Algebraic Milnor neighborhoods admit controlled sector transport

First refine as in [F2] to respect the zero fibre. On each irreducible component of a smooth stratum where $f$ is nonconstant, [F5] gives a nonempty open of the affine line over which its differential is onto. Its proper closed complement is finite by [F1]. There are finitely many stratum components, and constant strata have one value each, so excluding the finitely many nonzero exceptional values leaves no interior stratum critical point over a sufficiently small punctured disc.

For a nonconstant stratum $S$ of dimension $s$, close in ${\mathbb{C}}^M\times ({\mathbb{C}}^M{)}^{\ast }$ the locus $C_S^{\circ }=\{(x,\xi ):x\in S,f(x)\mathrm{\ne }0,df{\mathrm{\mid }}_{T_{x}S}\mathrm{\ne }0,\xi {\mathrm{\mid }}_{T_{x}S}\in \mathbb{C}d{f}_x\}$. Its covector fibre has dimension $M-s+1$, so every component of its closure $C_S$ has dimension $M+1$. The function $f$ is nonconstant on each base component, and its zero fibre is a proper closed subset there. Hence ${\mathrm{\Lambda }}_S=C_S\cap \{f(x)=0\}$ has dimension at most $M$. The closure includes base points in lower strata; no unproved relative Thom condition is being assumed.

Identify covectors with their coefficient vectors and consider $F:{\mathrm{\Lambda }}_S\times \overline{{\mathrm{\Lambda }}_S}\to {\mathbb{C}}^M\times {\mathbb{C}}^M$, $F(x,\xi ,y,\eta )=(x-\eta ,y-\xi )$. Its source has dimension at most $2M$: applying the generic-fibre formula to the projection of each product component gives the sum of the two component dimensions. For each dominant irreducible component [F1] gives zero-dimensional, hence finite, fibres on a nonempty target Zariski open; exclude the closed image closure of each nondominant component. A common nonempty open $O$ therefore has finite fibres. The real diagonal $a\mapsto (a,\bar{a})$ meets $O$ densely in every real open set: a nonzero polynomial vanishing on its proper algebraic complement remains nonzero after the invertible substitution $(a,b)=(u+iv,u-iv)$, and cannot vanish on a real open box, by induction on the number of variables using the finite-root property of each nonzero one-variable polynomial. Intersect these finitely many good parameter sets for all the strata.

On a complex tangent space $T_{x}S$ where $df$ is onto, put $\mathrm{\ell }=d\mathrm{\mid }x-a{\mathrm{\mid }}^2{\mathrm{\mid }}_{T_{x}S}$. At a transverse sphere point $\mathrm{\ell }\mathrm{\ne }0$. The restriction of $df$ to $\ker \mathrm{\ell }$ then fails to be onto exactly when $\mathrm{\ell }$ is a real linear combination of $\mathrm{Re}df$ and $\mathrm{Im}df$. Indeed a real covector on the target annihilating the image of $\ker \mathrm{\ell }$ pulls back to a multiple of $\mathrm{\ell }$, and the converse is immediate. A nontransverse sphere point is separately a bad boundary point: there $\mathrm{\ell }=0$, so it also satisfies the covector inclusion below. Identifying real covectors with real parts of complex covectors, the obstruction is $\bar{x}-\bar{a}\in N_{S,x}^{\ast }+\mathbb{C}d{f}_x$.

The good relative-conormal centre set of step 3.1 is Euclidean open and dense. Apply [F7] within its intersection with an arbitrarily small real open neighborhood of $p$, obtaining such a centre $a$ for which the ordinary distance has only finitely many critical points on all strata. In addition to $E_a$, avoid their finitely many critical distances when choosing the radius. Thus the sphere is transverse to all strata, including those inside $f=0$. The set $E_a=\{x\in X\cap f^{-1}(0):(x,\bar{x}-\bar{a})\in {\mathrm{\Lambda }}_S\text{ for some }S\}$ is finite: every such $x$ gives $(x,\bar{x}-\bar{a},\bar{x},x-a)$ in the finite fibre of $F$ over $(a,\bar{a})$. Choose $ϵ>\mathrm{\mid }a-p\mathrm{\mid }$ avoiding the finitely many distances $\mathrm{\mid }x-a\mathrm{\mid }$ for $x\in E_a$. This choice is cofinal. Inside a prescribed $B_R(p)$ choose $\mathrm{\mid }a-p\mathrm{\mid }<R\mathrm{/}4$ and $ϵ\in (R\mathrm{/}3,R\mathrm{/}2)$ avoiding those distances; then $p$ lies in the ball and its closed ball lies in $B_R(p)$.

