Positive scalar curvature forces rational inessentiality

Proof. The bounded-cohomology mapping theorem identifies real bounded cohomology of M with that of B\Gamma by pullback along c_M. The duality between bounded cohomology and the \ell^1-seminorm implies that \|M\|>0 supplies a bounded class whose ordinary cohomology image evaluates nontrivially on [M]. These are the mapping and duality results of [11]; neither requires a spin structure.

If (c_M)_*[M] vanished rationally, its real image would vanish. Every bounded class pulled back from B\Gamma would then evaluate trivially on [M], a contradiction. In dimension one, real bounded cohomology already vanishes and the assertion is vacuous; the direct circle argument below also suffices.□

Proof. Run the short-time Ricci flow starting at g. Short-time existence on closed manifolds and the scalar evolution equation give

See [29, Theorem 5.2.1 and Proposition 2.5.4] for these statements in arbitrary dimension, and [17] for the Ricci-flow framework. The maximum principle preserves nonnegativity. If the initial scalar curvature is positive at one point, the strong maximum principle makes it positive everywhere at any sufficiently small later time. If it is initially identically zero but the Ricci tensor is nonzero at a point, (2.1) gives a strictly positive initial time derivative there. At a sufficiently small positive time t_1, scalar curvature is therefore positive at that point. Applying the strong maximum principle from t_1 produces a metric of positive scalar curvature everywhere at a later time. Thus failure of this alternative forces \Ric_g=0.

For the final assertion, suppose that g is Ricci-flat, that M is oriented, and that n=\dim M>0. We use the orbit-packing argument of Milnor [21, Theorem 1]. Its complete universal Riemannian cover (\widetilde M,\widetilde g) has nonnegative Ricci curvature. Fix x\in\widetilde M and a finite generating set of \Gamma. There is a constant D such that

where |\gamma| is word length. Choose a positive radius r for which the balls B(\gamma x,r), \gamma\in\Gamma, are disjoint. To choose r, take a sufficiently small metric ball about the image of x contained in an evenly covered neighborhood. The radius-r balls about its lifts lie in distinct sheets and are therefore disjoint. All these balls have the same positive volume, say v_r. The orbit balls with |\gamma|\le k lie in B(x,Dk+r). Bishop–Gromov comparison [30, Theorem 4.1] therefore gives

Thus \Gamma has polynomial word growth. Write b(k)=\#B_\Gamma(1,k). There are integers k_j\to\infty such that b(k_j+1)/b(k_j)\to1. Otherwise, some \varepsilon>0 would satisfy b(k+1)/b(k)\ge1+\varepsilon for all sufficiently large k, forcing exponential growth. For each generator s,

These balls form a Følner sequence, so \Gamma is amenable. This uses only the growth bound, without a structural classification of groups of polynomial growth. Real bounded cohomology of an amenable group vanishes in positive degrees [11, Sections 3.0–3.1]. The mapping theorem and duality used in Lemma 2.1 now imply \|M\|=0.□

Proof. Choose a finite smooth triangulation of M by the Cairns–Whitehead theorem [5, 31]; see also [22, Theorem 10.6]. Label the vertex lifts equivariantly by elements of \Gamma, making one arbitrary choice for each vertex downstairs. Barycentric coordinates define the labelled simplicial map h_0. A smooth partition of unity subordinate to the open vertex stars, lifted equivariantly, gives h; weights with the same label are added. At each point the labels that occur belong to the vertices of one lifted simplex. Cocompactness bounds their word diameters by a fixed s, and it also bounds the \ell^1-operator norm of dh by a fixed H. Linear interpolation of the weights stays in \mathcal K_s.

For completeness, this entire homotopy is proper. If Q\subset \widetilde M is compact with \Gamma Q=\widetilde M, its image during the homotopy is compact. Given a compact B\subset\mathcal K_s, properness of the \Gamma-action allows only finitely many translates of that image to meet B. The inverse image of B is therefore a closed subset of finitely many translates of Q\times[0,1].

