# Equatorial cycles and the projective period test *Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).* This chapter compares the actual signed affine equatorial contour with the positive-normal-circle projective tube. It proves the center multiplicity two, the individual normal coefficient one, the affine scalar-fiber cohomology splitting and the whole-sphere logarithm needed to recover an affine null-homology statement from projective periods. Basic references are Atiyah, Bott and Gårding [A, B] for the classical hyperbolic contour and lacuna method, and Hatcher [H] for finite-chain homology and Thom constructions. The proof uses the complete internal rational-top-form and projective-tube theorems, with their exact smooth/chain hypotheses. The [complete angular-contour and polynomial-continuation reading](angular-contours-and-polynomial-continuation.md) supplies the analytic argument consumed in Section 3. The complete all-integer parity-sensitive angular Fourier identity is proved in [Angular distributions and their full Fourier transforms](../../AN02-L185.html#4-the-angular-fourier-identity-on-the-whole-frequency-space), Theorem 4.1, with arbitrary angular distributions and the full frequency origin retained. The positive-polar analytic tube bound, analytic differential decrease and ordinary-wavefront inclusion are proved in Analytic directions of holomorphic boundary values. The homogeneous Euler bound and sphere-conormal exclusion are proved in Dilation equations and tangential analytic singularities, Theorem 4.1 and Corollary 4.2. General smooth and analytic wavefront operations are proved in Pulling back and combining analytic singularities, Sections 4–8. Gaussian packets and homogeneous Fourier directions, Theorem 8.1, proves exact ordinary and analytic Fourier-wavefront interchange away from both origins for arbitrary complex degree, finite Euler chains and arbitrary origin extensions. [Derivative sequences and microlocal regularity](../../AN02-L190.html#complete-proof), Sections 1–8, supplies the exact admissible derivative-sequence convention, bounded localization and base regularity, all analytic-map operation bounds, homogeneous Fourier interchange away from both origins, and the uniform class estimates needed for limits. Ordinary normal continuity and hypocontinuity retain their stated hypotheses; stronger class estimates require the stated uniform constants. The component conclusion has the following precise scope. Let \(F\) be real homogeneous of degree \(m\geq1\), hyperbolic in \(N\), in \(n\geq2\) variables. Let \(C=\Gamma(F,N)^*\) and let \(W_F\) be the union of the tangent hyperbolicity polars defined in the analytic reading. **Component theorem.** At the analytic prerequisites just stated, on every connected component of \(\mathbb R^n\setminus(W_F\cup(-C))\), whether the canonical deformed affine equator bounds over \(\mathbb Q\) is constant. One such bounding cycle gives homogeneous polynomial values for every principal-power causal inverse, and entire exponential-type continuation for every hyperbolic lower-order completion and each positive power of that completion. The exact polynomial and entire estimates are proved in the accompanying analytic reading; the return from their periods to affine rational null homology is proved here. The relative-disc and fiber arguments below also stand on their explicitly stated geometric assumptions without requiring \(F\) to be hyperbolic. The hyperbolic hypotheses enter through the allowed fields and analytic contour formulas in the component application. Fix \(n\geq2\), a real homogeneous polynomial \(F\) of degree \(m\geq1\), a nonzero real vector \(x\), and \(L(\zeta)=x\cdot\zeta\). Set \[ H=\{L=0\},\quad M=H\setminus\{F=0\},\quad U=\mathbb P^{n-1}\setminus\{F=0\},\quad Y=\mathbb P(H)\cap U,\quad V=U\setminus Y . \tag{1} \] Let \(\Sigma=S^{n-1}\cap x^\perp\cong S^{n-2}\), with the inherited orientation: the boundary of the negative hemisphere of the outward-oriented \(S^{n-1}\). An allowed even equatorial field \(\Theta\) obeys \(x\cdot\Theta=0\), \(\Theta(-\xi)=\Theta(\xi)\), and the stated uniform deformation theorem makes the maps \[ k^-(\xi)=\xi-i\varepsilon\Theta(\xi),\qquad k^+(\xi)=\xi+i\varepsilon\Theta(\xi) \tag{2} \] take their values in \(M\), for a common small \(\varepsilon>0\). Write \(k=k^-\). Let \(\pi:\mathbb C^n\setminus\{F=0\}\to U\) be projectivization, let \(p=\pi|_M:M\to Y\), and put \(\beta=p_*[k]\). Here \([k]\) means the image of an actual finite oriented fundamental cycle of \(\Sigma\); in dimension zero it is the inherited two-point cycle, with total coefficient zero. Rational coefficients are used for the receiver. The integral orientation identities below hold without division. ## 1. The antipodal relation gives exactly two centers Let \(a(\xi)=-\xi\) on \(\Sigma\), and let \(S(\zeta)=-\zeta\) on \(M\). Evenness in (2) gives the exact map identity \[ k^+\circ a=S\circ k^- . \tag{3} \] The antipodal map of the sphere \(S^{n-2}\) has degree \((-1)^{n-1}\). For completeness, on the boundary of the unit ball in real dimension \(n-1\), the ambient map \(-I\) has determinant \((-1)^{n-1}\). It preserves the outward-normal prescription, hence induces that degree on the boundary. Reversing the chosen sphere orientation on both source and target leaves the degree unchanged. In the \(S^0\) case it interchanges the two points and sends their difference to its negative, again giving \(-1\). The scalar map \(S\) is homotopic to identity in \(M\), through \(S_t(\zeta)=e^{i\pi t}\zeta\), because homogeneity gives \(F(S_t\zeta)=e^{im\pi t}F(\zeta)\ne0\). Apply singular homology to (3). It follows that \[ [k^+]=(-1)^{n-1}[k^-]\quad\hbox{in }H_{n-2}(M;\mathbb Z). \tag{4} \] Thus the center combination used in the actual negative-exponent contour is \[ [k^-]+(-1)^{n-1}[k^+]=2[k]. \tag{5} \] Projectivization likewise gives \(2\beta\). This factor is two, independently of the degree \(m\) of \(F\). The degree-\(m\) cover \(F=1\) in the preceding finite-cover theorem supplied a vanishing group; it is not this hemisphere multiplicity. ## 2. A relative normal disc for one affine center cycle Take any smooth map \(k:\Sigma\to M\), with \(\Sigma\) a compact oriented closed \((n-2)\)-manifold, allowing a finite oriented zero-dimensional manifold. Compactness gives \(\delta>0\) such that \[ \Phi:D_\delta^2\times\Sigma\longrightarrow U,\qquad \Phi(z,\xi)=[k(\xi)+zx] \tag{6} \] is defined and avoids \(F=0\). Indeed \(F(k(\xi))\ne0\) has a positive minimum modulus on the compact image, and uniform continuity applies to the bounded vectors \(k(\xi)+zx\) for small \(|z|\). Also \(k(\xi)+zx\ne0\): its \(L\)-value is \(z|x|^2\), so if it vanished then \(z=0\) and \(k(\xi)=0\), impossible in \(M\). The inverse image of \(Y\) under (6) is exactly \(\{0\}\times\Sigma\), since \[ L(k(\xi)+zx)=z|x|^2. \tag{7} \] Hence (6) is a map of pairs \[ (D_\delta^2\times\Sigma,S_\delta^1\times\Sigma) \longrightarrow(U,V). \tag{8} \] Orient the complex disc positively and put it before the base cycle. Its relative fundamental class maps to a class \(A_k\in H_n(U,V;\mathbb Q)\). We claim that, with the exact disc-first homological Thom map \(\Theta\) in [Projective exhaustion and the finite-chain tube receiver](../../AN02-L124.html#tp6-the-included-normal-bundle-argument-and-the-tube-deduction), \[ A_k=\Theta(p_*[k]),\qquad \pi_*[C_\delta(k)]=\tau(p_*[k]), \tag{9} \] where the affine circle-first cycle is \(C_\delta(k)(z,\xi)=k(\xi)+zx\), \(|z|=\delta\), and \(\tau=\partial_{U,V}\Theta\). Here is the relative-disc verification. Choose a Hermitian metric on \(U\), by a smooth locally finite partition and local Hermitian metrics. Its real orthogonal normal representative of the quotient normal line is invariant under the complex structure. The local addition in [Manifold duality and tubular sections](../../reproduce/L124/context/providers/manifold-duality-and-tubular-sections.html), Section 3, has derivative \(DL_{(y,0)}(a,b)=a+b\), so