Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five

Proof. Finite maps and branch contacts. At u=0,

and at x=0,

Weierstrass preparation in w therefore makes both graph projections finite of degree four.

In the first graph d_0=u q_1q_2. Introduce invertible local coordinates x=u-w,\ t=u. Two branches of q_1=0 satisfy

The branch u=w^2 satisfies t=x^2+O(x^3), and u=0 gives t=0. The other two local branches arise from q_2=0. Substituting u=vw^2 gives

The two roots of v^2-v+1=0 are simple; hence the implicit-function theorem gives two branches

These exhaust the local branches of q_2=0: indeed q_2(u,0)=u^2(u-1), so Weierstrass preparation in u has degree two. Together with f=0,1, these are four branches t=B_f(x). All six are smooth and distinct.

In the second graph,

The four branches of \psi=0 are exactly the four t=B_f(x) above. Since \psi(0,w)=-w^4+O(w^5), the last two factors each have a unique smooth branch x=e w^8+O(w^9), e=1,2, and these read x=e t^8+\text{higher terms} in t=x+w. Thus both d_i are units times products of two x=A_e(t) factors and four t=B_f(x) factors with the same first distinguishing coefficients.

The two A-branches meet with multiplicity 8; any pair of B-branches meets with multiplicity 2; a branch from each group meets transversely. Hence

The integral plane-form comparison. Interpolate only the terms above the displayed leading coefficients and interpolate the nonvanishing units through units. This joins both germs to the same leading product

We now verify uniform isolation and boundary transversality along the interpolation, which yield an oriented isotopy of its Milnor tubes. In the sector |x|\le 2|t|, if x/t^8 stays bounded, weighted rescaling has the noncritical model t^4(x-t^8)(x-2t^8) at t=1; its factors have distinct simple roots, and its weighted Euler derivative excludes other critical points. If |x/t^8|\to\infty, all t-B_f(x) factors are t(1+o(1)), both x-A_e(t) factors are x(1+o(1)), and x\,\partial_x d/d\to2. In the sector |t|\le2|x|, the analogous rescaling for bounded t/x^2 gives x^2\prod_f(t-fx^2); for |t/x^2|\to\infty one has t\,\partial_t d/d\to4. These estimates and their first derivatives are uniform in the interpolation parameter after the ball is shrunk. The six branch parametrizations also meet small spheres transversely and remain disjoint there. The associated Milnor tubes and boundary open books consequently form an oriented isotopy. Transporting cycles and filling chains through it preserves the integral Seifert pairing. Thus D_0 and D_1 have isometric integral Seifert forms.

The required monodromy sectors. The plane-form comparison is now established. We finish by computing the two sectors used in the arithmetic argument and excluding Jordan blocks there. The three smooth branches of h have pairwise intersections 1,1,2; thus \mu(h)=6. The weighted rotation (X,T)\mapsto(\zeta^2X,\zeta T), \zeta^5=1, preserves h and has order five. Its nontrivial generator has no fixed point on F_h. Lefschetz therefore gives trace 1 on H_{\mathbb Q}. Over \mathbb Q, a representation of the cyclic group of order five is a sum of trivial and four-dimensional cyclotomic representations. Its rank and trace equations force ranks 2 and 4, respectively.

It remains to justify the asserted Jordan condition for d_\circ. On a sufficiently small sphere, write its argument in the sector |x|\le2|t| as the argument of

and in |t|\le2|x| as the argument of

The displayed correction products are uniformly close to 1 where used. Smooth cutoffs in the ratio |x|/|t| remove their small arguments by an isotopy of argument fibrations. To see regularity through that isotopy, use the angular fields of the weighted rotations with rates (8,1)/20 on the t-side and (1,2)/10 on the x-side. Each differentiates its corresponding leading phase at rate one. On their common sector, both differentiate 2\arg x+4\arg t at rate one; the derivatives of the small correction terms can be made arbitrarily small. Their convex angular interpolation therefore remains positively transverse to every intermediate phase. Near the binding only the appropriate weighted field is used and is tangent to the binding. Thus the isotopy preserves the open-book monodromy.

This periodic-piece description follows the plane-link calculus of [4, Sections 11 and 13]. The final phase in the overlap is 2\theta_x+4\theta_t. Each regular phase level there has two annuli, both parametrized by the radial ratio and the free angle \theta_t, with the same orientation. A single phase circuit exchanges their cores \gamma_1,\gamma_2 while preserving their angular orientations. Outside the overlap, twenty circuits of either weighted rotation return identically. Across each overlap annulus the \theta_t-return varies from one turn on the t-side to four turns on the x-side; hence the twentieth return makes the same signed threefold Dehn twist on both annuli. The oriented boundary of the t-side subsurface is \gamma_1+\gamma_2 modulo actual page-boundary circles. The latter lie in the radical of the surface intersection form. Write M_\circ for the monodromy of d_\circ. Then

Since M_\circ^{20}-I commutes with M_\circ, it vanishes on the generalized eigenspaces at 1 and at primitive fifth roots. The polynomial z^{20}-1 has simple roots; M_\circ is therefore semisimple on those sectors.□

Proof. For an analytic germ f that is not locally constant, an ordinary Milnor tube B_\epsilon\cap f^{-1}(D_\delta), with 0<\delta\ll\epsilon\ll1, has contractible total; this is the consequence of Milnor’s proof recorded in [10, Theorem 1.8 and Remark 1.9]. We compare the finite graph cuts used here with those ordinary tubes. Choose finite proper graph representatives over one sufficiently small closed base ball, and choose ordinary Milnor balls for h,d_i inside the corresponding cuts. The branch parametrizations above allow regular radii to be chosen so that the zero set in each closed intervening region is exactly a disjoint union of branch annuli. The cut boundaries are smooth near these annuli, and each annulus is transverse to both boundaries. The function is a complex submersion normal to each annulus. A field along the annulus directed from the inner boundary to the outer one therefore extends, by this submersion, to a field tangent to the nearby function levels. The disjoint flows give a value-preserving collar of a neighborhood of the entire collection of annuli. Properness makes the closed intervening region compact. Outside a neighborhood of these annuli the absolute value of the function has a positive lower bound, so a sufficiently small value disk confines the intervening tube to that collar. Attaching the collar to the ordinary tube preserves its homotopy type and gives regular lateral boundaries. Thus these selected cut totals are acyclic. Choose nearby pages whose base branch values lie in their interiors. Represent a base class by loops avoiding these finitely many values; the full preimage is a cycle upstairs. This operation is Poincaré–Lefschetz dual to pullback in relative first cohomology of the pages, so it depends only on the homology class of the loops. Counting the four sheets gives \pi_{i*}\pi_i^!=4.

For the first pairing identity, fill the base loop a on the reference page in the acyclic base total tube, and fill the positive neighboring-page transport of b upstairs. Pull the first oriented surface chain back through the finite holomorphic map, and push the second filling forward. Put both fillings and their boundary collars in general position. The base filling meets the generic branch curve at isolated points, and the other filling can be moved off their preimages; hence intersections relevant to the pairing occur away from ramification. At each such intersection the usual oriented projection formula counts the same local number before and after pushforward. The finite map is over the value disk, so it also commutes with positive nearby-page transport. This proves the first identity; reversing the two filling arguments proves the second. Finally, \pi_i^!H is rationally nondegenerate, since its restricted form is 4s_h by adjunction. Its two-sided orthogonal complement is \ker\pi_{i*}, and the displayed formula is the corresponding projection.□

Proof. The Chinese remainder theorem supplies arbitrarily large n_0 with the two congruences, so the strict lower bound can also be imposed. For large r we have r\ge2, and both n_j are 1\pmod4; therefore a,b are integers. The lower bound on n_0 gives n_0<n_1<2n_0, so a,b are positive, comparable, and grow without bound with r. A common divisor of n_0,n_1 divides 5\cdot2^r. Both integers are odd and 2\pmod5, proving \gcd(n_0,n_1)=1.