Choose a class in H^n(B\Gamma;\Q) pairing nontrivially with (c_M)_*[M], and represent it on the homogeneous bar resolution. On tuple chains, signed permutation averaging is an equivariant chain map lifting the identity of the augmentation. It is equivariantly chain homotopic to the identity on this free resolution. Antisymmetrizing the representative thus gives a cocycle \omega in the same class. It vanishes on repeated labels and restricts to a simplicial cocycle on every \mathcal K_a.

Give the vertices of M a global order. Sending an ordered lifted simplex to its tuple of labels gives an augmentation-preserving equivariant chain map into the bar resolution. Such a map is unique up to equivariant chain homotopy: inductively, the difference in each degree is a cycle in the exact target resolution, and freeness of the source chains permits equivariant fillings. Comparison with the chain map induced by a classifying map shows that evaluation of \omega on these labelled fundamental simplices is precisely its pairing with (c_M)_*[M], and hence is nonzero.□

Proof. A point in the realization of \mathcal O_L belongs to the interior of a unique minimal-dimensional ordered simplex. A group element fixing the point must preserve that ordered simplex, hence must fix its labels slot by slot. It is therefore the identity. This also proves freeness at points represented on faces; face identifications do not identify distinct orderings of the minimal simplex. Local finiteness and the word-metric bound make the action proper and its quotient finite. Thus \mathcal O_L\to\mathcal O_L/\Gamma is a principal cover, including when \Gamma has torsion. The quotient has a finite polyhedral subdivision. A simplex has at most \abs{B_\Gamma(1,L)} vertices after one of its vertices is translated to the identity, proving the dimension bound.

Embed the finite polyhedron \mathcal O_L/\Gamma in \R^{2v_0+1} and choose an open neighborhood retracting onto it. Choose a compact smooth domain inside that neighborhood containing the polyhedron in its interior, and double the domain. Folding the double to the domain followed by the neighborhood retraction gives a retraction from a closed oriented manifold C onto \mathcal O_L/\Gamma. Its dimension satisfies the stated bound. Pull back the principal cover along this retraction. The inclusion of the polyhedron lifts to i, while the pullback projection followed by \pi gives r. Their composition has the required value.□

Proof. Use a sufficiently fine smooth triangulation, so that both primal simplices and closed dual blocks have arbitrarily small mesh. If all images over a primal simplex share a label, every pair of labels in their union is within 2L. This gives the first assertion.

We describe the chain construction explicitly. The regular dual-cell decomposition has closed cells that are balls and cellular subcomplexes. Subdivide it barycentrically. A simplex s' of this subdivision is a flag of dual cells; let B(s') be the closure of its largest cell. This is the unique minimal closed dual cell containing s'. For a face t' of s' we have B(t')\subset B(s'), so these closed balls form a face-compatible acyclic carrier.

There is an augmentation-preserving chain map from the subdivision chains to dual-cell chains carried by B. To construct it, send a vertex to a dual zero-cell in its carrier with coefficient one. If the map has been defined below dimension j, the image of the boundary of a j-simplex is an augmented cellular cycle in its carrier, because all face carriers are contained there. Fill the cycle within this contractible cellular subcomplex. Make all choices on simplex orbits and translate. The deck action on the subdivision is free, so this is equivariant. Each carrier is finite, and hence each output is finite.

Now subdivide \mathcal O_L equivariantly and approximate the actual map i closely by a simplicial map into the subdivision of the dual decomposition. Choose the target stars inside sufficiently small evenly covered balls. Simplicial approximation on the finite quotients can then be joined to the original map by a homotopy staying in those stars. Lifting that homotopy from i gives an equivariant simplicial endpoint that remains uniformly close to i upstairs. Equivalently, one may choose the stars directly on free vertex orbits. Compose source subdivision, this simplicial chain map, and the carrier map just constructed. This defines \mathcal F. Contractibility of \mathcal O_L is not needed: the only fillings take place inside the target carrier balls.