its small-radius tubular map induces the identity on the quotient normal. Extending this to the full normal bundle uses the source's fiber map \(v\mapsto\rho(y)v/\sqrt{1+\|v\|^2}\), with \(\rho(y)>0\); its normal derivative at zero is positive multiplication by \(\rho(y)\). It therefore preserves the complex orientation. Shrink \(\delta\) so the image of (6), on its compact domain, lies in this tubular neighborhood. In bundle coordinates write the inverse tubular map as \[ \Psi(z,\xi)=(b(z,\xi),v(z,\xi))\in\nu . \tag{10} \] At \(z=0\), \(b(0,\xi)=p(k(\xi))\), \(v(0,\xi)=0\). Moreover \(v(z,\xi)\ne0\) for \(z\ne0\), by (7) and the tubular map's zero section. The normal derivative in the \(z\)-direction is the quotient of the complex vector \(x\), multiplied by the positive inverse radial scale \(\rho(p(k(\xi)))^{-1}\) if the full-bundle tube is used. Thus it is a complex-linear nonzero map \[ A_\xi:\mathbb C\longrightarrow\nu_{p(k(\xi))}. \tag{11} \] It is nonzero because the derivative of \(L\) in (7) is \(|x|^2>0\). Complex linearity makes it preserve the real normal orientation. The derivative here is the normal quotient derivative; tangential variation of the chosen representative causes no change to it. There is an explicit homotopy of pairs in the full vector bundle \(\nu\), for \(00\) values have nonzero normal vector, and the limit is nonzero by (11). Thus (12) is a homotopy of the pairs in (8) to the linear positive fiber-disc family (13). It uses the full normal bundle, so no upper bound on its scaled fiber radius is required. Here is the exact normalization and naturality calculation used in this step. Let \(f=p\circ k:\Sigma\to Y\), and define the oriented fiberwise complex-linear isomorphism \[ B:\Sigma\times\mathbb C\longrightarrow\nu,\qquad B(\xi,z)=(f(\xi),A_\xi z). \tag{14} \] This is a bundle map over \(f\), with the source interpreted as the pullback bundle trivialized by \(A_\xi\); it maps nonzero vectors to nonzero vectors. If \(u_\nu\) is the integral degree-two Thom class, \(B^*u_\nu\) is the positive trivial Thom class \(u_2\). Indeed its restriction to each fiber has generator one, because the real determinant of a nonzero complex-linear map is positive, and the already proved uniqueness of the Thom class gives the assertion. The right-cap operator in that complete Thom/Euler reading, Section 4 has the simplex formula \[ R_u\sigma =u(\sigma[j,j+1,j+2])\,\sigma[0,\ldots,j], \qquad \dim\sigma=j+2 . \tag{15} \] For a cochain representative \(u\), composition with \(B\) gives literal chain equality \[ B_\#R_{B^*u}c=R_uB_\#c . \tag{16} \] Both sides evaluate \(u\) on the same last three vertices and retain the same pushed-forward front face, with the same coefficient. They descend to the relative chains because \(u\) vanishes on the nonzero-vector subspace. If a different cocycle represents the same Thom class, the exact cap homotopy in the provider gives the same induced homology map. For the positive disc crossed with the fundamental cycle of \(\Sigma\), the trivial Thom calculation gives \[ (\operatorname{pr}_\Sigma)_*R_{u_2} [D^2\times\Sigma]=[\Sigma]. \tag{17} \] The product normalization in the provider puts the fiber last. Exchanging its real two-disc with the base of dimension \(n-2\) gives sign \((-1)^{2(n-2)}=1\), so (17) is exactly the disc-first class used here. The inclusion of the oriented disc pair \((D^2,S^1)\) in \((\mathbb C,\mathbb C^*)\) has positive relative fiber generator: radial retraction of the punctured disc onto its positive circle and the ball-pair boundary identify that generator. This is the same two-dimensional positive generator used in the supplied product-pair proof. Let \(\operatorname{pr}_\nu:\nu\to Y\) be its base projection. Equations (16)–(17) and \(\operatorname{pr}_\nu B=f\operatorname{pr}_\Sigma\) give \[ (\operatorname{pr}_\nu)_*R_{u_\nu} B_*[D^2\times\Sigma]=f_*[\Sigma]=p_*[k]. \tag{18} \] The left side is the exact homological Thom cap isomorphism. Its inverse therefore