The identities for \alpha,\beta follow from their definitions. Since b>a,

The right sides exceed four for sufficiently large r. In particular, the chosen a,b exceed four.□

Proof. The specializations in Lemma 2.1 give R_i(0,w)=w^4 and U_i(0)=1. Hence

These derivatives remain nonzero along the respective local root loci after shrinking. For s>0, the corresponding derivatives of R_{i,s} are multiplied by the nonzero factor s^4. At a point of C_{i,s} with y\ne0 or z\ne0, its defining equation has a nonzero vertical derivative; at y=z=0, one of the displayed base derivatives is nonzero. Thus the ambient germs are smooth.

The two ambient equations are equivalently the implicit equations

respectively. In the first, q_1(T,w/s) has ordinary order at least three. Since U_0(0)=1, analytic implicit solution gives

In the second graph q_2=q_1-u^2 has order two, and \psi=u(u-w^2)q_2 has order four. Therefore

Here s is fixed and O_{\ge j} denotes terms of ordinary total degree at least j; in particular all w^4 terms lie in these remainders. The chosen a,b exceed four, so w(y^a+z^b) has order greater than five. Substitution in h gives the nonzero initial forms

They have degrees five and four, respectively.

It remains to justify the isolated critical points claimed in the proposition. The input from the base curve is the following explicit gradient estimate. Along an analytic arc, let \nu denote order in the arc parameter, with \nu 0=\infty; for a vector, \nu is the minimum of its component orders. If an analytic arc p=(X,T)\ne0 has m_0=\nu p=\min(\nu X,\nu T), then

Indeed h_X=T(2X-3T^2) and h_T=X^2-9T^2X+10T^4. If \nu X<2\nu T, the first term of h_T has order 2\nu X\le4m_0. If \nu X>2\nu T, the term -3T^3 of h_X has order 3\nu T\le4m_0. In the equality case, put X/T^2=c+\text{higher terms}; 2c-3 and c^2-9c+10 have no common root, so one derivative has order at most 4\nu T=4m_0. The cases in which one coordinate vanishes identically are immediate.

The full multiplier argument, including the cases where a vertical variable vanishes, is given in Lemma 5.3; it uses the two strict bounds in Lemma 3.1 with the gradient exponent 4 just proved.

Finally, if a convergent hypersurface germ had a repeated irreducible factor, its gradient would vanish at generic nonzero smooth points of that factor. Lemma 5.3 therefore also implies reducedness.□

Proof. We first estimate the vertical faces and corners, then prove uniform regularity on the remaining outer strata.

At s=0 the vertical equation is yz=w^k. Its link on \partial Y_t is the cyclic quotient of a 3-sphere, hence a lens space. The cone is smooth away from its vertex. The y- and z-faces cannot meet: if both were active, then |yz|=M^2t^k>|w|^k. On the y-face,

After eliminating z=w^k/y, the free derivative is P_w=y^a+(kb+1)z^b, which is nonzero by the stronger choice of M. The z-face is symmetric, with P_w=z^b+(ka+1)y^a. These estimates persist for small positive s by uniform convergence of R_s and \partial_wR_s on the fixed outer face. At a y/w corner the bound

holds; similarly at a z/w corner. Choose \varepsilon+\delta below both lower bounds at t=\rho. On the w-face, near a zero of P with w\ne0, use y as the free coordinate and z=w^k/y. Then dP(\partial_{\log y})=w(ay^a-bz^b) =w(a+b)y^a\ne0 at P=0. On the compact outer w-face, this nonvanishing persists in one uniform small-value neighborhood; choose \varepsilon+\delta within it. The quotient expression \widetilde P=UV(U^{ka}+V^{kb}) also shows submersivity of P in the nonvertex cone interior: if U,V\ne0, its two partials cannot vanish simultaneously because that would imply (ka+1)(kb+1)=1; on each axis the transverse partial is nonzero. These calculations cover the vertical faces and corners near the small values used in (5.2).

To finish the regularity claim on the genuine outer strata, consider a sequence approaching the cone vertex while s\to0, and let \varrho=\max(|y|^{1/\alpha},|z|^{1/\beta},|w|). If \varrho/s\to\infty, weighted rescaling by \varrho gives smooth convergence at a nonvertex cone point; all lower Weierstrass terms vanish because A_j is bounded and (s/\varrho)^{k-j}\to0. If \varrho/s is bounded, rescale by s, with (Y,Z,u)=(y/s^\alpha,z/s^\beta,w/s); the limit is the smooth model YZ=R(p,u). For (Y,Z)\ne(0,0) the base projection is a submersion; at Y=Z=0 its differential contains the tangent image of the smooth root graph. The specified h,d Milnor cuts have the needed transversality on \partial\mathcal B, including corners with |h|=\varepsilon. For the actual function f_s, these arguments give the stated outer lateral regularity over the whole value disk. Indeed \lvert P\rvert\le\varepsilon+\delta whenever \lvert f_s\rvert\le\delta, so all the outer vertical-face estimates apply throughout that disk, also at their intersections with base faces because the free vertical directions keep the base fixed. On an outer stratum with no vertical face, a hypothetical failure with unbounded \varrho/s would contradict submersivity of P on the nonvertex cone after rescaling. If \varrho/s is bounded, then P=s^{1+a\alpha}u(Y^a+Z^b)=o(1) in the smooth rescaled model. The face \lvert h\rvert=\varepsilon is then absent over \lvert f_s\rvert\le\delta<\varepsilon/4. On the remaining base-ball face, the boundary fibrations of h and d over their full small value disks give rank two, respectively where (Y,Z)\ne(0,0) and on the smooth root graph. Compactness and the preceding smooth convergence preserve these ranks for small s. This assertion concerns the outer cuts of X_s; it does not assert disk-wide submersivity on the auxiliary J. The remaining interior critical-point possibility is excluded in Lemma 5.3. All statements here are open in compact families of smooth strata, so one uniform upper bound on s works for all strata.□

Proof. At the inner interface, (5.1) and |h+\theta P|=\delta imply

Thus h is regular there, including on its base-ball cut, and the |h|=\varepsilon cut is absent. On the inner w-face, (5.5) makes h regular, and \lambda can be decreased until P is small in C^1 there. The zero-parameter cone is smooth along this interface, so the same assertion holds throughout the (s,\theta) rectangle for small s. We use regularity of the horizontal boundary to isotope the sublevel totals as manifolds with corners; we do not assert that h is submersive through every value in the disk at \theta=0.

We first construct the prescribed circle family. For fixed (p_0,\phi,\theta,s), solve

for q near c, then set p=\Phi(p_0,q) and z=R_s(p,w_0)/y. The root bound gives \lvert R_s(p,w_0)\rvert\le(3\lambda/4)^k, so these circles lie on the y-face and strictly away from the z- and w-faces. Thus the small bound for P applies there. The solution is unique and smooth by the real implicit-function theorem: P is uniformly small there, and its real derivative D_qP\to0 as s\to0, since every p-dependent term of R_s(p,w_0) contains a positive power of s. Decrease s until \lVert D_qP\rVert<1/2 uniformly on this neighborhood. This lift keeps the h-page label p_0, w_0, and the exact oriented y-angle \phi fixed; the physical p may move. At \theta=0, the s-motion keeps p_0,y,w_0 fixed and only changes z=R_s(p_0,w_0)/y. Hence the family of anchored circles is \Hh_c\times S^1, with no base-to-circle shear. To make this family the restriction of the boundary isotopy, use the same coordinates near its compact trace and put

The real operator I+\theta D_q\mathcal P_s is uniformly invertible there. The local lifts that fix p_0,y,w have

Differentiating q+\theta\mathcal P_s verifies that both lifts preserve its complex value. They are tangent to the y-face and to the base-ball face, because y and the pair-trivialization label p_0 are fixed. Their restrictions to the circles are exactly the derivatives of the displayed implicit solution.