Consider a dual cell occurring on an original simplex e. Its carrier lies within a small mesh error of i(z) for some z\in e. The primal simplex \tau dual to that cell meets it at its barycenter; small primal mesh therefore puts every o\in\tau within another small error of such an i(z). Choose all meshes and the approximation close enough to apply the common-label property to the pair o,i(z). A common label v belongs both to r(o) and to r(i(z)). Since ri=\pi, the latter support consists of vertices of e. For a label \gamma of r(o) and a vertex w of e, we obtain

This proves the second assertion.□

Proof. On an oriented j-simplex e=[v_0,\ldots,v_j] define the averaged Alexander–Whitney diagonal by

Each ordering gives the usual diagonal on that simplex and its faces. Restriction of a uniformly chosen ordering to a face is again uniformly distributed, so the average commutes with the boundary. It is equivariant, and both factors in every term are faces of e.

Apply \omega to the first factor of degree n and \mathcal F\sigma to the second factor, and call the resulting graded map T. Its outputs are finite. In the tensor boundary, the sum of terms from the first factor vanishes after evaluation by \omega because \delta\omega=0. The boundary in the second factor carries the sign (-1)^n, proving (3.5). When j=n, each contributing second factor is a vertex. Its image under \mathcal F\sigma has augmentation one, while alternation makes every signed first-factor evaluation equal to \omega(e). Thus the augmentation of T_n(e) is exactly \omega(e).

Finally, every output of T_j(e) comes from a face of e. By Lemma 3.4, labels over its corresponding primal simplex are within 2L of every vertex of that face. Since e has diameter at most L, they are within 3L of every vertex of e.□

Proof. Cellular duality on the oriented cover identifies finite dual chains of degree k with compactly supported primal cochains of degree d-k. Write this identification as I, choosing the degree signs so that \delta I=I\partial. At degree zero choose the sign for which evaluation on the oriented locally finite fundamental cycle is the augmentation; the remaining degree signs are then determined successively by the boundary–coboundary identity. On product cells, define a cochain U of degree n+d by

and set it to zero outside the indicated chain degrees. The cochain is invariant under simultaneous translation and thus descends to D_L. The product-cell structure is valid even if the action on \mathcal K_L has inversions: the freely lifted second-factor cells give trivial stabilizers of product cells.

On a product cell of bidegree (j,n+d+1-j), the two terms in \delta U are a_{j-1}IT(\partial e) and (-1)^ja_jI\partial T(e). By Equation (3.5), their coefficients relative to IT(\partial e) are a_{j-1} and (-1)^{j+n}a_j, which cancel. Thus U is a cocycle.

Call a product cell far if all distances between vertices of its first simplex and labels of r over its closed second simplex are greater than 4L. This condition is inherited by faces and is translation invariant, so the far cells form a subcomplex D_{\mathrm{far}}\subset D_L. The support bound in Lemma 3.5 shows that U vanishes on it. Let K_* be the union of the closures of all non-far cells in the quotient. There are only finitely many such cells: after fixing a second-factor simplex from finitely many orbit representatives, non-farness puts some first-factor vertex within 4L of a label in a fixed finite set; the other vertices lie within L of that vertex. Local finiteness of the word metric gives only finitely many choices.

The same argument gives the required support bound. One pair of labels in a non-far product is within 4L, the first simplex has diameter at most L, and the union of second-factor labels has diameter at most 2L. Every pair in its closure is consequently within 4L+L+2L=7L. Furthermore D_L\setminus K_*\subset D_{\mathrm{far}}, so the relative cellular class of U induces the required class u.

It remains to compute the pairing. By proper homotopy invariance of compactly supported cohomology, replace h by h_0 and write \Phi_0 for the resulting graph map. On the product cell structure of X, each open cell maps to an open product cell, possibly of smaller dimension. Thus X_{\mathrm{far}}=\Phi_0^{-1}(D_{\mathrm{far}}) is a subcomplex. Every cell not in it lies in the compact set \Phi_0^{-1}(K_*), and there are finitely many such cells by local finiteness. The locally finite fundamental class maps successively to

In the latter group it is represented by the finite product fundamental cellular chain modulo X_{\mathrm{far}}. Its coefficient on each remaining top cell is the local orientation coefficient, as follows by restricting further to local homology at an interior point. Hence evaluation by the relative cellular cocycle computes the desired compact pairing.

Only bidegree (n,d) contributes. For each oriented fundamental top simplex of M, choose one lift and sum over all oriented fundamental top simplices of \widehat C. The latter sum evaluates IT(e) on the locally finite parameter fundamental cycle and equals the augmentation of T(e), namely \omega(e). Collapsed labelled top simplices contribute zero. The total is exactly the nonzero evaluation from Lemma 3.2, proving Equation (3.6).□

Proof. Continuity follows from the coordinate definition and local finiteness. At any point choose a chart whose cutoff equals one, and use its b=1 row. This row is the nonnegative average distance to its anchor g. For t\in\mathcal K_L and any \gamma\in\supp t, the triangle inequality gives

Over finitely many trivializing charts, a bound on V therefore restricts the first-factor labels to finitely many possibilities. Closed inverse images of compact subsets are consequently compact. On K_*, every input label is within 7L+L=8L of the active anchor. Each distance difference in Equation (3.8) is bounded by that distance to the anchor, giving \norm{V}_\infty\le8L, and hence the stated 24L bound.

For the second assertion, use a chart of weight one common to both probabilities and let g be its anchor. Every label of either probability is within 110L+s of g. In particular a label \gamma\in\supp t equals gb for one of the rows being used. In this row the average of d_\Gamma(\,\cdot\,,\gamma) over t is at most s. The row comparison implies that the average over t' is at most s+14R. Since \supp t' has diameter at most s, every \gamma'\in\supp t' satisfies

This applies to every label of t; the within-support distances already satisfy the bound.

Finally, each row of \mathcal V is locally a finite smooth sum. In a \widetilde M direction \sum_\gamma dh_\gamma=0. Choose one label from the current support and subtract its distance profile before differentiating. The remaining profile values have oscillation at most s, because the distance function to any fixed label is 1-Lipschitz for d_\Gamma. The derivative is thus bounded by s\abs{dh}_{\ell^1}\le sH. A smooth nonnegative coordinate that vanishes at the point has zero differential, so zero probability coordinates introduce no additional labels into this estimate.□

Proof. Poincaré–Lefschetz duality for the triad (P;E_R,E_B) [19, Theorem 3.43] sends u_P to a class

because \dim P=d+q. Its relative intersection with j_*[N,\partial N] is, up to the fixed orientation convention, Equation (3.15).

We spell out the relative realization using the absolute theorem of Thom [28, Theorem III.4]. Write k=q-n and clear denominators in \alpha. Choose an integral relative cycle z in (P,E_B) representing the resulting class. Form the double D=P\cup_{E_B}P' along the angular face. Compatible product collars at E_R\cap E_B give D the structure of a smooth compact manifold with boundary; the seam E_B is a separating hypersurface meeting the remaining boundary neatly. If the seam is empty, this double is the disjoint union of the two copies. Write z' for the reflected chain. Because the boundary chains on the seam agree, z-z' is an absolute cycle in D. The restriction and excision map

sends [z-z'] to [z].

The compact manifold D has finite polyhedral type. Transfer the doubled class to a finite polyhedron homotopy equivalent to D, apply Thom’s theorem there, and compose back to D. This supplies a closed oriented smooth k-manifold B and a map f:B\to D whose fundamental class maps to a nonzero integer multiple of [z-z']. Reverse the orientation if necessary to make this multiple positive. Push f away from the remaining boundary of D by an inward collar deformation, chosen tangent to the seam. Smooth f and perturb it transversely to the seam. These operations are homotopies in D, and compactness keeps the image away from the boundary of the seam. Thus Q=f^{-1}(E_B) is a closed smooth hypersurface, and B_+=f^{-1}(P) and B_-=f^{-1}(P') are compact oriented smooth manifolds with common boundary Q. Give B_+ the orientation restricted from B. The map H_k(B;\Z)\to H_k(B,B_-;\Z)\cong H_k(B_+,Q;\Z) sends [B] to [B_+,Q]: collar excision and the local orientation classes on the interior of B_+ characterize this restriction. Components wholly in B_- vanish in the relative group. Naturality for the map of pairs (B,B_-)\to(D,P'), followed by the displayed target excision map, therefore gives

This proves the required relative realization without prescribing boundary data or requiring the representing map to be an embedding.

Write a_+:B_+\to P for this representative and push its image off E_R by a collar deformation respecting E_B. At a corner one uses a field pointing inward from the radial side and tangent to the angular side; transversality of the two defining functions supplies such a field. Compactness now separates a_+(B_+) from E_R, and already separates j(N) from E_B. Consequently all coincidences of a_+ and j lie in a compact subset of the source interiors, with target values in \operatorname{int}P.

Perturb a_+ only near this compact coincidence set. More precisely, choose finitely many target coordinate neighborhoods with closures in \operatorname{int}P and use small flows of vector fields supported there to make a_+\times j transverse to the diagonal in P\times P. Compactness permits the perturbation to be small enough that it creates no coincidences outside these neighborhoods. The whole homotopy remains a map of pairs (B_+,\partial B_+)\to(P,E_B), is fixed near \partial B_+, and keeps the image separated from E_R. Thus it preserves the relative class and its pairing with j_*[N,\partial N]. The dimension sum is (q-n)+(n+d)=q+d, so the transverse intersections are isolated and have the asserted nonzero signed count. Set A_+=B_+ and write (o_0,y_0) for the perturbed map into E=C\times\R^q.

Finally take a sufficiently small boundary collar, free of intersections, and precompose the map there by a smooth collar retraction that is constant in the inward collar coordinate near the boundary and is the identity away from a slightly larger collar. The image on that collar stays in a region where the strict separation inequalities hold. The resulting map is smooth, has the same interior intersections, and has the claimed collar form.□

Proof of Theorem 3.1. Double A_+ along its boundary, reversing the orientation on the second copy, to obtain a closed oriented manifold A. Reflect both component maps o_0,y_0 across the seam. The collar conclusion of Lemma 3.8 makes the resulting maps smooth. Pull back \widehat C\to C by o_0:A\to C, obtaining \widehat A\to A, and form

The projection X\to C is a submersion, so Z is a smooth q-manifold even though o_0 need not be a submersion. Local sheet trivializations identify it with a product of the \widetilde M fibre directions and the A directions, with their product orientation. This also exhibits it as the stated cover of M\times A. Continue to write \mathcal V for the pulled-back profile map.

Choose once and for all a smooth function \eta:[0,\infty)\to[0,1] equal to one on [0,2] and zero on [3,\infty), and define

Properness of the profile map and compactness of A imply that its support is compact. It is smooth, including at \mathcal V=0, because the cutoff is constant in a neighborhood of that locus. Write a:Z\to A for the parameter projection, and let

This is relatively compact. On its angular boundary the defining boundary inequality for A_+ gives \norm{\mathcal V-y_0}_\infty>4R. On its other boundary,

and hence \norm{\mathcal V-y_0}_\infty>T_0/3. The cutoff equals one on both parts of the boundary. In view of T_0>100L=10000R, these bounds prove the boundary assertion in Equation (3.1).

The zeros in \Omega are exactly the intersections furnished by Lemma 3.8: if \mathcal V=y_0, their common p_R-norm is less than T_0, so the point belongs to N. We check orientations to identify their signed count. In oriented local lifts, the difference of the two map differentials, in tangent variables (v_M,v_C,v_A) and target rows (C,\R^q), is

Substitute v_C=v'_C+do_0(v_A). This substitution has determinant one. