Infinitely many closed geodesic images on every Riemannian sphere

Proof. It suffices to work on one connected component, so assume that M is connected. If \pi_1(M) is infinite, Rademacher and Taimanov [27, Theorem 2] give infinitely many geometrically distinct closed geodesics for the given metric. Their Riemannian convention identifies geodesics exactly when their images coincide [27, Section 3]. Taking the least positive period gives a prime representative with the same image. Their theorem includes the nonorientable case.

If \pi_1(M) is finite, the elliptization case of Perelman’s geometrization theorem says that M is diffeomorphic to a spherical space form S^3/\Gamma, where \Gamma is finite and acts freely and smoothly on S^3 [24, Corollary 0.2(b)]. Composing the quotient map with a diffeomorphism S^3/\Gamma\to M gives a finite smooth cover S^3\to M. Corollary 1.2 therefore applies. Only the smooth covering is used, so the given Riemannian metric remains arbitrary.□

Proof. Embed the sphere in \mathbb R^{n+1}. Pointwise exponential charts, the embedding H^1(S^1)\subset C^0(S^1), and the one-dimensional H^1 product rule make \Lambda a smooth Hilbert submanifold of H^1(S^1,\mathbb R^{n+1}). Its induced strong metric is complete. Indeed, an intrinsic Cauchy sequence is ambient Cauchy; its limit still takes values in the closed sphere, and a local Hilbert chart near that limit bounds intrinsic path distances by ambient distances. Time translations and reflection are isometries. Each acts smoothly, and their action is jointly continuous in the strong H^1 topology.

Extend g to a smooth positive symmetric matrix A(x) along the ambient tangent bundle, preserving the tangent–normal splitting. Then

is smooth. To check compactness, suppose that E(\gamma_r) is bounded and \| \dd E(\gamma_r)\|\to0. Pass to a subsequence converging weakly in H^1 and uniformly to \gamma. Let P_x denote Euclidean tangent projection. The tangent fields

are bounded in H^1. Testing \dd E(\gamma_r) on \eta_r, every term containing \gamma_r-\gamma without a derivative tends to zero: its uniform norm tends to zero, while the squared velocities have bounded integrals. This includes the terms from differentiating A(\gamma_r) and P_{\gamma_r}. Consequently

Uniform convergence of the coefficients and weak convergence of the velocities now give convergence of their energy norms. Uniform positivity of A proves strong H^1 convergence.

At a critical point the weak Euler–Lagrange equation says, in a coordinate interval, that the distributional derivative of the metric momentum is an L^1 quadratic expression in the velocity. The momentum, hence the velocity, is continuous. The geodesic equation then gives smoothness. Its periodic boundary condition follows from the first variation. Conversely, geodesics and constants are critical.

Normalize the negative gradient by

This is a smooth invariant vector field of norm at most one, strict away from the critical set. A bounded vector field on a complete strong Riemannian manifold has a complete flow: a trajectory of finite time has finite length and a limit, through which the local flow extends it.

Finally, near the constants use coordinates \gamma(t)=\exp_x \xi(t), where \xi\in H^1(S^1,T_xM) has mean zero. The inverse function theorem gives a uniform tube over the compact zero section. The normal Hessian of energy is positive coercive by the mean-zero Poincaré inequality. Taylor integration and the positive square-root change of normal coordinates make the energy an exact positive quadratic form there. These constructions are equivariant. Small energy forces both the diameter and the ambient L^2 derivative norm of a loop to be small, so all sufficiently small sublevels lie in this tube. Radial contraction gives the claim.□

Proof. Palais–Smale implies a positive lower bound on \|\nabla E\| in such an interval. The normalized flow therefore decreases energy at a uniformly positive rate there. A fixed sufficiently long time moves its upper sublevel into its lower sublevel, and the intervening flow gives the required homotopies. A small regular collar at an endpoint similarly compares the open and closed sublevels. Apply these constructions simultaneously at the endpoints of a pair. All maps are equivariant because they use the same invariant flow.□

Proof. Choose \rho>0 below the injectivity radius. The open manifold

is identified with the polygons formed by minimizing constant-speed segments; subscripts are cyclic. Its energy is

Choose N so large that B/\sqrt N<\rho. On each time cell, replace the central subinterval of proportion s, 0\le s\le1, by its minimizing segment. Cauchy–Schwarz makes all endpoint distances shorter than \rho, and shows that this replacement does not increase energy. At s=1 the loop is a polygon, and polygons are fixed throughout.

For positive subinterval width, strong H^1 continuity follows from smooth dependence of short geodesics on endpoints and continuity of restriction and rescaling in L^2. As the width tends to zero, the replacement energy is bounded by the original energy on the disappearing intervals. Those integrals tend uniformly to zero along any strongly H^1-convergent sequence. This proves continuity also at s=0. Centering the replacements proves the stated symmetries.

On the closed sublevel f_N\le B^2, every edge has length at most B/\sqrt N<\rho, so this sublevel is compactly contained in P_N. Its boundary is smooth at a regular value. The polygon first variation imposes matching velocities at every vertex, so its critical points are exactly the smooth closed geodesics in the band.

At such a geodesic, decompose an H^1 variation field into its piecewise Jacobi interpolation and a field vanishing at every vertex. Integration by parts makes the two summands orthogonal for the index form. On a cell the second summand satisfies the Dirichlet Poincaré inequality. A curvature upper bound, the speed bound B, and sufficiently small cell width make its index form positive definite. The polygon Hessian is the first summand, which therefore has the full Hilbert index and nullity. Sard’s theorem for f_N gives dense regular cuts, since its critical values in the band are exactly those of E. Compact smooth manifolds with boundary have finite CW models, completing the claim.□

Proof. A nonconstant geodesic has constant positive speed and is an immersion. Its periods form a closed discrete subgroup of \mathbb R, so there is a least positive period. Dividing by that period gives its prime root.

To compare two primes with the same image, first use unit-speed parameters. Cover the parameter circle of the first by finitely many compact intervals on which it is an embedded arc. The inverse images of those arcs under the second geodesic form a finite closed cover of its parameter circle. By Baire’s theorem, one contains an interval. Within this interval choose a point whose image is not an endpoint of the chosen arc; an immersion cannot take an interval into that finite endpoint set. Near this point both geodesics lie in the same embedded arc and have the same tangent line. Their unit velocities agree up to sign. Uniqueness for the geodesic equation identifies them globally up to translation and reversal. Their least unit-speed periods agree, giving the assertion for period-one parametrizations as well. This proof allows self-intersections of either prime.

If reversal preserved an oriented phase orbit, one would have \gamma(t)=\gamma(a-t). Differentiation at a fixed point of this parameter reflection gives \dot\gamma=-\dot\gamma, impossible for a nonconstant geodesic.□

Proof. Use the periodic index form over integer intervals at prime speed. In the form domain for the (m+k)-period problem, require that the value at time m equal the initial value. In the direct sum of the m- and k-period domains, require that the two initial values agree. Both are restrictions of codimension at most n, and the restricted forms identify. If the two indices drop by a,b\in[0,n], their original difference is a-b. This proves the first estimate. The indices are finite since the index form is a positive derivative form plus a compact zeroth-order perturbation.

The sequences I_{i,m}+n and I_{i,m}-n are respectively subadditive and superadditive. Their normalized limits agree; applying the inequalities along multiples of each fixed m gives

Nonnegative indices give \Delta_i\ge0. Equation (3.4) is the first estimate with two equal summands.

Let P_i be the real Jacobi Cauchy-data monodromy over one prime period. The periodic kernel is \ker(P_i^m-1). Take D_i to be the least common multiple of the orders of its root-of-unity eigenvalues, with value one if there are none. For s=\gcd(m,D_i), the same eigenvalues satisfy \lambda^m=1 and \lambda^s=1. On a contributing complex Jordan block, z^m-1 has a simple root, so its kernel is the ordinary eigenspace. All other blocks contribute zero. Hence

ODE uniqueness identifies the corresponding fields as repeated root fields. This computation is over \mathbb R or \mathbb C; it makes no division by multiplicity in \kk-cohomology.□

Proof. Choose disjoint neighborhoods W_\alpha of the compact critical sets, and smaller V_\alpha with positive distance from the complement of W_\alpha; the choices are equivariantly permuted. Work first in a fixed isolating band. Uniform drop outside the V_\alpha makes a sufficiently long segment remaining in that band visit one of them. Traversing the annulus between V_\alpha and the complement of W_\alpha takes positive time, since the speed is bounded and its width is positive. The traversal costs a uniformly positive amount of energy. Choose the final width b-a smaller than all these costs. Every segment of duration T with initial energy at most b and terminal energy at least a is then trapped in a single W_\alpha, before and after its visit to V_\alpha.

The first map in (3.6) is a pair homotopy equivalence: its inverse is the map induced by \phi_T. The intermediate flow preserves the relevant subspaces. The finite closed cover X=N\cup Y gives

by closed pasting. This does not require X to be compact. At f_T=a, the derivative of f_T along the flow is strictly negative. Transverse first hitting of this level is continuous, with hitting time zero on the lower side. Flowing for a fraction of that time gives a neighborhood retraction onto Y in X. On the upper side, stop at equality to obtain the corresponding retraction onto Q in N. The condition f\le b is preserved by descent. These are good pairs, so their quotient homeomorphism gives the second relative isomorphism. If a collapse set is empty, adjoin a disjoint basepoint.