We extend these local lifts to the sublevel continuations. Near J, vary \lambda slightly to obtain bicollars. For the \theta-motion the ambient C_s is fixed. On a collar of |h+\theta P|=\delta, the map consisting of h+\theta P and the active lateral defining functions is a submersion. Choose a parameter field V tangent to every active lateral face and satisfying d(h+\theta P)(V)+P=0 there. Near the compact anchor trace, patch it with the local \theta-lift above by a cutoff equal to one on that trace. The lift equation and all active-face tangency conditions are affine, so this patch remains a valid lift. Extend only its spatial correction by a cutoff supported in that collar and zero near the critical region, retaining the parameter direction, and integrate. This isotopes the smooth sublevel totals and preserves the complex value on their horizontal boundary. For the s-motion on the exterior, identify the smooth ambient family away from the cone vertex and make the same horizontal-collar correction, patched with the displayed local s-lift near the anchors. Uniqueness of the flows makes their restrictions exactly the anchored family, fixing p_0,w_0,\phi. On the interface rectangle (5.5) keeps h regular, so its collar lifts exist along every edge and path homotopy gives isotopic boundary identifications relative to these anchors. Locally this is explicit in the coordinates (p_0,q+\theta\mathcal P_s,y,w,\theta,s), in which the two anchored lifts are parameter coordinate fields. At s=0, the cone is independent of p, so on the entire page interface this normalization can be chosen explicitly as

The bound |P(v)|<\delta/4 keeps this formula in the chosen trivialization. It agrees with the anchored circles and identifies the interface with \Hh_c\times L while retaining the same base label. No value-preserving lift is used through the critical inner total.□

Proof. The gradient ideal of the isolated plane singularity h contains a power of the maximal ideal. For this h, use the same L_0=4 supplied by (3.1): on each nonconstant analytic arc p\to0, \nu(h_p)\le L_0\nu(p). For fixed s, at a constrained critical point in the ambient coordinates, write the spatial differential equation

After the change y=s^\alpha Y, z=s^\beta Z, w=su, put \xi=s^{1+a\alpha}. Then h+P=h+\xi u(Y^a+Z^b), while the constraint is s^k(YZ-R)=0. We write \chi=s^k\chi_{\rm orig} for the multiplier of the rescaled constraint YZ-R=0; the two symbols will be kept distinct. The rescaled spatial critical equations are

We first dispose of points over p=0, then of points with y=z=w=0 and p\ne0. Any remaining hypothetical critical sequence as s\to0 has, after a subsequence, either p\to p_0\ne0 or p\to0. We exclude the first alternative by the cone and root-graph calculations and the second by valuations.

At p=0, R_s(0,w)=w^k exactly, so the vertical constrained critical equations are those of P on yz=w^k. If w\ne0, then y,z\ne0. Multiplying the y- and z-equations by y and z, respectively, and eliminating \chi_{\rm orig}w^{k-1} from the w-equation gives

Their determinant is k(kab+a+b)\ne0, which would force y^a=z^b=0, a contradiction. If w=0, then yz=0; at a nonzero point on either axis P_w=y^a+z^b\ne0, whereas the w-derivative of yz-w^k is zero. Thus the vertical equations exclude every point over p=0 except the origin, without a condition on w/s. At y=z=w=0 with p\ne0, constrained criticality is exactly criticality of d on its smooth root graph and is excluded near the origin.

First exclude critical points whose base coordinate tends to p_0\ne0 while s\to0. Let \varrho=\max(|y|^{1/\alpha},|z|^{1/\beta},|w|). If \varrho/s\to\infty, weighted rescaling by \varrho leaves a nonvertex cone point: every lower term in R_s/\varrho^k tends to zero. The normalized vertical critical equations would make P critical on the smooth nonvertex cone, which is impossible. If \varrho/s is bounded, use the rescaled variables and multiplier in (5.6). The variables Y,Z,u are bounded and \xi\to0. On the bounded (p,u)-set, R_p is bounded, while h_p(p) stays bounded away from zero because h_p(p_0)\ne0. The base equation h_p=-\chi R_p therefore bounds \lvert\chi\rvert away from zero. The Y- and Z-equations in (5.6), with bounded u,Y,Z and \xi\to0, now force Y,Z\to0. After passing to a subsequence u\to u_0, the constraint gives R(p_0,u_0)=0. All roots over the selected base cut lie in the chosen local graph representative; only at this point do we use its unit component of R_p. The base equation then bounds \chi above, so a further subsequence has a nonzero finite limit \chi_{\rm lim}. The u-equation in (5.6) gives R_u(p_0,u_0)=0. Together with R(p_0,u_0)=0 and h_p(p_0)=-\chi_{\rm lim}R_p(p_0,u_0), this says h is critical on the smooth root graph R=0 away from its origin, contrary to isolation of d.

Thus a remaining critical sequence has p\to0. Real-analytic curve selection gives an arc; write \nu for its valuation, and set

The case \nu_{\rm v}=\infty was already covered by the root graph, so assume it is finite. Write \varrho for an arc scale with \nu(\varrho)=\nu_{\rm v}. If the lower Weierstrass terms all vanish under rescaling by \varrho, the vertical equation limits to yz=w^k at a nonvertex point. Normalize the original critical multiplier to \chi_{\rm cone}=\chi_{\rm orig}/\varrho^{d_P-k}, where d_P=1+a\alpha. A nonzero component of the cone gradient bounds this multiplier through the corresponding vertical critical equation; a convergent subsequence of all the equations then gives d(P|_{yz=w^k})=0, contradicting the submersivity just proved. Thus, for some j<k,

Since A_j(0)=0, \nu A_j(p)\ge m_0, and therefore

Return to the rescaled variables and multiplier in (5.6). By (5.8), p,u,Y,Z\to0 and \nu Y\ge\alpha m_0/k, \nu Z\ge\beta m_0/k. The unit component of R_p and the gradient bound give \nu\chi\le L_0m_0=4m_0. If Y,Z,u\ne0, the Y- and Z-equations in (5.6) give

Here \nu\xi,\nu u>0, and the strict signs use the two inequalities on a,b in Theorem 4.1, together with m_0>0. The two conclusions are incompatible. If exactly one of Y,Z vanishes, or if u=0 while both are nonzero, (5.6) forces \chi=0, hence h_p=0, contrary to p\ne0. If both vanish, then R=0 and the remaining critical equations make h critical on the root graph, contrary to isolation of d. This exhausts the arc.□

Proof. Replace h+P by h on the inner piece using Lemma 5.2. The base-with-w page is

and its root subpage \D is a page of d. For R_s\ne0, we retract the vertical hyperbola to a circle; over the root page, that circle collapses. We realize this model by balancing the two moduli inside the inner block. For fixed (p,w), write A=|y|^2/\sigma^2, B=\sigma^2|z|^2, so AB=|R_s|^2, and choose \sigma near \lambda^{(\alpha-\beta)/2}. For 0\le t\le1, replace A-B by (1-t)(A-B) while retaining AB and the circle angle. The positive solution for A,B is continuous at R_s=0; on an axis it shrinks the active coordinate to zero. Both moduli stay between their starting and final values. Since \max|R_s|\le(1+o(1))\lambda^k, the core |y|=\sigma\sqrt{|R_s|}, |z|=\sigma^{-1}\sqrt{|R_s|} stays strictly within the M-bounds. This is a deformation retraction, relative to the constant-angle section, onto

The constant-angle section splits off the homology of G. Quotienting that section gives (G/\D)\wedge S^1, so the remaining homology is H_{*-1}(G,\D), naturally for moving roots. The spaces G and \D are connected and have only first reduced homology. Their relative long exact sequence therefore gives

Thus the underlying groups are H_3(W)\cong K and H_2(W)\cong H/\pi_*D. We next fix the signs of these identifications and of the mod-k injection; these signs determine the residue condition in Theorem 4.1.