sends \(p_*[k]\) to the relative class of (13). Excision and the homotopy (12) give the first identity in (9). The argument fixes the induced map and its coefficient; it does not use an abstract equality of dimensions as a substitute for naturality. Finally the oriented product boundary is \[ \partial(D^2\times[\Sigma]) =S^1\times[\Sigma]+D^2\times\partial[\Sigma] =S^1\times[\Sigma]. \tag{19} \] Naturality of the pair boundary and (6) give the second identity of (9), with positive normal circle first and coefficient one. This establishes the geometric comparison for one specified affine center cycle at the declared tubular/Thom inputs. Apply (9) to both centers in (2). The actual signed affine contour \(\alpha_-\), using circle first and the weights of (5), has projectivized homology class \[ \pi_*[\alpha_-] =\tau\!\left(p_*[k^-]+(-1)^{n-1}p_*[k^+]\right) =2\tau(\beta)\quad\hbox{in }H_{n-1}(V;\mathbb Q). \tag{20} \] No factor \(m\), no additional hemisphere sign and no scalar arising from \(|x|^2\) is inserted: (7) is complex-linear with positive real determinant, and (19) fixes the normal boundary orientation. ## 3. Receiving the full all-powers period family Suppose a bounding canonical affine cycle at one point of a real connected component has already given, by the [complete angular-contour and polynomial-continuation proof](angular-contours-and-polynomial-continuation.md#one-bounding-cycle-gives-homogeneous-polynomials-on-the-whole-component), homogeneous polynomial values for every causal inverse \(E_k\) of \(F(D)^k\) throughout that component: \[ E_k|_{\mathcal C}=Q_k|_{\mathcal C},\qquad \deg Q_k=mk-n, \tag{21} \] with \(Q_k=0\) when this degree is negative. The conclusion (21) is provided by the complete analytic reading's contour argument; it does not assume component constancy of null homology. Fix another \(x\in\mathcal C\). For any integers \(k,s\geq1\), take a multi-index with \[ |\alpha|=mk+s-n,\qquad q=mk-n-|\alpha|=-s<0. \tag{22} \] When the requested degree is negative, there is no nonzero homogeneous numerator of that degree. Otherwise every such derivative of the polynomial in (21) vanishes. The exact negative-exponent contour formula, with its nonzero stated scalar and \((iL)^q=i^{-s}L^{-s}\), therefore gives \[ \int_{\alpha_-} \frac{\zeta^\alpha\,\omega(\zeta)} {F(\zeta)^kL(\zeta)^s}=0 . \tag{23} \] The numerator degree in (22) is precisely the projective descent degree for dimension \(d=n-1\). Its coefficient with the Kronecker form has weight zero and is horizontal, as proved in [Rational top forms detect cycles in a hypersurface complement](../../AN02-L182.html#8-the-projective-homogeneous-family-with-positive-pole-exponents). Thus (23) is the period of the descended projective form on (20). Linear combinations of the degree-\(|\alpha|\) monomials give every homogeneous polynomial numerator. Equation (20) and invertibility of two over \(\mathbb Q\) give zero periods against all those rational top forms on \(\tau\beta\). The general projective receiver in [Finite covers and normal circles in projective complements](../../AN02-L183.html#6-the-general-rational-period-receiver-now-has-both-inputs), Corollary 6.1, gives \[ \beta=p_*[k]=0\quad\hbox{in }H_{n-2}(Y;\mathbb Q). \tag{24} \] This is the exact projective consequence of the all-powers polynomial argument. It is still necessary to receive (24) back in affine \(M\), where a scalar-circle cohomology component can be present. ## 4. The affine fiber component and component constancy Here is the complete smooth decomposition needed from the affine circle-bundle argument: \[ H^r_{\mathrm{dR}}(M;\mathbb C) =p^*H^r_{\mathrm{dR}}(Y;\mathbb C) \oplus[\vartheta]\wedge p^*H^{r-1}_{\mathrm{dR}}(Y;\mathbb C), \qquad \vartheta=\frac{dF}{mF}, \tag{25} \] On scalar fibers \(\vartheta=d\lambda/\lambda\), of period \(2\pi i\). We prove (25), including injectivity and the primitive signs. Set \(r(\zeta)=m^{-1}\log|F(\zeta)|\) and \(S=\{|F|=1\}\subset M\). Homogeneity makes \[ S\times\mathbb R\longrightarrow M,\quad(s,r)\longmapsto e^r s, \qquad \zeta\longmapsto(e^{-r(\zeta)}\zeta,r(\zeta)) \tag{26} \] inverse smooth maps. Contracting the real factor is a deformation retraction. The fibers of \(S\to Y\) are circles with the action \(T_t(s)=e^{it}s\), and its vector field \(Z\) generates increasing \(t\). The real form \(\alpha=\operatorname{Im}\vartheta|_S\) is closed and invariant, with \(\alpha(Z)=1\); moreover \(\vartheta|_S=i\alpha\). These facts follow by differentiating \(F(e^{it}s)=e^{imt}F(s)\) and \(d|F|=0\) on \(S\). For a smooth form \(w\) on \(S\), define \[ Aw=\frac1{2\pi}\int_0^{2\pi}T_t^*w\,dt,\qquad Kw=\frac1{2\pi}\int_0^{2\pi}\int_0^t T_v^*(\iota_Zw)\,dv\,dt . \tag{27} \] Differentiating pullbacks and the coordinate identity \(\mathcal L_Z=d\iota_Z+\iota_Zd\) give \(A-I=dK+Kd\), by integrating first in \(v\) and then in \(t\). The parameters are compact, so these are smooth coefficientwise integrals, with no compactness assumption on \(S\). Thus averaging replaces each closed form by a cohomologous invariant one, and averaging a primitive of an invariant exact form keeps it a primitive. For an invariant closed form \(w\), put \(b=-i\iota_Zw\) and \(h=w-i\alpha\wedge b\). Both are invariant and horizontal: \(\iota_Zb=0\), and \(\iota_Z(i\alpha\wedge b)=ib=\iota_Zw\) cancels the vertical contraction of \(w\). Cartan's identity gives \(d b=0\), and \(d\alpha=0\) then gives \(d h=0\). In a circle product chart, horizontal invariant coefficients are independent of the angular variable and contain no angular differential; they descend uniquely to forms on the base. Therefore \[ w=p^*\eta+i\alpha\wedge p^*\gamma \quad(d\eta=d\gamma=0). \tag{28} \] This proves surjectivity of the two summands in (25). For injectivity, suppose the expression in (28) is \(du\). Average \(u\), then decompose its invariant coefficients, without requiring them closed, as \(u=p^*u_0+i\alpha\wedge p^*u_1\). Differentiation and comparison of the horizontal and vertical parts give \(\eta=d u_0\), \(\gamma=-d u_1\). Thus both base classes are zero. In degree zero the vertical summand is absent. Under (26), the global form is \(\vartheta=dr+i\alpha\); \(dr\) is exact, so the same decomposition transfers from \(S\) to all of \(M\). This proves (25). For \(n\geq3\), that bridge also constructs a logarithm on the canonical cycle by extending its allowed even field to the entire \(S^{n-1}\), using [Tangent cones that permit an imaginary push](../../AN02-L118.html#lc035-6-the-exact-field-adapters-used-in-c7-c8-and-ah3), the complete proof sections on compact permitted families and the full-sphere field extension. Those sections supply the exact compact-cone and uniform zero-free deformation statements used here. The resulting nonvanishing function \(\Phi(\xi)=F(\xi-i\varepsilon V(\xi))\) has a global logarithm \(g\) on the sphere, giving \[ k^*\vartheta=m^{-1}d(g|_\Sigma). \tag{29} \] We spell out the logarithm receiving step. Near the equator project a sphere point orthogonally onto \(H\), normalize that nonzero projection, and compose with \(\Theta\). This is a smooth even extension in a collar. The precise compact local-cone stability input keeps it allowed on a smaller collar. Blend it with the constant allowed vector \(N\) by an even cutoff supported in that collar and equal to one on the equator. Convexity of every relevant open cone keeps the blended field \(V\) allowed; outside the collar it is \(N\). The exact uniform zero-free deformation input, on the compact whole sphere, supplies one small \(\varepsilon\) for which \(\Phi\) is nonzero everywhere. If that parameter is smaller than the initial one, the already proved allowed-field/parameter homotopy identifies the same canonical cycle. The form \(a=d\Phi/\Phi\) is closed. Every piecewise smooth loop on \(S^{n-1}\), \(n-1\geq2\), misses a point: its finitely many compact smooth arcs are Lipschitz, and subdivision into \(O(\delta^{-1})\) pieces covers their images by balls of radius \(O(\delta)\); their \((n-1)\)-dimensional volume tends to zero as \(O(\delta^{n-2})\). Stereographic projection from a missing point identifies the loop with one in \(\mathbb R^{n-1}\), where straight contraction gives a piecewise smooth contraction. Stokes for that contraction makes its integral of \(a\) zero. Define \(g\) by integrating \(a\) from a fixed point. The zero loop periods make it independent of the path. In local coordinates it is a smooth primitive, so \(dg=a\). Therefore \(d(\Phi e^{-g})=0\); the connected sphere makes that function a nonzero constant. Adding a chosen logarithm of that constant gives \(\Phi=e^g\). Restricting to \(\Sigma\) gives (29). The full-sphere extension matters in \(n=3\): a nonvanishing function on the equatorial circle alone need not have zero winding. Equation (24) kills all periods of the first summand in (25), by the finite smooth-cycle/de Rham comparison. Equation (29) kills those of the second: for a closed base form \(\eta\), \[ k^*(\vartheta\wedge p^*\eta) =m^{-1}d\bigl(g|_\Sigma\,k^*p^*\eta\bigr). \tag{30} \] Stokes gives zero on the closed \(\Sigma\). [When zero smooth periods mean that a cycle bounds](../../AN02-L116.html#cd7-detection-noncompactness-degree-zero-and-rational-coefficients) then gives \([k]=0\) over \(\mathbb C\), and its faithful rational coefficient comparison gives \([k]=0\) over \(\mathbb Q\). For \(n=2\), \(H\) is a complex line. Existence of the nonzero zero-free cycle means \(F|_H\) is a nonzero homogeneous polynomial on that line, so its complement is \(\mathbb C^*\). It is path connected: any two nonzero points are joined by radial and angular paths avoiding zero. The inherited two-point cycle has total coefficient zero and hence bounds a finite path chain in \(M\). There is no negative-degree fiber summand, and no ordinary positive-point generator is substituted for this reduced zero-cycle. Suppose the canonical affine cycle bounds at one point of the component. The complete analytic reading's polynomial continuation theorem gives (21) throughout that component. Equations (20)–(24), the affine splitting (25), the logarithm (29)–(30) and the two-point argument therefore prove that it bounds over the rationals at every component point. If it bounds nowhere, the null-homology condition is false at every point. Thus its truth is constant on a connected component, at the exact stated analytic prerequisites. The analytic reading also supplies the homogeneous-polynomial values of all principal-power inverses and the entire exponential-type continuation for every hyperbolic lower-order completion and each of its positive powers. The constants of that exponential bound may depend on the power. A polynomial lacuna need not be a zero region: the two-dimensional wave inverse has the nonzero constant value \(-1/2\) in the future interior with \(D=-i\partial\). The proof concerns rational null homology; it does not infer integral component constancy or remove integral torsion. ## References [A] Michael Atiyah, Raoul Bott and Lars Gårding, *Lacunas for hyperbolic differential operators with constant coefficients, I*, Acta Mathematica **124** (1970), 109–189. [Primary article](https://www.its.caltech.edu/~matilde/HypPDEAtiyahBottGarding.pdf). [B] Michael Atiyah, Raoul Bott and Lars Gårding, *Lacunas for hyperbolic differential operators with constant coefficients, II*, Acta Mathematica **131** (1973), 145–206. [Primary article](https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6156-11511_2006_Article_BF02392039.pdf). [H] Allen Hatcher, *Algebraic Topology*, Cambridge University Press (2002), §§3.3 and 3.G. [Author's freely readable text](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf). The exact relative-cap and tubular arguments used here are the complete linked internal proofs. Original exposition, examples, solutions and figures are CC0-1.0. Linked historical works retain their own rights and are not reproduced. The rational theorem detects no integral torsion.