The constructions are equivariant. Finite invariant closed covers remain closed covers after Borel construction, and the neighborhood homotopies descend. This proves the equivariant version without asserting compactness of a Borel space. Trapping makes N\cap W_\alpha and Q\cap W_\alpha clopen label pairs. Restriction to a fixed manifold intersects every space in (3.6) with that manifold, using the original flow. Finally, the maps in that diagram are inclusions and hence respect ambient cup products and connecting homomorphisms.□

Proof of the decomposition, naturality, and cup assertion. Choose an even polygon order adequate for the event. Start with one complete K-invariant strict descending gradient flow for the original f_N. Its gradient is tangent to \operatorname{Fix}(\tau). Thus a critical point of the restriction is exactly a full critical point there, and the restricted flow is strict away from them.

By Lemma 3.4, the critical circles come in disjoint reversed pairs. The half-shift preserves each orientation circle and fixes it pointwise exactly when m is even; for odd m, choose its neighborhood disjoint from the fixed locus. Use Lemma 3.6 with disjoint equivariantly permuted neighborhoods of these oriented circles. Reflection induction identifies a block with the ordinary pair of one orientation, while K induction retains its half-shift action. Literal intersection of the original lenses with the fixed manifold gives the full-to-fixed map. In particular, the actual diagrams are

with N^\tau=N\cap\operatorname{Fix}(\tau) and likewise for Q. They commute with ambient relative maps and the associated exact sequences. No alternative local normal form is being used to identify these maps.

The finite union of prime images does not fill S^n, for n\ge2. Choose q outside it, and choose all critical neighborhoods inside the invariant open set

The entire lens therefore evaluates into the punctured sphere. Evaluation at zero is reflection invariant, so it induces ER\times_R N\to S^n\setminus\{q\}. The top sphere class pulls back to zero. Hence its cup product on H_R^*(N,Q) is zero. The inclusion isomorphisms of Lemma 3.6 transfer this vanishing to the actual crossing. This uses only reflection invariance of evaluation; half-translation invariance is not needed. All choices are common to the finitely many labels at a simultaneous level. The uniform rank statements are proved next.□

Proof. Normal coordinates. Take N divisible by k and sufficiently large. The fixed manifold Z=P_{N/k}\subset P_N of the 1/k-shift has restricted energy h=k^2f_{N/k}. Near the selected orbit the bump vanishes. Use a cyclically invariant tubular metric. Symmetry makes the normal first derivative vanish along Z. The Hessian has reducing fixed and normal summands, and Lemma 3.5 places its full kernel inside the fixed tangent space. The normal block is therefore nonsingular, of negative rank

We record the normal coordinate construction to keep the degenerate base unchanged. Taylor integration writes

where A_0(z)=A(z,0) is symmetric and invertible. Normalize by |A_0|^{1/2}, putting

Near the zero section, B is close to J. The near-identity series D=(JB)^{1/2} is J-self-adjoint, and D^*JD=B. Thus the fiber inverse function theorem applied to D|A_0|^{1/2}\nu gives

The construction is uniform near the compact orbit. The positive and negative bundles can be nontrivial; no trivialization of them is required.

The local quotient. Let \psi_t be a complete strict descending flow for the unmodified h. In (3.7), lift it using metric connections, expand u exponentially, and contract y exponentially. The energy derivative is

off the critical set. Convex interpolation with a normalized negative gradient gives a complete strict full field agreeing with this one near the orbit.

Choose a full trap W_f whose base projection has compact closure inside the base chart and avoids all other h-critical points. The exact-field region contains uniform fiber balls over this closure. A smaller selected base trap W_h lies under uniform small fiber balls contained in W_f; all other base traps lie outside its projection. First choose a common sufficiently large time T for full and base trapping, then sufficiently tight regular cuts a<c<b. In particular, \sqrt{b-a} is smaller than the available fiber radius.

Apply Lemma 3.6 to the isolated full crossing. On its quotient N/Q=X/Y, shrink y to zero. The new first T-paths have the same base and negative coordinate, and a scaled positive coordinate, so they stay in the exact region. Both initial and terminal energies decrease. Thus the maps take N into X and Q into Y, and descend continuously to the quotient; some points may collapse. This retracts onto the y=0 subset together with the basepoint.

Apply the full forward flow to this subset. While a point remains uncollapsed, every shifted T-segment is trapped in the exact region, so y=0 persists and the base follows \psi. On the compact base projection where h\ge b>c, its drop has a positive lower bound. A single further time therefore sends all survivors into h\le b. This last subset is preserved up to collapse, so its inclusion is a homotopy equivalence. The homotopies are continuous on the pointed quotient; products of a quotient map with the compact interval remain quotient maps.

Write h_T=h\circ\psi_T. The invariant subset h\le b used in this homotopy equivalence, with its equality boundary before collapse, is described by

For one direction, the projected segment stays in the base band and the projection of the full trap, so it belongs to the selected base label. Conversely, a selected base path is trapped in W_h, and e^t|u|\le\sqrt{b-a} for 0\le t\le T. Its lift therefore stays in the exact region and satisfies the full lens inequalities. Equality in the radius bound is precisely its exit.

Let D_- and S_- be the unit disk and sphere bundles of the negative bundle over P. Scaling by the continuous radius in (3.8) gives a homeomorphism from

onto the final pointed subset. Indeed it is bijective away from the collapsed union; its domain is compact and its target is Hausdorff. The radius vanishes exactly over A_P. This also handles rank zero and the disjoint-basepoint convention.

Relative Thom calculation. For completeness, we prove the relative Thom calculation for this lens, without requiring a manifold structure on its base. On P, let t_p be the first forward time until h_T\le a, truncated at a fixed T_0>0. It is continuous. Finite hits are transverse since a is regular; at a point that has not hit by T_0, either the hit occurs transversely at T_0, or the strict terminal inequality persists in a neighborhood. The base flow stays in its clopen label until equality.

For a unit-disk coordinate put

This is continuous also at u=0, where the radial entry is inactive for |u|<e^{-T_0}. Its zero set is \mathcal A=S_-\cup(D_-)_{A_P}. On the open neighborhood t<T_0, flow the base for time \lambda t, parallel transport u, and multiply its norm by e^{\lambda t}. Neither the base exit nor the sphere is passed before the stopping time. The residual minimum time is (1-\lambda)t, so the neighborhood is preserved, the union is fixed, and at \lambda=1 it is reached. This proves that (D_-,\mathcal A) is a good pair. Rank zero simply removes the radial alternative.

The sphere pairs are good by radial collars. Closed pasting identifies

Under the natural absolute mod-2 Thom isomorphisms, restriction from H^*(D_-,S_-) to this group is the restriction H^{*-b_-}(P)\to H^{*-b_-}(A_P). The exact sequence of the triple S_-\subset\mathcal A\subset D_- gives

The base-pair exact sequence has the same kernel and cokernel. Thus

This argument compares vector spaces, rather than subtracting possibly infinite dimensions.

Absolute Thom applies to these compact metrizable bases as follows. Pull the bundle back along the CW weak equivalence |\operatorname{Sing}P|\to P. The disk and sphere bundle projections are fibrations, so their induced total-space maps are weak equivalences. They give singular cohomology isomorphisms, including the relative disk/sphere groups and their restriction maps. Apply the usual mod-2 Thom isomorphism on the CW model, naturally under the same construction for A_P. The rank-zero case is the base pair itself.

Finally, divide the base cuts by k^2 in (3.1). The selected base label is the root label at c_i^s, so (3.10) proves the asserted normal shift. Iteration is a homeomorphism onto the repeated-loop subspace, including its oriented phase circles; this identification uses no covering transfer or division by multiplicity.□

Completion of Theorem 3.7. There are only finitely many root levels (s\ell_i)^2, with s\mid D_i. Around each distinct level choose a fixed isolated interval with regular endpoints. Choose one coarse polygon order N_0 adequate up to every upper endpoint.

The total crossing pairs in P_{N_0} have finite total cohomological rank: their regular sublevels are compact smooth manifolds with boundary, and the pair exact sequence applies. Their relative cohomology vanishes outside 0,\ldots,nN_0+1. Every selected root label is a direct summand of such a total crossing by Lemma 3.6. This includes coincident root levels.

At an arbitrary event, choose the fine polygon order divisible by kN_0 for every label, where k=m/\gcd(m,D_i). For every even label also impose divisibility by 2k'N_0, where

These are finitely many conditions. After the normal shift, polygon retractions through the Hilbert pair identify each finer root crossing with its coarse pair. Tight cuts within the fixed isolating intervals give the same groups by regular-strip deformations. The finite maximum of their total ranks is therefore a bound B for every iterate label.

If the root group in degree a is nonzero in the shift calculation, then

For example,

is a uniform support constant. Enlarge the constants if necessary for the fixed computations: on \operatorname{Fix}(\tau) start with 4f_{N/2}, apply the same argument to m/2, and divide cuts by 4(k')^2. The support is centered at I_{i,m/2}.

The normal shift changes no total rank. All degeneracy remains in finitely many root crossings, so the proof assumes neither nondegeneracy nor analyticity. It never divides by \Delta_i; zero-mean-index primes satisfy the same conclusions.□

Proof. Write v(t)=|\dot\gamma(t)| and L=\int_0^1v(t)\,\dd t. When L>0, put

For u<1, the absolutely continuous map \phi_u is an increasing homeomorphism with Lipschitz inverse. Set \Psi_u\gamma=\gamma\circ\phi_u^{-1}. At u=1 the inverse need not exist. Instead, define \Psi_1\gamma(s) to be the value of \gamma on the fiber \sigma^{-1}(s). Each fiber is an interval on which v=0 almost everywhere, so absolute continuity makes this definition unambiguous. For L=0 leave the constant path fixed for every u.

For s_1<s_2, choose ordered preimages t_1<t_2 under \sigma. Then

Thus \Psi_1\gamma is L-Lipschitz. Since \gamma=(\Psi_1\gamma)\circ\sigma and \sigma is a nondecreasing surjection, comparison of partitions gives \ell(\gamma)\le\ell(\Psi_1\gamma). The Lipschitz estimate gives the reverse inequality. Consequently \Psi_1\gamma has length L and speed L almost everywhere, and its energy is L^2.

For u<1, change of variables by \phi_u gives

On \{v>0\}, convexity of the reciprocal bounds the integrand by (1-u)v^2+uLv; on \{v=0\} it is zero. This proves (4.1), including u=1 by the preceding argument. The same change of variables preserves length for u<1. Endpoints are unchanged, and constant-speed paths have \sigma(t)=t.

We verify joint strong continuity, including the possibly flat fibers at u=1. Fix a smooth embedding of M in a Euclidean space and extend its metric to a smooth positive definite matrix on a neighborhood of M. Suppose \gamma_i\to\gamma strongly in H^1 and u_i\to u. The one-dimensional Sobolev embedding gives uniform convergence of the paths. Smoothness of the metric gives v_i\to v in L^2, and hence L_i\to L.

If L>0, the maps \sigma_i and \phi_{u_i} converge uniformly to \sigma and \phi_u. Given s_i\to s, choose t_i in the inverse fiber defining \Psi_{u_i}\gamma_i(s_i). Every subsequential limit t satisfies \phi_u(t)=s. The path \gamma has one value on that fiber: for u<1 the fiber is a singleton, and for u=1 this was proved above. Therefore

A compactness argument makes this convergence uniform in s. If L=0, the image of each output is contained in the image of its input, so uniform convergence to the constant path follows directly.

The energies also converge. If L>0 and u<1, the reciprocals in the integral formula are uniformly bounded and converge in measure; combining this with v_i^2\to v^2 in L^1 proves convergence of the integrals. If u=1, Cauchy–Schwarz and (4.1) give

If L=0, the upper bound E(\Psi_{u_i}\gamma_i)\le E(\gamma_i) tends to zero.

For completeness, energy convergence here implies strong, rather than merely weak, convergence of derivatives. Uniform convergence and the energy bound give weak H^1 convergence to the already identified output. In the squared derivative norms, replace the metric matrix at the varying outputs by its value at the limiting output. The resulting error tends to zero, because the matrices converge uniformly and the derivatives are bounded in L^2. The derivative norms thus converge in one fixed positive weighted L^2 Hilbert norm. Weak convergence together with convergence of these norms gives strong convergence. This proves the asserted joint continuity.

Finally, for the reversed path the arclength map is 1-\sigma(1-t). This identity proves commutation with reversal, also at the quotient endpoint u=1.□

Proof. The space of such pairs is compact in C^0 by the Arzelà–Ascoli theorem. Consider one pair (\alpha_0,\beta_0) from this space, with endpoints x_0,y_0. Simple connectivity gives a continuous homotopy between them, fixing the endpoints. Replace its paths by polygons on one sufficiently fine, fixed subdivision of the path parameter. Uniform continuity of the original homotopy ensures that all consecutive vertices are within the injectivity radius. Polygonal interpolation then gives an H^1-continuous homotopy with a finite length bound. Short pointwise geodesic interpolation joins its two endpoint polygons to \alpha_0,\beta_0. These interpolations also have a finite length bound, since the original endpoint paths are Lipschitz.

This construction extends to nearby pairs with varying common endpoints x,y. Attach the short geodesics from x to x_0 and from y_0 to y to each prototype path, using fixed, small positive time intervals at its ends. First choose those intervals small enough that the retimed prototype endpoints are uniformly close to the original endpoints; this uses their Lipschitz bounds. Then restrict x,y and the input paths to a sufficiently small C^0 neighborhood. Short pointwise geodesic interpolation joins the actual first path to the adjusted prototype first path, and similarly at the other end. Both interpolations fix x,y. Their derivatives are bounded by a fixed multiple of the derivatives of the two interpolated paths. Consequently their lengths have a common bound on this neighborhood. Every operation just described is continuous in the strong H^1 topology. A finite subcover of the compact pair space, and the maximum of its finitely many bounds, give K.□

Proof. Set D=\operatorname{diam}(M)+1. For a fixed prototype choose shortest paths, called spokes, from its initial point to all its piece vertices. For nearby inputs extend these choices by short endpoint connectors, using fixed thirds of the unit interval. Shrink the neighborhood so that each spoke has length at most D and Lipschitz constant at most 3D. At the initial vertex use the constant spoke.

Proceed through the pieces in order. Assume the preceding prefix has already been replaced by its spoke. Retime that spoke and the next piece to halves of the enlarged prefix. This retiming preserves length and is continuous in H^1: every nonconstant path has a positive time interval throughout the retiming. In the first step, where the prefix is empty, insert a constant wait at the initial point and retime the first piece. The two paths to be compared are now the spoke followed by the next piece, and the spoke to the next vertex. On a unit interval both have Lipschitz constant at most

Apply Lemma 4.2 to replace the first by the second, and leave the remaining suffix unchanged.

During a replacement the active prefix has length at most K(M,L_0); during a retiming it has length at most D+1. The suffix still consists of original pieces. Thus C=\max\{K(M,L_0),D+1\} bounds the increase over the length of the entire original path at every stage. At the end the path is its final spoke, of length at most D. For a particular input there are finitely many pieces, hence finitely many local choices. Intersecting their neighborhoods makes the entire construction continuous on one neighborhood. This affects the neighborhood, but neither C nor D.□

Proof. Normalize the family by Lemma 4.1. A loop of length L<a now has constant speed L. Let C,D be the constants of Lemma 4.3. Choose B so large that

For sufficiently large a, the integer k=\lfloor a/B\rfloor is at least j+1. Divide the loop into k equal time blocks, and each block into \lceil a/k\rceil equal pieces. Every piece, rescaled to a unit interval, has Lipschitz constant at most one. Every block has length L/k.

For each parameter point, Lemma 4.3 supplies local recipes for the first j+1 blocks. There are only finitely many blocks and pieces in this particular family. Choose a common neighborhood for these recipes, and cover P by finitely many such neighborhoods. Subdivide a triangulation finely enough that the closed stars in a final barycentric subdivision are subordinate to this cover. Color each vertex by the dimension of the simplex of the preceding triangulation whose barycenter it is. The colors lie in \{0,\ldots,j\} and are distinct on each simplex.

At a vertex v of color h, use its local recipe on block h+1. If \lambda_v is its barycentric coordinate, at homotopy time s run that recipe for the fraction

Two positive-coordinate stars of the same color are disjoint; at the boundary of either star the fraction is zero and the recipe is the identity. Recipes of different colors affect different blocks, keeping their endpoints fixed. They therefore paste to one continuous H^1 homotopy. At each parameter at most j+1 blocks are changed, and the length throughout is less than a+(j+1)C.

At the terminal time some barycentric coordinate is at least 1/(j+1), so one block has been completely shortened. The final length is consequently at most

If L\ge a/2, then k\le a/B gives L/k\ge B/2, and (4.2) makes (4.3) less than a-1. If L<a/2, the rough bound L+(j+1)C<a/2+(j+1)C has the same consequence once a>2((j+1)C+1). Choose a_j larger than this threshold and large enough that k\ge j+1. We may take H_j=(j+1)C.□

Proof. Normalize by Lemma 4.1; the resulting cycle has F=\ell<b. We describe a single finite model for all such normalized cycles. Choose an integer N large enough that paths of energy at most (b+1)^2 move less than a fixed small injectivity radius on intervals of length 1/N. Sampling at these times and joining successive vertices by minimizing geodesics gives a continuous polygonal map. For the discrete energy

Cauchy–Schwarz gives E_N\le E on sampled loops. Choose b'\in(b,b+1) such that (b')^2 is a regular value of E_N. The set

is contained strictly inside the open set of vertex tuples whose successive distances are smaller than the injectivity radius. It is a compact smooth manifold with boundary and therefore has finite-dimensional homology in every degree. Polygonal interpolation maps Q to loops of length at most b'.

There is also a homotopy from each normalized loop to its sampled polygon, with one uniform energy bound depending only on b and M. Indeed, on each interval both paths stay in a small normal neighborhood of its initial vertex. Pointwise short-geodesic interpolation is H^1-continuous, and the derivative of this interpolation is bounded by a fixed multiple of the two path derivatives. Its square-root energy is thus at most 2C_0b for a fixed metric constant C_0. This also gives a uniform length bound for the homotopy.

The kernel of H_j(Q;\kk)\to H_j(\Lambda;\kk) is finite-dimensional. Choose finite cycles representing a basis and finite global fillings of their polygonal images. Their supports are compact, so their lengths have a common finite bound. The sampled image of any cycle in the lemma represents a linear combination of this basis. The difference bounds in Q, whose entire image has length at most b'. Adding that comparison chain, the finitely many chosen fillings, and the two normalization and polygonal homotopies proves the claim with one R_{b,j} strictly larger than all these bounds. No bound on the number of simplices in the original cycle is needed.□

Proof. A finite singular cycle over \kk can be regarded as the image of a cycle on a finite polyhedron of dimension at most j. One explicit construction takes the finitely many singular simplices, pairs equal faces that cancel in their boundary, and glues these faces by their ordered affine identifications. The first barycentric subdivision has embedded closed simplices and vertices distinguished by their original face dimensions; a second subdivision removes multiple simplices with the same vertex set. This gives a finite polyhedron with its top-dimensional cycle and the original map to \Lambda. The usual subdivision and prism chains identify its image with the original singular cycle. All these chains have image in the union of the original simplices. Thus the number of parameter simplices plays no role in a length estimate.