Orient this S^1 by increasing \arg y. For an arc I from w_0 to a root, use the circle-first orientation [S^1_y]\times I on its sweep; the collapsed root end leaves boundary +\gamma at w_0. Write \mathfrak S(C)=[S^1_y]\times C for circle-first suspension. The cross-product boundary rule gives

in a product chart over an oriented base interval a_0, with its endpoint faces taken relatively. The root part of the boundary of a_0\times I is -[a_{0,\rm root}]; write \partial_{\D} for this root-endpoint boundary, and for the resulting relative connecting map on relative cycles. Define \Psi as the negative of the relative root-endpoint connecting map under this circle-first suspension, namely -\partial_{\D}\mathfrak S^{-1}. Thus the local chain -a_0\times(S^1_y\times I) has signed root contribution +[a_{0,\rm root}]. The interval endpoint faces and the remaining w_0-boundary must cancel before a homology class is defined. The next section supplies this cancellation by a lens chain with a cap at the seam, whose local root vector is diagonal. Then \Psi applies to the resulting mod-k class with the sign just computed; no individual root is assumed to close around the whole base loop. This fixes the sign of \Psi throughout. We use the same suspension convention for H_3(W)=K. In degree two, identify H_2(W) with H/\pi_*D by the negative of circle-first unsuspension. Künneth products on J\simeq\Hh\times L are ordered with the base factor first: for a closed oriented base loop a, [a\times\gamma]=-\mathfrak S(a). This normalization therefore sends the interface torus [a\times\gamma] to +a\bmod\pi_*D. The quotient is defined on H/kH because kH\subseteq\pi_*D, by \pi_*\pi^!=k. With \Z/k coefficients, the same relative sequence injects H_2(G,\D;\Z/k) into H_1(\D;\Z/k)=D/kD. Under suspension, the signed injection from H_3(W;\Z/k) is the negative relative boundary just fixed, namely \Psi in (6.1).

It remains to prove acyclicity of the large total. The same collapse argument for total disks first shows that \widehat W is acyclic, because the total tubes of h,d are acyclic. We control the exterior attachment using the explicit contraction of the cone model.

At s=0, put d_P=1+a\alpha=1+b\beta. The explicit contraction

scales both h and P by t, and scales both yz and w^k by t^{k/d_P}. It preserves the base ball, its |h|-cut, the vertical block cuts, and the |h+P|-cut. Hence \widehat\T_0 and its inner part are contractible. Mayer–Vietoris therefore makes \widehat J_0\to\widehat M_{{\rm ext},0} a homology equivalence. Transport by Lemma 5.2 gives the same statement at positive s. Attaching the acyclic \widehat W along \widehat J leaves \widehat\T_s acyclic.□

Proof. Fix s>0 after Lemmas 5.1 and 5.3. In a holomorphic chart on the smooth germ C_s, choose a closed ordinary Milnor ball B_{\rm m} inside the interior of every outer cut of X_s. Choose 0<\delta'\le\delta so that its sphere is transverse to all fibers over \{|v|\le\delta'\}. The corresponding ordinary analytic total

is contractible; this is the ordinary tube consequence of Milnor’s proof recorded in [10, Theorem 1.8 and Remark 1.9]. It does not apply by itself to the custom outer cuts.

Let \widehat\T_s' be the large total with value radius \delta', and set

Write N_s,A_s,S_s,\T_s' for the corresponding pages at a fixed \lvert c'\rvert=\delta'. Stratify \widehat A_s by the genuine outer faces and the new Milnor-sphere face. The auxiliary cut J remains ordinary interior in this comparison. Lemma 5.1 gives rank two on the outer strata over the whole disk, the Milnor choice gives it on the sphere, and Lemma 5.3 gives it in the intervening interior. The cuts are compact, so f_s is a proper stratified submersion on \widehat A_s and on \widehat S_s over this disk, stratified by its interior and boundary circle. The face-preserving Ehresmann argument over the contractible disk therefore makes A_s\hookrightarrow\widehat A_s and S_s\hookrightarrow\widehat S_s homotopy equivalences. Their relative homology groups vanish. Relative Mayer–Vietoris for

now identifies the large total/page relative homology with the ordinary local one.

The disk-wide outer regularity and isolation away from value zero also permit the large radius to shrink from \delta to \delta' through all positive intermediate radii: horizontal collar fields are tangent to every outer face and vanish near the critical region. Lemma 6.1 proves that \widehat\T_s is acyclic at the original radius; this isotopy therefore makes \widehat\T_s' acyclic. The ordinary total \widehat N_s is contractible as above. The connecting maps in the relative long exact sequences of these two acyclic totals identify their relative H_4-groups with the page H_3-groups, proving the homology assertion. A local filling can be used unchanged in the large total. Transporting the two page cycles from the smaller to the original value circle along two disjoint angular rays adds collars at disjoint complex values, hence no new intersections. Thus the filling pairing is identical.□

Proof. Use Lemma 5.2 to compute the exterior at s=0. Normalize c>0, and put q=h. Cut at \operatorname{Re}q=c/2, as in Figure 1, calling the left and right pieces A and B, respectively. On A, |P|=|c-q|\ge c/2, so the removed inner block is absent. The allowed P-values form a convex region avoiding zero, and the boundary estimates of Lemma 5.1 trivialize its P-pages with their outer cuts. Contracting the base by the positive weights of h and carrying these pages through the trivialization retracts A to Q.

On B, choose one boundary-respecting trivialization of the h-pages over the convex regular-value region \{|q|\le\varepsilon,\ \operatorname{Re}q\ge c/2\}. It can agree near c with the page labels used in Lemma 5.2. For v=(y,z,w), define the radius of its block by

The exterior condition is \lambda\le r(v)\le\rho. Scale v by the cone weights, from t=1 down to t=\lambda/r(v), while retaining its h-page label and setting

This keeps \lambda\le r(v_t)\le\rho, and q_t stays in the same convex h-region. The base-ball cut is preserved by the trivialization. The homotopy therefore stays in B and stops at the actual interface J, fixing J pointwise. At its endpoint v_L\in L, the value is q=c-P(v_L), generally different from c. The s=0 interface continuation of Lemma 5.2 retains v_L and the base label while changing this value to c. It identifies the retract with \Hh\times L.

The seam values form the contractible interval I=\{q=c/2+iu:|q|\le\varepsilon\}. Trivializing both page factors along I identifies the overlap C=A\cap B with \Hh\times Q\times I. After contracting I, its map to J\simeq\Hh\times L is homotopic to \mathrm{id}_{\Hh}\times(Q\to L), where Q\to L is weighted radial projection. Both \Hh,Q have homotopy dimension one, so overlap H_3=0, while overlap H_2=H\otimes E. Moreover H_2(J)=H\otimes\Z/k=H/kH and H_3(J)=\Z. The latter generator maps to zero in H_3(W)=K: it is the full inner fiber over a regular base point, whose H_3 vanishes in the collapse model. We record the orientation of the remaining interface map. For a full-page 3-cycle write its exterior chain as e=e_A+e_B, with the inner chain w satisfying \partial w=b_J,\ \partial e=-b_J. Define t_C=-\partial e_A\in Z_2(C). Then

This fixes the sign of the Mayer–Vietoris coordinate. Let r_t:C\to B, 0\le t\le1, be the stopped radial homotopy just constructed, reparametrized to end in J. Its endpoint is the actual interface, with the same h-page label under the compatible parameter rectangle. For the cycle t_C, the interval-first prism chain S=r_*([0,1]\times t_C) has \partial S=r_{1*}t_C-t_C. Consequently

Under the degree-two normalization fixed after (6.2), this is the positive sign of the last arrow in (6.4). The Mayer–Vietoris sequence now gives (6.4), with final map the quotient of 1\otimes\tau. Before W is attached, exterior H_3 differs from its H\otimes\ker\tau contribution by precisely the image of H_3(J); that image dies after attachment. This proves the last statement.□

Proof. First note that the ambiguity in \eta is reduction of an integral class in H_3(J)=\Z, and this class dies in H_3(W) by Lemma 6.3; thus the left-hand side is unambiguous. It suffices to test on loops in \Hh avoiding the finite branch discriminant. Take w_0=\lambda/2, with every root in |w|<\lambda/4. On the \theta=0, h=c model of the interface lens fiber, for p\in\Hh_c the y-angle loop

is the positive generator of H_1(L)=\Z/k. Indeed, at s=0 the quotient (U,V)\mapsto(U^k,V^k,UV) lifts one positive turn of the y-angle to a path ending at the deck transform (U,V)\mapsto(\omega U,\omega^{-1}V), where \omega=e^{2\pi i/k}. This generates the deck group. The anchored s-continuation preserves the same oriented loop. Let \widetilde\gamma_p denote its image on the actual f_s=c interface under the explicit lift in Lemma 5.2. That lift preserves the h-page label p, w_0, and the exact y-angle, though physical p moves. We compute below in the \theta=0 model and transport all relative classes to the actual J and W by this isotopy. In the model, the exact sequence of the pair (L,\gamma_p) reads

It supplies a unique relative class S_p with \partial S_p=k\gamma_p.