If no sufficiently small punctured tube avoided that obstruction on the sphere, there would be $x_j\in S\cap \mathrm{\partial }{B}_{ϵ}(a)$ with $f(x_j)\mathrm{\ne }0$, $f(x_j)\to 0$ and the obstruction of step 3.2. Compactness of $X\cap {\bar{B}}_{ϵ}(a)$ and finiteness of the strata give a subsequence in one $S$ converging to $x_0\in X\cap \mathrm{\partial }{B}_{ϵ}(a)\cap f^{-1}(0)$. Step 3.2 gives $(x_j,{\bar{x}}_j-\bar{a})\in C_S$, so its limit belongs to ${\mathrm{\Lambda }}_S$. Thus $x_0\in E_a$ and $\mathrm{\mid }{x}_0-a\mathrm{\mid }=ϵ$, contradicting step 4.1. There is consequently $\delta >0$ with no sphere obstruction for $0<\mathrm{\mid }f\mathrm{\mid }<\sqrt{2}\delta$, and shrink it also as in step 1.1.

Over the punctured rectangle, the sphere is transverse to every input stratum and $f$ is submersive on both its interior and boundary intersections by steps 1.1, 3.2 and 5.1. Transverse intersection with a smooth hypersurface preserves Whitney conditions: the limiting tangent planes intersect its tangent kernel in the expected dimension, and the first-order Taylor formula puts limiting fibre secants in that kernel. Thus the interior/sphere intersections are a finite Whitney stratification. The closed-ball tube is proper over this base since inverse images of compact base subsets are closed in the compact $X\cap {\bar{B}}_{ϵ}(a)$. The actual hypotheses of [F3] therefore hold, and its transports preserve the boundary stratum and hence the interior.

On a contractible half-rectangle, fixed ordered coordinate paths and [F3]'s compact-path transports give local products and inverses. The coefficient reconstruction of [F4] transports $A$ on these products; finite interval subdivisions glue the comparisons, and reversing transport gives inverse cochain maps. These local coefficient comparisons preserve the marked boundary and therefore restrict to the open-ball interior; the global finite fibre models are constructed in the final step below. Step 4.1 supplies a cofinal family; intersecting with arbitrarily small function rectangles retains cofinality at $p$. Smooth affine charts reduce the last assertion to this construction. This proves the transport assertion.

Replace the rectangle by a small circular function disc and consider $W=X\cap {\bar{B}}_{ϵ}(a)\cap \{\mathrm{\mid }f\mathrm{\mid }\le \delta \}$. The sphere is transverse to every input stratum by step 4.1, and the function-circle boundary and its sphere intersection are transverse by step 6.1, since $df$ restricted to the sphere is onto at nonzero values. Cutting the finite Whitney strata by these transverse boundary hypersurfaces gives a compact finite real Whitney space with its outer faces unions of strata. By [F8] it has a finite triangulation respecting those faces; removing them gives a finite deleted-subcomplex model for $U=X\cap B_{ϵ}(a)\cap \{\mathrm{\mid }f\mathrm{\mid }<\delta \}$. The compact Milnor fibre with its sphere boundary has the same finite controlled Whitney construction. Its interior therefore has a finite deleted-subcomplex model. By step 7.1 the positive-half-disc part $U_{+}$ is a constructible fibre bundle with that interior fibre over a contractible half-disc, giving its finite cochain model. The homology and cohomology long exact sequences for $(U,U_{+})$ then give bounded finite-dimensional relative homology and cohomology. In particular these rational groups are finite free. No triangulation of the real cut $\mathrm{Re}f=0$ or of an arbitrary map has been claimed. Together with step 7.1 this proves the stated conclusions, with AC exactly through [A1].