Apply Lemma 4.4 repeatedly, normalizing between successive applications. Starting at level a, the successive starting levels are a,a-1,a-2,\ldots, until the family lies below one fixed level b_j>a_j+1. The prism chains at the successive stages all lie in \mathcal A_{a+H_j}: the peak at stage r is bounded by a-r+H_j, so these bounds are not added over the stages. Only finitely many stages are needed for each input cycle.

The resulting cycle is still globally null-homologous. Apply Lemma 4.5 at the fixed level b_j. Taking D_j>\max\{H_j,R_{b_j,j}\} gives the required filling after adjoining the shortening prisms. If the original a is already at most b_j, the bounded-level lemma applies immediately. This proves the assertion for every positive level, without a regular-value assumption.□

Proof. The resonance statement is Theorem 1.1 of [19]. Its cohomological critical levels may be computed in singular cohomology: the open energy sublevels and \Lambda are paracompact, locally contractible Hilbert manifolds, where singular and Čech cohomology agree. Their Lemma 5.3 gives the rank bound; Lemma 5.4 and Equation (12) give the powers and the pairing, with the index in that equation replaced by m+1. These statements apply for every n>2 and every field. The indicated powers and pairing may also be taken integrally and reduced modulo two: the integral pairing is one, so its reduction is one in both parities of n. The absolute class of degree 3n-2 has a unique relative lift because 3n-2>n+1. Finally evaluation retracts the inclusion of constant loops, so [M]\ne0; the rank bound gives the last assertion.□

Proof. Extension by zero defines a cochain, although generally not a cocycle. The ordinary cochain identity nevertheless gives

If its front U-face is zero, the evaluation is zero. Otherwise the entire parent lies in O, and every face entering the remaining coboundary is an actual pullback of T. They cancel because \delta T=0. This proves the assertion without discarding any extension-by-zero terms separately. Each block or skip face is contained in its parent image and has the same property.

In a contributing term the U-face first puts the whole rectangle in the closed intermediate tube, where h is defined and uniformly continuous on the compact parameter set. Its nonzero T-face cannot have its entire image in \{\ell<L\}, and hence contains a point with \ell(h)\ge L. Refining the factors so that \ell\circ h oscillates by less than \varepsilon on each such rectangle proves the estimate. The tube conditions follow in the same way from the gap between support and intermediate radii. Include all homotopy intervals among the factors before refining. The finitely many conditions then hold simultaneously. All later subdivision prisms and block or skip faces stay in the original rectangles, so both the domain and length conditions persist.□

Proof. The cut degree is 1-n-p=-q. The cap boundary formula applies by Lemma 5.2; its time-end terms vanish because (x(0),x(t)) is exactly diagonal at t=0,1. Applying the same construction to a finite filling proves null-homology preservation. On a surviving parent the tested first loop has length at least L-\varepsilon, while

This gives (5.10). If its upper level is nonpositive, no term survives. The join boundary formula uses the closed cocycle U_j. Its output length is at most (1+\kappa)\ell(x)+\ell(y), and \eta_j may be chosen after all finite length bounds have been specified.

For the unit, consider a shuffle of c\times x. A constant-factor step remaining after the n-cap makes the output degenerate: the rotated constant is exactly x(0), so the output is independent of that factor. The remaining terms have all n constant-factor steps first. Their coefficient is the Thom evaluation on [M]\times\{x(0)\}, at the initial parameter vertex of the x-simplex, and equals one. Thus the output is a constant half followed by x. Removing the waiting half by reparametrization gives (5.11) without changing length.□

Proof. Write J=J_{x(0),y(0)} and Z=(Jx)\#y. On x\times I_s\times y, first cap with \sigma=U_j(x(0),y(0)), then with U_c(Z(0),Z(s)), then with T(C_1(Z,s)), and output C_2(Z,s). Call this cycle P(x,y). Its time endpoints vanish by (5.6). We prove

All homologies will have the bound in (5.12).

The second half vanishes. Split I_s at 1/2. At the seam Z(1/2)=Z(0), so each piece is separately a cycle after capping. For s\ge1/2, both the cut evaluation and the second-cut output depend only on (y,s). A shuffle contains j steps from x. Its join face uses at most n of them, and its T-face at most p. Since j>n+p, another such step belongs either to the cut Thom face or to the retained output face. In the first case the normalized Thom evaluation is zero; in the second the output simplex is degenerate. In fact projection drops the unused x-step, and the output factors through the corresponding standard degeneracy map. Deleting a repeated projected vertex uses a face of the original simplex, so this factorization stays within the cut domain. The second-half cycle therefore bounds in its image sublevel.

The first half gives the native cut. Put t=2s on the first half. The first cut is exactly JC_1(x,t). The second cut consists of an arc, the rotated x-tail and y, with respective durations

Interpolate them linearly to 1/4,1/4,1/2. The result is (JC_2(x,t))\#y. Length is unchanged, and continuity at t=1 follows because the shrinking tail contributes at most a fixed multiple of \int_t^1|x'|^2 to the energy.

Keep this output fixed and move the cut evaluation and tested first loop from those of Jx to those of J^u x, with u decreasing from 1 to 0. Cross with this parameter before capping. Round rotations preserve the cut distance, so all surviving parents support the fixed output throughout. The cap boundary formula gives a homology: it requires continuous supported maps, not an identity between the fixed output and an intermediate tested cut. Its time endpoints remain diagonal. Small rotation distortion keeps the first-cut plus output length within a fixed small error of \ell(x)+\ell(y).

At u=0 put

using extension by zero as in Lemma 5.2. This normalized cochain has degree n+p and is independent of y. Impose the native cut smallness even before the join cap. We may then commute \sigma and \gamma using

For completeness, if the degrees of cochains u,v are r,s, their mod-two cup-one on a simplex of dimension r+s-1 is

The coboundary formula follows by pairing deletions outside and inside the block; the shared-end deletions cancel in adjacent summands except for the two outer cups. This gives (5.14) since \delta\sigma=0.

Lemma 5.2 makes \delta\gamma zero on every block involved, including all extension-by-zero terms. Each nonzero summand of the primitive has a join face and a cut/test block. Its parent is therefore in both tubes and has high first-cut length. On either time endpoint the cut part of every \gamma-face vanishes. Capping with the primitive and pushing to the fixed output supplies the required homology between the two cap orders.

With \gamma first, a y-step in its front face gives a degenerate evaluation. The surviving shuffles start with n+p steps from x\times I. Their suffixes are exactly the shuffles of the retained cut face with y. The native second cut starts at x(0), so the remaining join cap and output are precisely those of \mu(D_Tx,y). Common factor refinements compare any independently chosen join subdivision. This proves the first comparison in (5.13).

Symmetry of the joined cut. Keep the cut interval whole during the following homotopies. Rotate the entire loop by J^{-u}, obtaining

Then translate the circle parameter of x\#(J^{-1}y) through half a period, ending at (J^{-1}y)\#x. Translation is jointly continuous in H^1 and preserves length. At every stage the evaluations at cut times zero and one are diagonal. An intermediate half-time seam need not be diagonal; any seam terms cancel because we use the full interval.

The swapped join cochain represents the same Thom class relative to F_{\eta_j}, and hence differs from \sigma by the coboundary of a cochain relative to that far set. A nonzero face of this primitive forces its whole adequate parent into the join tube. Cap with the primitive followed by the cut and test cochains and use the boundary formula; the cut kills the time endpoints. Shuffle symmetry now identifies the result with P(y,x). Repeating the first-half argument in the opposite order proves the last comparison in (5.13); the second half vanishes because N>n+p.

One length bound for all these homologies. Choose the join radius so that each partial rotation changes the length of every path with the prescribed finite bounds by less than \varepsilon. This is possible after including the bounds for any additional finite fillings. In the whole-loop homotopies a surviving test saves at least L-\varepsilon from the common joined length, at a cost of two short arcs and small rotations. In the first-half homotopy with fixed output, compare the tested J^u C_1 with C_1; each differs in length by less than \varepsilon. Together with the two cut arcs and the output rotation this gives a cost less than 8\varepsilon. Timing changes preserve length. The cup-one and primitive comparisons satisfy the same bound by whole-parent support and oscillation. Thus each individual homology lies already below a+b-L+1 after sufficiently small choices. Their sum, together with the image-preserving refinement and normalization homologies, satisfies the generous common bound a+b-L+4. The number of chain terms does not add lengths: the support of a sum is contained in the union of their supports. Neither a larger degree nor a finer subdivision traverses an additional long loop. This proves the claimed uniformity.□

Proof. Hingston–Wahl give singular chain models on H^1 loops by capping with the evaluation pullback of a diagonal Thom cocycle, then adding short arcs and concatenating or cutting [20, Theorem A, Definitions 1.2 and 1.4]. The dual relative operation is the Goresky–Hingston product [20, Theorem 2.13]; all sign differences disappear over \kk. We use these models with the auxiliary round metric and compare the actual maps and cochains. Choose their support radius to equal ours and their map domain to contain our larger tube; both choices can be made within their injectivity-radius requirement.