Let F_p be the inner yz=R_s(p,w) fiber above a regular p, and let r_1,\dots,r_k be the roots in the w-disk \Delta. Let X_p=X(\Delta,\{r_1,\dots,r_k\}) be its balanced collapse core, and let \gamma_p^{\rm core} be the circle over w_0 in that core. The balancing retraction does not fix the outer circle \gamma_p pointwise. It gives a map of pairs

that is an absolute homotopy equivalence and restricts to the orientation-preserving circle homeomorphism retaining the y-angle. The long exact sequences of the two pairs therefore show that r_p induces an isomorphism on relative homology. The constant-angle section of X_p is a disk meeting \gamma_p^{\rm core} in one point. Adjoining that disk to the relative subspace does not change relative homology; quotienting it and \gamma_p^{\rm core} gives the circle suspension of \Delta/(\{r_1,\dots,r_k,w_0\}). Hence there is a natural isomorphism

Use r_p to identify the oriented boundary classes of \gamma_p and \gamma_p^{\rm core} in this calculation. The class of an arc from w_0 to r_i, swept by the y-angle, has root coefficient +1 and \gamma_p-boundary +1, fixing the convention. The image of S_p has a_0=-k. To compute this one fiber class, let r_i(t), 0\le t\le1, be any braid of distinct roots in |w|<\lambda/4, starting at the roots of R_s(p,\cdot) and returning to the same unordered root set. Put Q_t(w)=\prod_{i=1}^k(w-r_i(t)), so Q_1=Q_0, and consider the compact fibers F_t=\{yz=Q_t(w)\}\cap Y_\lambda, their boundaries L_t, and the circles \gamma_t defined by the same w_0 and y-angle formula as \gamma_p, with R_s(p,w_0) replaced by Q_t(w_0). Choose a smooth disk velocity v_t(w), supported in |w|<\lambda/4, equal to \dot r_i(t) near each moving root. Then

extends smoothly across the roots. Indeed, near r_i(t), writing Q_t=(w-r_i)B_{i,t} gives \ell_t=(\dot B_{i,t}+\dot r_i B'_{i,t})/B_{i,t}, with B_{i,t}\ne0.

The y- and z-faces of these fibers never meet, since |Q_t(w)|\le(5\lambda/4)^4<M^2\lambda^4. Choose a smooth real cutoff \chi, depending on |y|^2,|z|^2, that is zero near the y-face and one near the z-face. The time-dependent lift

is smooth also at the roots and axes, and differentiation gives \frac{d}{dt}(yz-Q_t(w))=\ell_t(yz-Q_t(w)). It is tangent to the y-face because \dot y=0 there, to the z-face because the coefficient of z is purely imaginary there, and to the w-face because v_t=0 there; these statements also cover all corners. Compactness therefore gives transport of the full triples (F_t,L_t,\gamma_t). On \gamma_t, it fixes w_0 and y=M\lambda^\alpha e^{i\phi} exactly, and gives \dot z=\dot Q_t(w_0)/y. Thus the return fixes the parametrized circle \gamma_p pointwise.

The return maps of (L,\gamma_p) and (F_p,\gamma_p) commute with inclusion. The former fixes the unique class S_p with boundary k\gamma_p; hence the latter fixes its image in H_2(F_p,\gamma_p). To compute that same return map in (7.2), observe that the lift covers the disk flow \psi_t of v_t and preserves the y-angle wherever defined. It therefore intertwines the balancing collapses with the map (w,e^{i\phi})\mapsto(\psi_t(w),e^{i\phi}) on the circle models; collapsed root fibers are carried to collapsed root fibers. The endpoint isomorphism in (7.2) consequently sends its action to the permutation of the root coefficients a_i, with a_0 fixed. Every root permutation is available, and no additional term can occur because the endpoint map is an isomorphism. Invariance of the image of S_p, together with a_0=-k, now gives a_1=\cdots=a_k=1. Reducing S_p modulo k closes its boundary and yields the diagonal root vector in the absolute mod-k collapse homology.

Now work over an oriented base loop a\subset\Hh_c missing branch values, still in the \theta=0, h=c model. Write \D_a=\pi^{-1}(a). Let W_a be the restriction of the inner family F_p to p\in a, and let J_a\subset W_a be its restricted lens-interface family, with inclusion i_a:J_a\hookrightarrow W_a. The compatible isotopy of Lemma 5.2 transports their inclusions to the actual pieces J\subset W; the vertical maps below use this transport. The same circle-first suspension and negative relative root boundary define

The target remains the root homology in the h-model, with its inclusion into H_1(\D;\Z/k). The actual covering monodromy over a may be a smaller braid subgroup, with several orbits. Over a small arc I\subset a, the same transport tracks roots and preserves the circle angle. Collapse relative to the constant-angle section, take the relative root boundary, and slice over (I,\partial I). Each operation is natural under this product trivialization: on chains its relative map is the fiber operation times [I,\partial I]. Restriction and inclusion give the following naturality diagram:

The common orientation of the normal circle fixes the sign of the two right-hand maps. The fiber endpoint calculation gives the diagonal root vector after slicing the top row at p. We construct a Bockstein lift on J_a by first closing a swept chain at the seam. The anchored circles trace the base-first torus a\times\gamma_p over a. Cut a at one point, choose a representative S_0 of S_p with \partial S_0=k\gamma_p, transport it along the resulting interval, and call its endpoint S_1. The anchored circle is fixed at the seam, so \partial S_0=\partial S_1=k\gamma_p. Uniqueness of the relative class makes S_1-S_0 a null-homologous 2-cycle in L; choose a seam cap C_a with \partial C_a=S_1-S_0, using H_2(L)=0. Let T_a be the interval-first sweep. Then

The second boundary is divisible by k, so the reduction is a cycle. Set

and let \eta be its image in H_3(J;\Z/k) under inclusion and transport. The Bockstein convention gives \beta_{\rm Bock}\eta_a=[a\times\gamma_p] on J_a, and \beta_{\rm Bock}\eta=[a\times\gamma_p] on J, with the torus transported in the second identity. Changing the cap changes \eta_a by the reduction modulo k of an integral class from H_3(L), and changes \eta by its image. The integral class maps to zero in H_3(W) by the fiber calculation used in Lemma 6.3, so its mod-k image dies as well. Thus both \Psi i_*\eta and the full transfer \pi^![a] are images of classes in H_1(\D_a;\Z/k). Choose the slice away from the seam. Slicing at p\in a sends each oriented component circle of the unramified cover \D_a\to a to the sum of the roots in its monodromy orbit. Distinct orbits have disjoint supports, so this slice map is injective on H_1(\D_a;\Z/k). The fiber calculation gives the diagonal root vector for \Psi_a i_{a*}\eta_a, and the full transfer has the same vector. Their classes agree already in H_1(\D_a;\Z/k), hence in D/kD. Such base loops generate H_1(\Hh), proving (7.1).□

Proof. Fix x, with z,b,u as above. Orient J as the outgoing boundary of W, and represent x by page chains w\in C_3(W) and v\in C_3(M_{\rm ext}) with \partial w=b_J and \partial v=-b_J. The oriented prism calculation (6.5)–(6.6) gives [b_J]=b=(1\otimes\tau)z in H_2(J)=H/kH. Because this group is killed by k, choose an integral A\in C_3(J) with \partial A=k b_J. Write i_{\rm ext}:J\hookrightarrow M_{\rm ext} for the exterior inclusion. The chains

are closed in their respective pieces. Their sum in the full page is kx: the two copies of A cancel on J. The exterior class [v_k] has second coordinate kz, hence is x_{\rm ext}(kz) in the full page; its possible H_3(J) ambiguity dies after attaching W. Consequently u=[w_k]\in K. Modulo k, [w_k]=-i_{W*}[A\bmod k]. The coefficient convention gives \beta_{\rm Bock}[A\bmod k]=[\partial A/k]=[b_J]=b. With the \Psi orientation fixed after (6.2), Lemma 7.1 gives \Psi i_*[A\bmod k]=+\pi^!b. The preceding minus sign in [w_k]=-i_{W*}[A\bmod k] therefore gives

