplex intersections follow from the fitting contract and a common subdivision. Transport the resulting entire link triangulation radially from ${\rho }_X=d$ to ${\rho }_X=d^{\mathrm{\prime }}$; its projection to $X$ does not change and its intersections with all previously treated collar faces agree with the previous partial extension, because the $X$-radial transport commutes with those transports and preserves their radii. Glue along those common full subcomplexes. This constructs $C_{(n,p)}$ using only smaller-dimensional induction and the proved smooth bundle-fitting interface.

For $B_n$ use the completed $C_{(n,p)}$ to triangulate every top-stratum boundary at an intermediate parameter $d^{\mathrm{\prime }\mathrm{\prime }}$, and extend this boundary triangulation to parameter $d^{\mathrm{\prime }}$ by the collar operator. The remaining portion of each top stratum is a smooth compact manifold with corners; apply the relative manifold triangulation theorem of [F3] and the ordinary finite-fitting clause of [F4] to that core, retaining its already triangulated closed boundary collar, so no bundle structure on the generic top core is required. The shared collar has the star contract and a buffer, and every face intersection is a genuine corner face; a finite simultaneous subdivision makes the overlaps full subcomplexes. Gluing the finite core and collar complexes gives the interior triangulation, the already lower-dimensional pieces and their boundary projections retain their maps, and only finitely many $(n,p)$ assertions occur, so finitely many decreases of $d$ leave a positive final parameter.

Fix the resulting interior triangulation $g_n:J_n\to W$; its image is the disjoint union of all stratum cores. Descend in $i$, with the invariant that the image of $g_i$ is this union plus $W\setminus T_{i-1}(d)$, that all stratum portions and future lower level sets are subpolyhedra, and that the projection from every future lower level to its current base portion is PL. Only these future projections need be retained; no claim is made that every old higher projection becomes a PL map on the completed space.

For a stratum $X$ of dimension $i-1$ let $A_X=g_i^{-1}(S_X(d))$ and $B_X=g_i^{-1}(X)$; the base portion is precisely $X_d^0$ because the smaller-dimensional frontier tubes have not yet been filled, and the normal level portion is correspondingly truncated over this base, hence compact, with PL projection $f_X:A_X\to B_X$ by the invariant. Finite subdivision makes it simplicial by an explicit finite polytope construction: cut the target by the affine-hull and facet hyperplanes of the images of every face of every source simplex on which the map is affine, so every such image is a union of resulting target cells; cut each source simplex by the inverse images of these target cells. Every face of a source cell is a source face intersected with the inverse image of a target face, so its image is one target cell or empty, because the target refinement has already made every original source-face image a union of cells. For each nonempty source cell choose an interior point mapping to the chosen interior point of its image target cell, which exists because an affine surjection of a convex polytope onto its image sends relative interior onto relative interior. Cone the face subdivisions using these points and barycentrically cone the target cells; nested source faces have nested image target faces, with repetitions allowed, so the resulting map is simplicial. Mark all intersections with future lower levels and stratum closures, and apply one further barycentric subdivision to make all these marked subcomplexes full.

Use the mapping-cylinder helper [F2] for $f_X$, allowing simplex vertex identifications; its finite domain and explicit homeomorphism with threshold inverse have already been constructed. Map its topological cylinder to $W$ by $(q,t)\mapsto R_X(td)g_i(q)$ for $t>0$, with the zero end mapped to ${\pi }_X{g}_i(q)$. [F1] supplies continuity at that zero end and the homeomorphism onto the closed $X$-collar over $X_d^0$. The new collar meets the old image only in its outer level $A_X$ and base $B_X$: every old higher collar began at a core with ${\rho }_X\ge d$ and its radial transports preserve ${\rho }_X$, so no old point has $0<{\rho }_X<d$ over this base; two distinct same-dimensional $X$-collars are disjoint because their tubes are incomparable. Thus gluing along the two boundary copies is injective; the domain is finite compact, the target $W$ is Hausdorff, and the radial inverse proves surjectivity onto exactly the newly filled collar, so the map onto the new image is a homeomorphism.

For a future lower stratum $Y<X$ the functions ${\rho }_Y$ and ${\pi }_Y$ factor through ${\pi }_X$ on the $X$-collar, so the part of $S_Y(d)$ inside this new cylinder is exactly the cylinder over the corresponding marked source and base subcomplex pair; the mapping-cylinder helper makes it a subpolyhedron, and its projection to the future $Y$-base is the cylinder retraction followed by the existing base projection ${\pi }_Y{\mathrm{\mid }}_X$, both PL. This proves the descending invariant. Stratum-closure intersections are retained by the same subcylinder compatibility, because the radial maps preserve strata, so an upper stratum or its closure contributes its marked top subset together with the appropriate zero-end base subset. No dimension used here is a complex perversity dimension.

After all $i$ have been processed the image is $W$, and all strata and prescribed unions of strata are unions of open simplices. The complex dimension is at most the maximal real stratum dimension: smooth core complexes have their manifold dimension, link dimensions drop by one, and the mapping-cylinder dimension bound of [F2] adds at most one to the source-link dimension while retaining the smaller base dimension. The finite simplicial cylinder maps are smooth inside positive-height cells, and radial transport is smooth on each smooth stratum; at a zero end the lower stratum already has its smooth core triangulation, and no ambient smooth immersion through a singular end is asserted. The finite list of glued complexes therefore supplies the claimed finite compatible stratified triangulation by the preceding interior construction and the exact relative-supplier premises.