For joining, partially rotate the first input by J^u and apply their short-arc retraction, whose values have equal starts throughout. Writing z=x(0) and z'=y(0), the residual mismatch satisfies d_{\rm rd}(J^u z,z')=(1-u)d_{\rm rd}(z,z'), so all inputs remain in its domain. At u=0 this is their retraction; at u=1 the inputs already have equal starts, so their retraction is the identity and gives our rotation retraction. This is a homotopy inside equal-start pairs. Interpolate their energy-optimal concatenation times to halves. For path pieces of square-root energies P,Q, the optimal time for the first piece is P/(P+Q); during this interpolation its energy contribution is bounded by

This tends to zero if that piece collapses. The other piece is handled in the same way, and if both collapse both contributions vanish. Thus the timing homotopy is continuous also at constant pieces.

For cutting, their short-arc retraction followed by cutting gives exactly the two paths in (5.7), with optimal instead of half timings. The same interpolation compares them. At either time endpoint one of the two paths remains constant. Pairing with T and b evaluates on z\times I the cochain

Both tests are relative to constants. Hence \chi vanishes at t=0 through its first factor and at t=1 through its second factor.

Replace U_c by the reduced ordinary Thom representative used in that model. They differ by a coboundary relative to the far set. Although its primitive need not vanish on the diagonal, all of its time-end contributions are multiplied by \chi and vanish. If a cap convention reverses the cup order, the cup-one formula supplies the comparison. On a small simplex in the tube, \chi is a closed pullback; on a simplex in the far set, every face of the accompanying Thom cochain or relative primitive vanishes. Thus extension-by-zero errors contribute nothing, exactly as in Lemma 5.2. The same argument permits replacement of either relative test by a cohomologous representative. Join Thom representatives are compared in the same way, without interval-end terms.

Finally the Alexander–Whitney map in the coproduct evaluates T\otimes b as the displayed cup of the two pullbacks. Their small-chain replacement and our refined products represent the same relative chain class: both are homologous to the input, and small-chain inclusion is a quasi-isomorphism, also relative to the time ends, by subdivision. This proves (5.15) and the asserted join identification. The argument compares two relative tests; it makes no assertion about a canonical absolute extension of the coproduct.□

Proof. Represent \Theta^{\bullet(m+1)} by a finite cycle z. Its cut has degree n+2(m+1)d-q=3n-2. Lemma 5.5, commutativity of the relative product over \kk, and (5.3) give \langle\theta,D_Tz\rangle=1. The rank-one calculation therefore yields [D_Tz]=\Theta.

Both |z| and |S|=n+q exceed n+p=q+1. Apply Theorem 5.4 to these cycles, now forgetting the finite output length bound. Using the join identification,

If D_TS were null-homologous, its join with z would bound, by Lemma 5.3. Thus D_TS\ne0 in H_n(\Lambda)=\kk[M], proving (5.16). This determines the absolute constant component by interchange and nonvanishing, rather than by the relative scalar test alone.

The cut bound gives support in \mathcal A_{l-L+1}. Its difference from a constant fundamental cycle is globally null in degree n; Proposition 4.6 fills that difference in \mathcal A_{l-L+1+D_n}. By (5.5), this is one fixed level B.□

Proof. We first prove the absolute assertion for the length filtration \mathcal A_a, then pass to square-root energy and to the relative pair.

Choosing the degree reduction. Let d=n-1 and let \alpha,C_H be the constants of Proposition 5.1. For sufficiently large j put

Then

Choose an integer J>2n so large that, whenever j\ge J, these choices have m\ge1, p>n+1, and L>0 in (5.4). Take S,T,l,L as in that equation. Write

where D_s is furnished by Proposition 4.6. All the remainder degrees in (6.4), and the degree n, occur in this finite maximum.

Filling a high-degree null cycle. Let x be a finite globally null j-cycle in \mathcal A_a, with j\ge J. Its cut D_Tx is a globally null r-cycle in \mathcal A_{a-L+1} by Lemma 5.3. If a-L+1>0, Proposition 4.6 gives a chain W such that

If a-L+1\le0, the cut has no surviving term and we take W=0.

Fix a fundamental constant-loop cycle c. By Proposition 5.6, the cycle D_TS-c is globally null in degree n and is supported in \mathcal A_{C_{\mathrm{gap}}+1}. Fixed-degree filling gives a chain V with

This length bound is independent of j, although the chain V itself may depend on the chosen power.

Choose the join radius after these finite length bounds have been specified. Lemma 5.3 and Theorem 5.4 permit one choice for the interchange homology and both joined fillings. Refinement comparisons are included in the same bounds. The relevant chains have the following supports:

The first assertion is omitted when W=0; otherwise it follows by adding the input bounds a-L+1+D_r and l and the join allowance 1. The second follows from (6.6) in the same way. Both chains have degree j+1.

The degree inequalities (6.4) are precisely the two hypotheses needed for the interchange homology. It gives

The chain \mu(W,S) has boundary \mu(D_Tx,S), while \mu(V,x) has boundary \mu(D_TS,x)-\mu(c,x). Finally (5.11) joins \mu(c,x) to x in \mathcal A_a. Adding these chains and the interchange homology therefore fills x.

Thus one permissible length increment is

It exceeds every bound just used. For j<J, the conclusion follows directly from Proposition 4.6, since D_j\le D_*<D. We have proved that every globally null finite cycle in \mathcal A_a, in any degree, fills in \mathcal A_{a+D}. The constants in (6.9) depend only on the metric and n, and every construction used above is independent of a finite-prime hypothesis.

Returning to square-root energy. A cycle x in X_a is in \mathcal A_a because \ell\le F. Fill it by the length assertion and apply \Psi_1 to that filling. Lemma 4.1 gives F(\Psi_1\gamma)=\ell(\gamma), so the new filling lies in X_{a+D} and has boundary \Psi_1x. The normalization prism from x to \Psi_1x lies entirely in X_a by (4.1). Adding this prism proves (6.1) with the same D. Strict sublevel inequalities are preserved throughout; no approximation by regular levels is involved.

Passing to constants-relative homology. Let i:M\to\Lambda include constants and let \ev_0:\Lambda\to M be evaluation at zero. Since \ev_0\circ i=\operatorname{id}_M, the chain map

annihilates C_*(M) and preserves C_*(X_a) for every a>0. If z represents a relative cycle in (X_a,M), then \partial z\in C_*(M), so Pz is an absolute cycle representing the same relative class. If z is globally null relative to M, there are finite chains U in \Lambda and c' in M with \partial U=z+c'. Applying P gives \partial(PU)=Pz. Hence Pz is globally null absolutely. By (6.1) it fills in X_{a+D}, and passage to the relative chain quotient proves (6.2). This argument includes degree zero and uses exactly the same increment.□

Proof. For a real rank-r representation of H, its associated bundle over Y_H restricts to a bundle over A_H. Suppose that its unit sphere is H/H'. The sphere-bundle pair is then the Borel pair for H'. The relative Thom sequence gives

Here is a cochain construction that also justifies (7.6) when the inclusion A_H\subset Y_H is not a cofibration. For a vector bundle over Z\supset A_0, let D,S denote its disk and sphere bundles, and let D_{A_0},S_{A_0} be their restrictions. In the singular chain complex of D, put

The two summands intersect in C_*(S_{A_0}), since singular chains have the singular simplices as a basis. Restriction of cochains gives degreewise exact sequences

Surjectivity follows by extending linear functionals on chain subspaces. Cup with a Thom cocycle sends a relative base cochain into \operatorname{Ann}\mathcal L: on S the Thom factor vanishes, and on D_{A_0} the base factor vanishes. Comparison of the first sequence with the base-pair sequence, using the absolute Thom isomorphisms, proves

The second sequence and the disk contraction give (7.6). The map back to the base is cup with the zero-section pullback of the Thom class, namely w_r. Pullback of Thom classes gives naturality and the projection formula. The absolute Thom isomorphism used here is the ordinary mod-two one [17, Section 3.2, pp. 88–89]; for a general base it follows by pulling back to its singular CW replacement and using the disk and sphere fibrations. This argument uses the sum of two chain subspaces and does not require an excision assertion for their union.

The standard G-plane has sphere stabilizer R and Euler class U, proving (7.2). Let \rho_2 be the representation that doubles rotation angles and sends the chosen reflection to \operatorname{diag}(1,-1). Its sphere stabilizer is K. On K it has a fixed line, so its Euler class restricts to zero. The injectivity in (7.1) makes that Euler class zero already on BG, giving (7.3). Finally, the sign representations with kernels 1\subset R and R\subset K have Euler classes w and v, respectively.□

Proof. Along (7.7), put

A tangent field is a(t)e_t+b(t), where b(t)\perp\operatorname{span}\{x,u\}. Half the Hessian is

In each of the d normal directions, the negative Fourier frequencies are |k|<m. Their real dimension is 2m-1. The null modes are the two normal modes at frequency m, together with the constant tangential mode. They have dimension 2d+1 and are precisely the variations of the frame (x,u). The nonzero Fourier eigenvalues, divided by the corresponding squared H^1-norms, are bounded away from zero. Thus the Hessian is invertible on the orthogonal complement of its kernel. At constants, the Hessian is 2\int|V'|^2, positive and invertible off its constant kernel.