Choose b_H\in H lifting b and put l=(u+\pi^!b_H)/k. The congruence makes l\in D, and \pi_*l=b_H, \pr l=u/k. Conversely, if l,z obey the condition in (4.1), the last arrow of (6.4) kills z. Choose any x with that z, and put

Here u_x is the integral numerator of the rational inner coordinate u_x/k. By (7.5), u_x-u_l\in K\cap kD. This intersection equals kK: if kv\in K for v\in D, then k\pi_*v=0, and H is torsion free, so v\in K. Adjust x by (u_l-u_x)/k\in K through the first arrow of (6.4). The adjusted class has rational coordinate (\pr l,z). Thus every pair in (4.1) occurs. Replacing b_H by b_H+kh changes l by \pi^!h and does not change \pr l, so the coordinate description is independent of the lift.□

Proof. The inner classes bound in the acyclic \widehat W. For an exterior page class representing an element of H\otimes\ker\tau, its image in H_3(\widehat M_{\rm ext}) may be nonzero. The compatible sublevel continuations take \widehat J to the product of the acyclic base total of h with L. Its integral H_3 is therefore generated by one oriented lens fiber. Lemma 6.1 then gives H_3(\widehat J)\cong H_3(\widehat M_{\rm ext})\cong\Z. Under the interface continuation, the inclusion J\hookrightarrow\widehat J sends the oriented lens-fiber generator of H_3(J) to a generator of H_3(\widehat J). Subtract the corresponding integral multiple of the page-interface generator, which is zero in the full page. The adjusted exterior class bounds in \widehat M_{\rm ext}. This can be done compatibly under a short value-circle transport arc. Put the two fillings on opposite sides of a bicollar of \widehat J; their interiors are then disjoint, proving orthogonality.

Continue the exterior piece to s=0. Its full lift to (U,V) lies off the cone vertex, where the cyclic map is an orientation-preserving unramified k-cover. Thus intersections of full lifts are k times intersections downstairs. This covering count uses the original fillings off the cone vertex. In the contractible upstairs total, the filling pairing is unchanged when those fillings are replaced by the radial fillings used below. The smooth join calculation follows Sakamoto’s product formula [21, Theorem 2, p. 715] and the product construction of [8]; the cone extension and integral gluing were established here. The lifted cycles also live in the full smooth tube for h+\widetilde P, whose total is contractible by weights. Its page Mayer–Vietoris coordinate is H\otimes H_1(\widetilde Q); the lift of z\in H\otimes E has coordinate (1\otimes\tr)z. For pure tensors, represent this class by the join of loops in the two factor pages, varying their values as (tc,(1-t)c), 0\le t\le1. It meets the equal-value cut in the product of the two loops. Fill the join by the two factor radial cone fillings with values in \{(t,t')c:t,t'\ge0,\ t+t'\le1\}. The two ordered fillings meet only at the common origins of the factors because their neighboring rays are disjoint away from zero. Near each intersection they are products of two real 2-dimensional filling intersections. Their Koszul sign is (-1)^{2\cdot2}=+1; any uniform join-orientation sign occurs in both fillings and cancels. Hence the upstairs form is s_h\otimes\widetilde s. Division by k gives s_h\otimes e downstairs. Since kE\subseteq\ker\tau, exterior classes span their rational summand, so the identity holds on all of it.□

Proof. Starting from two plane fillings, we build relative chains and sweep them by the normal circle at two distinct balancing radii. The sweeps will be separated away from the root locus, so each remaining intersection is an intersection of the plane fillings multiplied by the sign of two normal disks. We first construct the sweeps as filling chains and then compute that normal sign.

Use the h-model of Lemma 5.2 over the value disk \{|v|\le\delta\}. For |v|\le\delta, set

and let \widehat\D=\widehat G\cap\{R_s=0\} be its root total. By the selected plane cut calculation used in Lemma 6.1, \widehat\D is acyclic; the same holds for \widehat G, the base total times a disk.

Choose the initial chains as finite integral sums of mapped oriented simplices, piecewise smooth after compatible subdivision. The swept chains will be continuous at the conical strata described below and piecewise smooth elsewhere. Intersections mean transverse intersections of the maps on their smooth common contacts, counted with their integer coefficients; we retain set notation for their images. For two classes l_1,l_2\in K, take c_1 as the reference page and c_2 as its positive neighbor, and represent l_i by cycles on the root page in G_{c_i}. Their images vanish in H_1(G_{c_i}) because l_i\in K=\ker\pi_*. This vanishing and the acyclicity of \widehat\D let us choose surface chains \Sigma_i\subset\widehat\D with \partial\Sigma_i=l_i and V_i\subset G_{c_i} with \partial V_i=-l_i. Arrange \Sigma_1,\Sigma_2 transversely in the real four-dimensional \widehat\D, with generic V_i and disjoint page-boundary collars. Since H_2(\widehat G)=0, the closed chain \Sigma_i+V_i bounds a 3-chain \Gamma_i\subset\widehat G. Prescribe a collar of the \Sigma_i-face on which R_s=t\ge0 is a positive-real normal coordinate. Near l_i, make its normal lift tangent to h=c_i, possible because (h,R_s) is a submersion there, and join it to a collar of V_i. Keep these collars fixed. Perturb only the remaining interiors of \Gamma_1,\Gamma_2 relative to their boundary chains, and make

transverse there. Each I_i is a real one-dimensional chain, with any endpoints on the prescribed V_i contacts. Relative transversality gives I_1\cap\Sigma_2=\Sigma_1\cap I_2=I_1\cap I_2=\varnothing: the respective dimension sums are 1+2<4 and 1+1<4. Separate their finite boundary contacts in the distinct page-value collars. Thus the only common root contacts are the prescribed \Sigma_1\cap\Sigma_2 intersections.

Choose distinct \sigma_1,\sigma_2 sufficiently near \lambda^{(\alpha-\beta)/2} that the strict balancing bounds in Lemma 6.1 hold for both; their circles then stay inside Y_\lambda. Sweep each \Gamma_i by the y-angle using |y|=\sigma_i\sqrt{|R_s|}, |z|=\sigma_i^{-1}\sqrt{|R_s|}, and z=R_s/y away from the root. This is the chain-level relative suspension C_3(\widehat G,\widehat\D)\to C_4(\widehat W). To realize it by mapped chains, triangulate the domains compatibly with the prescribed collars, the transverse root curves, and their boundary faces. Sweep each oriented top simplex, collapse its circle over the root, and push the resulting top chain into \widehat W. The internal faces cancel with the original chain coefficients. The \Sigma_i-face is zero in the relative complex, so it contributes no boundary after collapse. Write \mathfrak S also for the circle-first suspension of Section 6 followed by this collapse and pushforward. Give the sweep the negative of the raw product orientation. In the relative quotient chain complex the two relevant signs are

The second identity says that this boundary is the positive K-page class in the convention fixed after (6.2). The same reversal for both fillings does not change their intersection product.

We now verify that the mapped sweeps have the claimed filling boundary at root contacts. Along the prescribed normal collar, the collapsed circle is the center of a disk. At an incidental transverse root curve I_i, the local quotient is I_i\times\operatorname{cone}(T^2): its torus link is closed and oriented. The oriented top simplices of a triangulated cone have cancelling internal faces, and its tip is not a codimension-one face; hence this local model contributes no additional chain boundary. The map extends continuously to y=z=0 there.

We next locate the intersections of the two sweeps. Where R_s\ne0, the sweeps are disjoint at a common base point because their |y| differ. Where R_s=0, their possible common contacts are (\Sigma_1\cup I_1)\cap(\Sigma_2\cup I_2) =\Sigma_1\cap\Sigma_2. Thus all intersections occur above the prescribed transverse surface intersections. In particular, the conical tips over I_i are disjoint from the other swept chain. A small general-position approximation in disjoint neighborhoods of those tips, fixed on the prescribed collars and with each boundary kept in its assigned page, preserves the filling classes and adds no intersections. Thus the filling intersection is computed entirely on the smooth prescribed collars. Only the intersection sign of the normal disks remains. In complex normal coordinates y,z there, each swept collar is the real disk z=\sigma_i^{-2}\overline y. In real coordinates y=x+it, its graph map has matrix \mathrm{diag}(\sigma_i^{-2}, -\sigma_i^{-2}), so the ordered intersection determinant is

Both collar disks have the same orientation relative to the complex y-orientation (both are reversed by the chosen sweep), so their common reversals cancel in the intersection product. Also, interchanging a disk past a root surface contributes (-1)^{2\cdot2}=+1. Thus every root-surface intersection sign is reversed. The root-surface filling intersections are exactly s_d(l_1,l_2) by the common nearby-page convention, proving the lemma. Changing the uniform sign of the K-coordinate does not change this bilinear restriction.□

Proof. Both n_0,n_1 are 2\pmod5, so (n_j^2-1)/4\equiv2\pmod5. Thus a,b\equiv2\pmod5 and 5\nmid g. Also 2^r\mid n_j-1 and 2\mid n_j+1, whence 2^{r-1}\mid a,b and g\ge2^{r-1}>5.

Write a=ga', b=gb', with \gcd(a',b')=1. The desired index equations, on writing i=j+4t, become

Reduction modulo five gives m=4ab+a+b\equiv0\pmod5; since 5\nmid g, we have 5\mid m/g. Let d=\gcd(a'+b',4b'). The equality \gcd(a'+b',b')=1 gives d\mid4, and d also divides m/g=4ga'b'+a'+b'. Hence d\mid m/(5g), so the displayed equation has integer solutions for q=1,4. Its homogeneous solutions change j by 4b'/d; choose a solution with 1\le j\le4b'/d\le4b'. For q=1,

because g>5. Thus i=(qm/(5g)-a'j)/b'>0 for both values of q.□

Proof. The weighted rotation preserves each axis, hence its boundary homology class. Transfer of either axis boundary is a single boundary circle of the upstairs plane-curve page. In a positive plane-curve Seifert convention, the pairing of two distinct complex branch boundaries is their positive local intersection multiplicity. The sum of all oriented boundary classes is zero, so the self-pairing of the V=0 boundary is -(1+4a)=-n_0^2, and that of U=0 is -(1+4b)=-n_1^2. The two axes meet once. Reversing the second orientation gives the mixed value -1 upstairs. Division by the covering degree proves the first two formulas.

The integer-valued functional x\mapsto4e(b_j,x) reduces modulo 4 through \tau. Indeed a loop whose \tau-class is zero lifts closed; its full transfer consists of four deck translates, and the transferred axis class is deck invariant. The four Seifert pairings are equal, so their sum is divisible by 4. A class in \ker\tau can be represented by such loop systems, and \tau(b_j)=1 generates \mathbb Z/4. Evaluation at b_j gives 4e(b_j,b_j)=-n_j^2\equiv-1\pmod4, proving the last congruence.□

Proof. The coefficient of x\in E_{\mathbb Z_\ell} on v_j is c_j(x)=-4e(v_j,x). It is integral because 4e(b_j,x) is integral and n_j is a unit in \mathbb Z_\ell, and it sends v_j to 1 by (9.5). Its kernel U_{j,\ell} is therefore an integral orthogonal summand. The local congruence above gives \tau_{\mathbb Z_\ell}(U_{j,\ell})=0. At \ell=2, c_j(x)\equiv\tau_{\mathbb Z_2}(x)\pmod{4\mathbb Z_2} because n_j\equiv1\pmod4; at odd primes the target of \tau_{\mathbb Z_\ell} is zero. Thus the projection x\mapsto c_j(x)v_j is integral also at \ell=2: all divisions are by the unit n_j.

The image of (9.6) satisfies the lattice congruence (9.2). Conversely, write the second coordinate of an arbitrary lattice element as z=x_H\otimes v_j+u. Its congruence says x_H-\pi_{\nu*}l\in4H_{\mathbb Z_\ell}. Replacing l by

is legitimate even at \ell=2, because the numerator belongs to 4H_{\mathbb Z_\ell}. It does not alter \operatorname{pr}_\nu l and makes its image under \pi_{\nu*} equal to x_H. This proves surjectivity; injectivity follows from the rational direct sum.

The H\otimes U_{j,\ell} summand is orthogonal to the image of D_\nu. On two vectors from D_\nu, transfer adjunction gives

The product-form contribution on v_j is the negative of the last fraction, by e(v_j,v_j)=-1/4. Combined with the leading minus sign on K_\nu in (9.2), the result is -s_{d_\nu}(l,l'), as claimed.□

Proof. The two v_j have the same norm -1/4, while e(v_0,v_1)=-1/(4n_0n_1). Therefore

The vector w is monodromy fixed and hence pairs symmetrically for e. Reflection

is consequently an e-isometry commuting with monodromy, and it exchanges v_0,v_1. Define \rho_\nu on F_\nu as identity on (K_\nu)_{\mathbb Q} and 1_H\otimes R_w on (H\otimes E)_{\mathbb Q}. This is a rational Seifert involution. Let \mathcal R_\nu be identity on D_\nu and 1_H\otimes R_w from H\otimes U_{0,\mathbb Q} to H\otimes U_{1,\mathbb Q}. Equation (9.6) gives \rho_\nu\alpha_{\nu,0}=\alpha_{\nu,1}\mathcal R_\nu. The same R_w occurs for both coverings, so the square formed by \mathcal R_0,\mathcal R_1 and \mathcal I\oplus1 commutes. Consequently

The minus space of \rho_\nu is W_\nu=H_{\mathbb Q}\otimes\mathbb Qw. Its restricted Seifert form is e(w,w)s_h, so its Seifert operator is A_h. In particular it lies entirely in F_{\nu,1}\oplus F_{\nu,5}. Both factors of (9.8), and therefore \Delta, act identically on every other rational primary summand.

The minus spaces within the two primary sectors are

The canonical map x\otimes w\mapsto x\otimes w from W_0 to W_1 preserves the restricted Seifert and symmetric forms and intertwines A_h. It therefore restricts to isometries on the individual sectors displayed above. These restrictions are nondegenerate: e(w,w)\ne0, and s_h+s_h^{\mathsf t} is nondegenerate on both H^{(1)} and H^{(5)}. Since the Seifert isometry J_0 intertwines A_0 and A_1, composition with J_0^{-1} identifies, separately on each sector, the minus spaces of \rho_0 and J_0^{-1}\rho_1J_0. On F_{0,5} the isometry \Delta commutes with the L-action and preserves the Hermitian form recovered from B_0. The two involutions in (9.8) have the same L-dimension of their minus spaces, so their L-determinants cancel: \det_L\Delta=1. It is therefore an L^+-rational special-unitary element.

On F_{0,1} the same comparison gives determinant 1. The spinor norm of an involution with nondegenerate minus space is the product, modulo rational squares, of the quadratic values of an orthogonal basis of that space. The minus spaces just identified are isometric, so their spinor norms are equal and cancel in the product. Thus \Delta has trivial spinor norm. The standard exact sequence for \operatorname{Spin}\to\operatorname{SO} now supplies a rational spin lift [1, Section C.5, paragraph before Example C.5.5 and Lemma C.5.8].□

Proof. By Lemma 2.1, the rational primary parts H^{(1)} and H^{(5)} have dimensions two and four, respectively, and together equal H_{\mathbb Q}. Their eigenvalue sets are stable under inversion, so they are nondegenerate orthogonal summands for s_h. The restriction to H^{(1)} is symmetric, and each complex primitive fifth eigenspace is one-dimensional and pairs nondegenerately with its inverse eigenspace.

To obtain both Hodge signs in the primitive fifth eigenspaces of E_{\mathbb C}, we construct holomorphic forms with inverse primitive fifth characters. Their conjugate antiholomorphic classes will give opposite cup-product signs. Set m=4ab+a+b, and write a=ga', b=gb' with \gcd(a',b')=1. The weights of U,V in (9.3) are b/m,a/m. For q=1 and q=4 there are positive integers i,j, congruent modulo 4, with

The existence and positivity of these two index pairs were proved in Lemma 9.1.