We describe why the resulting attachment respects the full group. Use the strong metric induced by the Euclidean H^1-norm. Each parameter symmetry is a smooth isometry, and the action is continuous. Pointwise exponential gives a chart on the Hilbert normal bundle of the compact manifold C_m, at a uniform small radius. In these coordinates Taylor’s formula writes the energy difference as \langle A(z,y)y,y\rangle, with A(z,0) an invertible symmetric operator on the normal fiber. Normalize first by |A(z,0)|^{1/2}, and let J=\operatorname{sign}A(z,0). For the resulting operator B near J, the norm-convergent power series C=(JB)^{1/2} satisfies C^*JC=B. The change y\mapsto C(z,y)y is a diffeomorphism after shrinking the tube. All these constructions commute with the parameter isometries. They give the exact normal form

on the negative and positive bundles.

For completeness, the sublevel attachment uses the original Hilbert space, rather than a polygon model with only finite rotational symmetry. Patch a complete normalized descending gradient field to the split field

inside the normal tube, using an invariant squared-radius cutoff. Both fields strictly decrease E_0 away from C_m within the isolating energy band. The coordinate derivatives are uniformly bounded in a small tube over the compact zero section, so the patched field remains bounded. The ambient strong metric is complete; hence this field is complete. Palais–Smale and regular-band deformation (Lemmas 3.1 and 3.2) give a uniform decrease outside any smaller tube in an isolating energy band. The flow-lens construction of Lemma 3.6 therefore computes the crossing in that smaller tube. Use its notation X,Y,N,Q, with lower energy cut t_-<c_m. In split coordinates, contraction of u_+ decreases both endpoint energies of a trapped flow segment. It defines maps N\to X taking Q into Y, even when a contracted point leaves N through the lower endpoint. Via N/Q\cong X/Y, these maps induce a contraction to the negative disk with its boundary collapsed. For flow time T, the surviving disk before applying the flow has radius e^{-T}\sqrt{c_m-t_-}; its boundary flows to the lower cut. The flow multiplies u_- by e^T, identifying this disk and sphere with the usual negative-bundle disk and sphere. The disk is closed in the lens and meets Q exactly in its sphere. These statements remain true after Borel construction. The disk/sphere quotient consequently embeds in the lens quotient: the saturation of a closed subset of the disk is that subset, possibly together with the closed collapse set. The contraction fixes its image. The disk/sphere inclusion is a good pair. This proves (7.8) by the mod-two Thom isomorphism.

All pair maps just used are G-equivariant. The negative bundle carries its actual derivative action; no triviality of that action is assumed. Restriction to a subgroup and cup by an absolute class therefore commute with the attachment isomorphism. Wider regular cuts give the same assertion by regular-band deformation. Finally, sufficiently small energy places a loop in the positive normal chart of the constants, where contraction of the normal coordinate proves the last assertion.□

Proof. The G-action on frames is the right action through \rho_m, up to the convention for inverse precomposition. For odd m, the map from the critical Borel space to \mathrm{Gr}_2(\mathbb R^{n+1}) has fiber the classifying space of the cyclic group of order m. Local choices of frames give local trivializations. That fiber has no positive-degree mod-two cohomology, by transfer, so the map induces a cohomology isomorphism. The tautological plane pulls back to the plane of \rho_m; its classes are w,U by (7.13).

These two classes generate the Grassmann cohomology. Indeed, projectivize the tautological plane. On the resulting ordered-line flag space, let s,t be the first Stiefel–Whitney classes of its two lines. Over the Grassmannian, the projective-bundle formula gives a free module with basis 1,s. Over \mathbb{RP}^n, the same flag space is the projectivization of the first line’s perpendicular bundle, so its ring is generated by s,t. Since s+t=w, st=U, and s^2=ws+U, every polynomial in s,t reduces to P(w,U)+sQ(w,U). Uniqueness of the free-module expansion implies that every class pulled back from the Grassmannian is a polynomial in w,U. The complementary bundle has rank d, with total Stiefel–Whitney class (1+w+U)^{-1}. Its components are the polynomials A_i, proving (7.15) and the odd-stage assertion.

If m is even, \tau acts trivially on the critical manifold. Consequently

The same statement holds for the crossing modules after the Thom shift, with the actual equivariant negative bundle retained. The polynomial U=v(v+w) is monic in v, hence acts injectively even though the coefficient ring in (7.11) has zero divisors. The injection in (7.3) makes U injective on the G-crossing as well. Sequence (7.2) therefore identifies the quotient of the G-crossing by U with the R-crossing. Choose a lift y_m of ex_m; the bottom x_m already lifts. Their w-multiples generate modulo U, by (7.12). Subtracting such a combination and dividing the remainder by U reduces degree by two, so induction from the bottom degree proves generation.□

Proof. First use H=R, inducting on p. By (7.12), the earlier interval module is generated over the ambient classes w,e by chosen extensions of its bottoms x_j, m_0\le j<p. The connecting homomorphism sends such a bottom into the new crossing in degree a_j+1<a_p, where the crossing is zero. It is linear over w,e, hence vanishes on the whole earlier module. This proves (7.16) for R, and also justifies the inductive generation assertion.

For G, Lemma 7.4 gives the same induction with the bottom generators and, at even stages j, the chosen top generators of degree a_j+n. Bottoms again have zero connecting image. For a top, the only potentially relevant degree is a_j+n+1. If d>2, then n+1=d+2<2d, so this degree is less than a_p. If d=2, equality can occur only when j=p-1 is even. In this case p is odd, and the target is its one-dimensional bottom G-crossing. Restriction to the corresponding reflection bottom is injective: both bottom Thom classes restrict to the nonzero bottom class. The reflection connecting map has already been proved zero. Naturality therefore kills the G-connecting map in this last case as well. This proves (7.16) in every dimension n\ge3.

We pass to the global pair degree by degree. Every singular chain in the Borel construction is finite and has bounded round energy, since energy descends to a continuous invariant function. Thus

Universal coefficients over the field \kk identify its dual with the inverse limit of the degree-j cohomology groups. The local crossing groups vanish below a_p, which tends to infinity. Sequence (7.16) consequently stabilizes in every degree. For j<a_{m+1}, no subsequent crossing contributes, proving (7.17). All limits in this argument are taken in a fixed degree; the total cohomology module is the graded direct sum.□

Proof. The recurrence (7.14) implies

Indeed, substitution of the recurrence makes the left side for i\ge2 equal to U times the left side for i-1; the initial value is w^2+A_2=U.

At an odd crossing, (7.15) gives

The class A_dx_m is nonzero: on restriction to R, it becomes w^dx_m, which is nonzero in (7.12). The relevant part of the standard-plane Gysin sequence is

Its source is one-dimensional, spanned by ex_m. Since its image contains the nonzero class A_dx_m, exactness over \kk forces

Apply the same integration map to e\,\res_R\widetilde\zeta_m in the pair (\Lambda,Z_{m-1}). Naturality and (7.22) identify its restriction to the crossing with A_dx_m. Both sides have offset d<2d, so (7.17) gives

The next map in Gysin is multiplication by U, hence

Also A_{d+1}\widetilde\zeta_m=0, because its local value is zero and d+1<2d. Multiplying (7.21) at i=d by U\widetilde\zeta_m therefore gives

Its image in H_G^*(\Lambda,M) is U^n\zeta_m=0.

For n=3, one has d=2, A_2=w^2+U, and A_3=w^3. The two intermediate global relations are exactly

Their proof used uniqueness only at offsets two and three, both strictly less than the next-bottom gap four. It did not extend an offset-four relation across that gap.□

Proof. A critical root doubles to a critical loop, so regularity of 2a implies regularity of a. Doubling identifies X_a with X_{2a}^{\tau} as an R-space. Since \tau acts trivially there, a product universal model and the field Künneth formula give

The same identification holds for fixed crossing pairs.

Choose a common even polygon model through the cut, and double its compact regular sublevel along an invariant collar. Denote the resulting closed smooth K-manifold by \mathcal D. Inclusion of the original sublevel and the fold map make its restriction arrow to the fixed locus a retract of the corresponding arrow for \mathcal D. The fixed submanifold \mathcal D^\tau has an invariant open tubular neighborhood O_0 which retracts equivariantly onto it. The compact complement of O_0 is covered by finitely many invariant open sets O_1,\ldots,O_s, each admitting an equivariant map to K/H_i with \tau\notin H_i. Indeed, use a small stabilizer-invariant neighborhood disjoint from its translates by other cosets, and then saturate it.

The possible groups H_i are 1, R=\langle r\rangle, and \langle\tau r\rangle. On each, U=v(v+w) restricts to zero. Thus U lifts to H_K^2(\mathcal D,O_i). Cup these lifts with a class relative to O_0. The relative cup product for the open cover has target relative to \bigcup_{i=0}^sO_i=\mathcal D, so U^s annihilates H_K^*(\mathcal D,O_0). The exact sequence of this pair proves localization at U. Passing to the retract proves it on the original sublevel. The exact sequences for a pair of regular sublevels, and for the pair relative to M, give the corresponding assertions for pairs.

The class w is zero on each ambient local block, because that block is induced from the \langle\tau\rangle-pair of one orientation. It is therefore nilpotent on each finite-cut relative group by induction over the crossing exact sequences. The same argument applies to the fixed reflection groups and to crossing pairs. On these modules localization at U is equivalent to localization at v: after inverting v, the factor 1+w/v in U=v^2(1+w/v) is invertible; after inverting U, the element (v+w)/U is an inverse for v.

Apply the sign-line Gysin sequence of Proposition 7.1 to one orientation of a local block. The total ordinary rank bound and index support in Theorem 3.7 imply the claimed per-degree bound, vanishing below the index window, and that multiplication by v is an isomorphism above it. The fixed local group has the product description in (8.1). Its root support is within a fixed distance of I_{m/2}, and

Consequently both source and target have already stabilized above I_m+A, with a uniform enlarged A. The localized restriction isomorphism is then an actual isomorphism there: clear a power of v and invert the intervening, already invertible multiplication maps. For an odd iterate the fixed block is empty, so its stabilized source is zero. Labelwise naturality follows from the common full/fixed lens in Theorem 3.7.

Finally, the sign-line Gysin sequence from K to R and the finite total dimension of B_{2a} make multiplication by v an isomorphism on the ambient relative group in all sufficiently high degrees. The fixed group has the same property by (8.1). The same clearing argument upgrades localization to actual restriction in high degrees at this cut.□

Proof. Grow the ambient cut through its critical events, using primes for the groups just after an event. Let C^q be its actual local K-group and \mathcal W its total fixed root reflection group. The comparison maps fit into the following diagram, whose rows are exact:

The bottom row is the fixed exact sequence after inverting v and normalizing powers consistently in neighboring degrees. Its maps f,i preserve root labels, and \delta raises a root label by one. In particular, the repeated terms denote the total graded groups in an exact sequence, not an assertion of ungraded periodicity.

Fix j\ge1 and put J=2j. Call this cutoff far from the event if every participating iterate satisfies |I_m-2j|>C_0, for a sufficiently large fixed C_0. Lemma 8.1 and the index comparison then give

Indeed, the blocks below the window are already in their actual stable range and have all root labels below j; those above it have zero actual source in these three degrees and no low root labels.

Write P=P_j, L=P\mathcal V, L'=P\mathcal V', and

There is no connecting term across the projection edge, so low-label exactness gives

We next justify the projected-image calculation, including mixed low and high components. For z\in E_q, commutativity and (8.5) place \delta z in P\mathcal W. Also \delta(Pz) is low, since \mathcal W^{j+1}=0, while \delta((1-P)z) has labels greater than j. These disjoint label ranges imply

Given x\in PE_q\cap\ker\delta, choose z\in E_q projecting to x and an actual representative in A_q. Its actual connecting class restricts to zero. The target \lambda_{q+1} is injective, so actual exactness lifts the representative to A'_q. Projecting its fixed restriction proves

The kernel of this map is exactly A_0: stable exactness gives the inclusion in A_0, and every element of P\mathcal W has an actual local lift. Therefore

The actual connecting ranks are precisely r_q, again by target injectivity. Hence

Using H_{j,J}+t_J=\dim A_J-\dim PE_J and (8.6) gives

Together with (8.7), this proves the exact change

The preceding-degree deficit cancels the possible increase in the high-image and kernel terms. This is why all four terms were retained.

For a cutoff which is not far, local restriction need not be injective. Nevertheless the change is uniformly bounded. Choose M_0 bounding \dim\mathcal W and all the relevant actual local dimensions. Since i preserves labels, i(PE'_q)\subset PE_q. Its kernel has dimension at most M_0 by stable exactness. Its cokernel is bounded using the surjections

The first source has dimension at most \dim C^{q+1}\le M_0. Thus |\dim PE'_q-\dim PE_q|\le M_0. The same kernel and cokernel argument bounds |\dim L'-\dim L| by M_0, and actual exactness bounds |\dim A'_J-\dim A_J| by M_0. Since

we obtain |\Phi'_j-\Phi_j|\le6M_0.

Each prime contributes at most one iterate to a given positive event. The finite prime list therefore bounds the number of participating blocks, and only a bounded number of integers j lie in their fixed index windows. Thus every event increases the sum of the potentials by at most a fixed constant. The number of events below 2a is at most \sum_i\lfloor2a/\ell_i\rfloor. The potentials start at zero below the first positive event and are unchanged on no-critical strips. Their finite support at each cut permits summation of (8.8) and the exceptional bounds, proving (8.3).□

Proof. We first prove the ordinary comparison away from fixed-width bands. Choose a regular endpoint a_+\in(a+D,a+D+1). A local group at an event m\ell_i\in[a,a_+] can contribute in degree j or its neighboring degrees only if

By (9.1), this implies

The right-hand side is bounded uniformly over the finite prime list. Thus, outside fixed-width bands about the \beta_i a, the ordinary map

is injective, by the crossing exact sequences. A class in its source that is zero globally is already zero in X_{a+D} by Theorem 6.1. Injectivity therefore makes the map H_j(X_a,M)\to H_j(\Lambda,M) injective. Field duality gives surjectivity of ordinary cohomology restriction.

In degrees above a fixed low threshold, the ordinary groups relative to M identify with the absolute groups, by the evaluation retraction and the cohomology of the constant sphere. In these degrees Proposition 5.1 gives global rank at most one. Its resonance estimate implies that every nonzero global class restricts nontrivially to X_a when j<\alpha a-c_0, and restricts to zero when j>\alpha a+c_0, for a fixed c_0. Together with the preceding surjectivity, this proves

provided j also lies outside the fixed low-degree interval and the bands in (9.3). When the global group is zero, surjectivity gives the first assertion as an isomorphism between zero groups. Increase c_0 to absorb all these restrictions.

Consider a present gap and put P_a^j=H_R^j(\Lambda,X_a) for this proof. Ordinary triple exactness and (9.4) give H^j(\Lambda,X_a)=0 whenever the needed two neighboring ordinary restrictions are isomorphisms. The sign-line Gysin sequence of Proposition 7.1 consequently makes w:P_a^j\to P_a^{j+1} an isomorphism in the corresponding interior. After allowing n further degrees below, every element of P_a^j is divisible by w^n there. Its image in B_\infty^j is zero, since w^nB_\infty=0. Applying this at j and j+1 to the triple (\Lambda,X_a,M) gives a natural short exact sequence

The quotient identification commutes with w. It gives both the claimed injection and the isomorphisms on Q_a in a present gap.

In an absent gap, (9.4) and sign Gysin make w:B_a^j\to B_a^{j+1} an isomorphism in the interior. Allowing n further degrees above makes w^n injective there. The image of B_\infty^j is killed by that power, so it is zero. Hence Q_a^j=B_a^j and the claimed isomorphisms also hold in an absent gap.

Choose the margin c larger than c_0+2n+6 and the fixed low-degree threshold. Then all neighboring degrees and all powers just used lie in the required untrimmed ordinary ranges whenever the statements of the lemma apply. This is one fixed choice for all cuts and gaps. Finally, any reflection local contribution below cut a has

The relative groups start at zero below the first event. The local crossing sequences therefore give B_a^j=0 above s_Na+c, after one final increase of c.□

Proof. Fix consecutive slopes s<s' and write

There is a fixed \delta>0 such that r\ge\delta a for every such gap, since the list of distinct slopes is finite. Suppose that some Q_a^j in this gap is nonzero. The isomorphisms in Lemma 9.2 make it nonzero throughout the integer gap. For large a, choose

and a class y_{l_0}\in B_a^{l_0} with nonzero image in Q_a^{l_0}. Set y_l=w^{l-l_0}y_{l_0} through l\le\lfloor A+0.90r\rfloor. Every resulting quotient class is nonzero. In particular, with

the classes y_l, l\in\mathcal L, are independent in \bigoplus_j B_a^j, as are their powers w^q y_l. Indeed they occupy distinct root degrees, and their quotient classes remain nonzero between degrees A+0.65r-O(1) and A+0.70r. Figure 2 displays the intervals used below.

At the ambient cut 2a, let E_J and P_k be the actual normalized restriction image and root-degree projection from Theorem 8.2. Recall its nonnegative terms

In particular that theorem gives

Average D_{k,2k} over the integers k\in[A+0.30r,A+0.35r]. There are a fixed positive multiple of r such integers for large r, so some k satisfies D_{k,2k}\le C_1, independently of a, because r\ge\delta a. The space

lies in the root degrees at most k. Thus

for a fixed c_1>0 and sufficiently large r.

Choose a basis z_1,\ldots,z_\nu of this intersection and actual ambient lifts

The normalized fixed restriction of u_\mu has the form z_\mu+c_\mu, where every root label in c_\mu is greater than k. Multiply the actual classes by w^q. The selected terms w^q z_\mu remain independent, since w^q is injective on W_0. Their labels are at most A+0.70r, whereas the other terms have labels greater than

or vanish. The latter terms cannot cancel any of the selected ones. Moreover all labels of these normalized images exceed A+0.64r for sufficiently large r.

For any integer k'\in[A+0.61r,A+0.64r], there is enough actual degree to multiply further by a nonnegative power of v, since

The classes

have unchanged normalized fixed images: multiplication by v changes the actual degree and leaves the root coefficients unchanged. Multiplication by w raises each root label by one, as used above. Thus the images are independent and lie entirely above root cutoff k', proving

There are at least c_2r such cutoffs for large r. The left side of (9.6) is therefore at least c_1c_2r^2\ge c_1c_2\delta^2a^2, contradicting its linear upper bound. This proves the assertion for the chosen gap. The finite number of gaps allows one common lower threshold for a.□

Proof of Theorem 9.1. First let a be large with 2a regular. By Lemma 9.3, transient classes can occur only within distance c of one of the finitely many degrees s_i a. Lemma 9.2 excludes degrees above the last such band. Thus only a bounded number of integer degrees can contribute. We now bound the dimension in each of them uniformly.