On the smooth affine curve \widetilde P=1, the differential

is regular and nonzero. The second expression is used where \partial_V\widetilde P=0. Its deck character is \omega^{i-j}=1, so it descends to the quotient, and its weighted rotation character is e^{2\pi iq/5}. To obtain the needed holomorphic classes, we check that these differentials extend across every end of the normalized compactification. At an end put u=\operatorname{ord}U, v=\operatorname{ord}V. If 4au<4bv, then u<0 and v=-(4a+1)u; the order of (9.10) is (i-(4a+1)j)u-1\ge0. Here i<4a+1, because (9.9), q\le4 and j\ge1 give bi\le4m/5-a<b(4a+1). The case 4bv<4au is symmetric, using the second expression and j<4b+1. In the remaining case 4au=4bv<0, leading terms in U^{4a}+V^{4b} cancel and \operatorname{ord}(\partial_V\widetilde P)=u+4bv. There is a positive integer c with (u,v)=-c(b',a'); the order is

These are all possible ends, since one of U,V has a pole.

The two differentials for q=1,4 thus give nonzero holomorphic forms on the smooth compactification with characters \zeta and \bar\zeta. Complex conjugation gives antiholomorphic forms of the opposite characters. The cup-product Hermitian form has opposite signs on holomorphic and antiholomorphic forms. Poincaré duality exchanges reciprocal characters and gives opposite signs for the Hermitian intersection form on the corresponding homology classes. Since both characters have been constructed, each of the two homology eigenspaces has both signs. With complex coefficients, the map from affine-page homology to compactification homology is surjective and preserves intersections. Average a linear section of this map over the finite group generated by the commuting deck transformations and weighted rotation. The resulting equivariant section lifts the classes with their characters and intersection numbers unchanged. Transfer over \mathbb C identifies the deck-invariant part with E_{\mathbb C} and preserves the monodromy characters. Since e-e^{\mathsf t} is, up to our common Seifert convention, the page intersection form, the Hermitian form i(e-e^{\mathsf t})(x,\bar x) has both signs in each of E_\zeta and E_{\bar\zeta}.

We transfer these signs to the fixed quadratic sector by tensoring a primitive base eigenspace with the inverse eigenspace of E. After complexifying, choose \lambda\in\{\zeta,\bar\zeta\} and a nonzero x\in\ker(A_h-\lambda)\subset H_{\mathbb C}. This eigenspace is one-dimensional and pairs nondegenerately with its inverse eigenspace. The preceding statement that both signs occur applies to \ker(A_e-\lambda^{-1})\subset E_{\mathbb C} under either inverse monodromy convention. For y in this eigenspace, put

These are real, and I_H\ne0. The relation s^{\mathsf t}=sA in each factor, together with the vanishing of s_h(x,x) and e(y,y) when \lambda^2\ne1, gives

The vectors in this display lie in F_{\nu,1}\otimes_{\mathbb Q}\mathbb R, and the two signs of I_E prove that B_\nu|F_{\nu,1} is indefinite. These tensors and their conjugates already contribute at least four real dimensions. The invariant part of H has dimension two, and b_0 is a nonzero invariant vector of E by (9.4); its tensor contributes at least two more dimensions.

For the Hermitian sector, we tensor the primitive eigenspace of E with a fixed base vector. Choose x\in H\cap H^{(1)} with s_h(x,x)\ne0. Such an integral vector exists: the nondegenerate symmetric restriction to H^{(1)} has a rational vector of nonzero norm, and clearing denominators puts it in H. Tensor x with E_\zeta. This lies in the \zeta component of F_{\nu,5} for one of the two inverse monodromy conventions. If the eigenvalue of A_e on E_\zeta is \lambda\in\{\zeta,\zeta^{-1}\}, then on this eigenspace

and the displayed ratio is a nonzero real number. The two signs found above therefore give both signs for the Hermitian form of B_\nu at this real embedding. Its dimension is at least two.□

Proof. By Lemma 9.5, G_1=\operatorname{Spin}(F_{0,1},B_0) is a connected, simply connected, absolutely almost simple group over \mathbb Q of quadratic dimension at least five [1, Proposition C.3.10, Lemma C.4.1, Proposition C.4.10, and Example C.6.6]. The proved indefiniteness makes it noncompact over \mathbb R. The second correction group G_5=\operatorname{SU}(F_{0,5},\mathfrak h_{0,5}) is connected, simply connected, and absolutely almost simple over L^+: after algebraic closure it is \operatorname{SL}_d, with d\ge2 [12, Proposition 24.40(b), equation (158), and Summary 21.96]. It is noncompact at the real place found in Lemma 9.5. We use G=G_1\times\operatorname{Res}_{L^+/\mathbb Q}G_5, acting on the indicated primary spaces and as identity on their s_0-orthogonal primary complement. The Spin action preserves B_0 and commutes with A_0=1 on F_{0,1}. The special-unitary action preserves the Hermitian form, hence B_0, and commutes with A_0 on F_{0,5} by L-linearity. On both sectors 1+A_0 is invertible and

Consequently this action of G preserves the Seifert form s_0. We obtain the density assertion for G by treating its two factors separately.

The strong-approximation theorem of Kneser and Platonov, in the precise form of [19, Theorem 2.3], says that for a connected, simply connected, absolutely almost simple group over a number field, noncompactness at a place in the omitted finite set implies density of rational points in the adeles away from that set. Apply it to G_1 over \mathbb Q and to G_5 over L^+, omitting all archimedean places. Restriction of scalars and a finite product give density of G(\mathbb Q) in G(\mathbb A_f).

Write \rho\colon G\to\operatorname{GL}(F_0) for this rational action. Lemma 9.4 gives a rational lift \widetilde\Delta\in G(\mathbb Q) with \rho(\widetilde\Delta)=\Delta. For each rational prime \ell, put a_\ell=\widetilde\Delta if \ell\mid n_0, and a_\ell=1 otherwise. The tuple \mathbf a=(a_\ell)_\ell has finite support and lies in G(\mathbb A_f). We seek one rational point g_{\mathrm{rat}}\in G(\mathbb Q) whose localizations g_{\mathrm{rat},\ell} differ from these selected corrections by automorphisms of the full source lattice:

To construct a suitable open neighborhood of \mathbf a, define the exact source-lattice stabilizer

Each C_\ell is compact open. Here the spin action has finite central kernel, so it need not be faithful; compactness follows from finiteness of the covering onto its algebraic image. For the adelic topology we only need compact opens inside these stabilizers. Spread out G and its rational action on F_0 to a smooth affine group scheme \mathcal G over \mathbb Z[1/N], enlarging N so that the action preserves \Lambda_0\otimes\mathbb Z[1/N]. At every \ell\nmid N its standard compact open \mathcal G(\mathbb Z_\ell) lies in C_\ell. For each of the finitely many primes \ell\mid N, choose a compact open subgroup K_\ell\subset C_\ell; at the other primes put K_\ell=\mathcal G(\mathbb Z_\ell). Then \mathcal K=\prod_\ell K_\ell is compact open in G(\mathbb A_f) and \mathbf a\mathcal K is an open coset contained in the desired set of local integral corrections.

At \ell\nmid n_0, Lemma 9.3 makes J_0 an integral lattice isometry. At \ell\mid n_0 we have \ell\nmid n_1, so J_0\rho(a_\ell)=J_1 is integral. Consequently

Density supplies g_{\mathrm{rat}}\in G(\mathbb Q)\cap\mathbf a\mathcal K. Writing g_{\mathrm{rat},\ell}=a_\ell k_\ell with k_\ell\in K_\ell\subset C_\ell shows that J_0\rho(g_{\mathrm{rat}}) induces the same local lattice equality at every prime. A full \mathbb Z-lattice in a rational vector space is the intersection of its localizations at all primes; applying this to both sides yields J_0\rho(g_{\mathrm{rat}})\Lambda_0=\Lambda_1. Since J_0 and \rho(g_{\mathrm{rat}}) preserve their respective Seifert forms, this is the asserted integral congruence.□

Proof of Theorem 1.1. Proposition 3.2 and Lemma 5.3 give smooth ambient fourfold germs and reduced isolated functions of orders five and four. Theorem 4.1 identifies their actual integral Milnor Seifert forms, and Proposition 9.6 gives an integral congruence. The Kato argument above identifies their embedded links. The analytic pair-cone construction above extends that equivalence to an ambient homeomorphism germ, proving the assertion.□