For high degrees, the reflection local ranks of Theorem 3.7 give

This follows by summing the crossing exact-sequence bounds, starting with the zero relative group below the first event. If \Delta_i>0, the number in braces is bounded by 1+2(C_{\rm loc}+n)/\Delta_i, up to an integer rounding constant. If \Delta_i=0, it is zero once j>C_{\rm loc}+n. The finite prime list therefore gives a uniform bound on \dim B_a^j in every sufficiently high degree.

The fixed low-degree interval needs a different argument, which also accounts for all zero-mean-index iterates. Choose a regular a_+\in(a+D,a+D+1). By ordinary death,

The kernel on the right is bounded in dimension by the sum of local ordinary ranks in this interval of cut values of width at most D+1. The number of its events is at most

independently of a. Uniform local ranks bound the kernel, and the global-image rank is bounded by the fixed global degree rank. Consequently \dim H^j(X_a,M) is bounded uniformly in a for each fixed low degree j.

The sign-line Gysin sequence gives

Induction, including degree zero, bounds B_a^j uniformly throughout any fixed low-degree interval. Combining this with (9.8), every degree in the finitely many remaining bands has uniformly bounded dimension. Since the number of such degrees is also bounded, their sum bounds T(a) as asserted.

Cuts a in a bounded interval have only finitely many crossing stages, so they satisfy a common bound as well. Finally, if a is regular but 2a is not, choose a nearby a' in the same no-critical interval such that 2a' is regular. The critical values and their halves are locally finite. The no-critical restriction isomorphism between the root sublevels is natural with respect to global restriction, and hence preserves both B_a and its global image. In particular T(a)=T(a'). This proves (9.2) for every positive regular cut.□

Proof. The standard-plane Gysin sequence in Proposition 7.1 embeds H/UH into \bigoplus_j B_{2L}^j, which has finite dimension. Lift a finite homogeneous basis of H/UH. These lifts generate H: subtract their contributions modulo U, divide the remainder by U, and induct down the nonnegative degree. This is an induction decreasing degree by two at each step.

If a polynomial p(U) has nonzero constant term, it acts injectively on H. Indeed, the lowest nonzero homogeneous component of a nonzero element survives in its product with p(U). Factoring an annihilating polynomial as U^s p(U) therefore shows that every torsion element is killed by a power of U. The ring \kk[U] is Noetherian, so \mathcal T has finitely many generators and is killed by one common power of U. It is consequently finite-dimensional. Rank-nullity on \mathcal T gives

Moreover UH\cap\mathcal T=U\mathcal T, since Uh torsion implies that h is torsion.

Set \mathcal F=H/\mathcal T. Multiplication by U is injective on \mathcal F. Lift a homogeneous basis of \mathcal F/U\mathcal F. Degree induction again proves generation. These lifts are free: in a homogeneous relation cancel the least power of U and reduce modulo U. Independence of the selected basis excludes the terms with that least power, a contradiction. Thus \mathcal F has a homogeneous free basis of size \rho, and

Summing the standard-plane Gysin sequence over degrees gives the first identity in (10.1), since its two contributions are H/UH and a shifted copy of \ker U.

For sufficiently large D, the torsion no longer contributes in degrees D,D-1. The dimensions of those two free degrees add to \rho, one contribution from each free generator according to parity. The weight-two-plane Gysin sequence gives

By Lemma 8.1 and (8.1), the last dimension equals \sum_j\dim B_L^j in high degree. This proves the second identity without requiring surjectivity of full-group restriction to the fixed locus.□

Proof. Use the notation of Lemma 10.1, and put

The selected odd bottoms have global G-equivariant U-torsion lifts by Theorem 7.2. Their restrictions to H/UH are independent, since that quotient injects into reflection cohomology and their bottom restrictions are independent. They thus occupy an N-dimensional subspace of \mathcal T/U\mathcal T. Write

The selected even bottoms and tops also have global full-group lifts. Their classes modulo U are jointly independent with the selected odd bottoms: this follows from their reflection restrictions and the rule that a top is counted only modulo all bottoms in its degree. In the projection

the kernel is \mathcal T/U\mathcal T, of dimension N+J_0. The odd span already occupies N of these dimensions. Hence the image of the even span has dimension at least V_e(2L)-J_0.

Choose a maximal homogeneous independent family among these even projections; its size is at least V_e(2L)-J_0. Multiply the corresponding global lifts by powers of U to put them in two common large degrees D,D-1, according to parity. Independence persists in \mathcal F: from a relation in one of the common degrees, cancel the least power of U and reduce modulo U, then repeat if necessary. Thus the sum of the ambient full-group global-image ranks in these two degrees is at least V_e(2L)-J_0.

We compare these images with the fixed locus, including the image-level Gysin maps. The subspace X_{2L}^\tau is G-invariant. Under its identification with X_L, rotations act with doubled parameter and reflection acts as before. In sufficiently high degree the map

is injective. Indeed, compose it with the injective fixed G-to-K restriction. The same composite factors through the ambient G-to-K injection and the high actual K restriction isomorphism of Lemma 8.1.

For H'=G,K, set

Naturality places the ambient global images inside these fixed global images. Apply the weight-two-plane Gysin sequence to the fixed global and fixed finite pairs. Its global push is surjective, so on images the finite push gives a surjection

The finite G-to-K injection identifies I_G^D with a subspace of \ker\pi. Consequently

On the fixed K-spaces the global restriction is the coefficientwise map B_\infty[v]\to B_L[v]. Since B_L has finite total dimension, \dim I_K^D=V(L) for sufficiently large D. Choose D large enough for all the preceding comparisons. We have proved

Subtract global-image dimensions from (10.1) and use (10.5)–(10.6):

All summands in the final line are nonnegative. Equality therefore forces (10.4) for each individual odd bottom.□

Completion of the proof of Theorem 1.1 for n\ge3. Suppose that there are only finitely many prime geodesic images. By Theorem 9.1, T is a bounded nonnegative integer on regular positive cuts. It is constant on every no-critical interval, so its maximum is attained on a nonempty bounded open regular interval I\subset(0,\infty).

Every regular point x\in2^kI has regular successive halves: a critical half would double to a critical point. Iterating (10.3) therefore gives T(x)\ge T(x/2^k), which is already the global maximum. Thus every regular point in every expanded interval attains that maximum. For k\ge1, equality holds between x and x/2, so (10.4) holds at every such x. This uses monotonicity under doubling only; no monotonicity between arbitrary cuts is asserted.

Take k large and choose regular x_1<x_2 in the lower and upper quarters of 2^kI. The interval of degrees strictly between \alpha x_1 and \alpha x_2, after removing the fixed endpoint margins of Lemma 9.2, has length tending to infinity. Remove also the fixed-width bands about every \beta_i x_1 and \beta_i x_2. They remove a bounded total number of possible degrees. The odd bottom degrees

form an arithmetic progression of spacing 4d. For sufficiently large k, one such degree j=a_m remains, beyond the fixed low-degree range. Lemma 9.2 makes \xi_{m,0} invisible at x_1 and visible at x_2. At x_1 the degree lies in an absent gap, where the global image is zero, or above the last band, where the sublevel group is zero. At x_2 it lies in a present gap where global reflection restriction is injective.

A global class nonzero at a smaller sublevel cannot become zero at a larger one, since restriction to the smaller sublevel factors through the larger. There are only finitely many events between x_1 and x_2. Choose the first event at which \xi_{m,0} becomes nonzero, and tight regular cuts x_-<x_+ around it, still inside 2^kI. The exact sequence of the crossing puts its post-event restriction in the image of

because its pre-event restriction is zero. By Theorem 3.7, cup with e vanishes on this whole reflection crossing, including simultaneous prime labels. Naturality of the cup action therefore gives

Hence b_{m,0}(x_+)=1 while h_{m,0}(x_+)=0. Since m is odd, this contradicts (10.4) at the regular point x_+ of the expanded interval. The finite-prime-image assumption is impossible.□

Proof of Corollary 1.2. Let \pi:\widetilde M\to M be a finite smooth covering of degree r, where \widetilde M is diffeomorphic to S^n. Equip \widetilde M with the pullback of the given metric. The map \pi is then a local isometry, so it sends every nonconstant closed geodesic upstairs to a nonconstant closed geodesic downstairs. Taking the least positive period of the latter does not change its image.

We show that at most r upstairs prime images project to any one downstairs prime image. Choose an oriented unit-speed representative \sigma:\mathbb R\to M of that image, of least period L>0, and put p=\sigma(0). For each q\in\pi^{-1}(p) there is a unique lift of \sigma starting at q, with initial velocity (\dd\pi_q)^{-1}\dot\sigma(0). Lifting one period permutes the finite fiber \pi^{-1}(p); lifting the reversed path gives the inverse permutation. After m\le r periods a lift therefore returns to its initial point. Its velocity also returns, because the downstairs velocity returns and \dd\pi_q is invertible. The lift is a closed geodesic. If m is the first return of that fiber point, its least period is mL: any period upstairs projects to a period of \sigma and hence is an integer multiple of L.

By Lemma 3.4, unit-speed geodesics with the same image agree after a parameter shift and possibly reversal. Consequently any upstairs prime geodesic whose projected image is the chosen one can be parametrized so that its projection is exactly \sigma. It is then one of the lifts starting at the r points above p. This proves the finite-fiber bound. The argument permits self-intersections and does not require the covering to be regular; phase or reversal can only identify some of these finitely many images.

Theorem 1.1 gives infinitely many upstairs prime images for the pullback metric. Since every downstairs prime image has at most r preimages in this set, there are infinitely many downstairs prime images as well.□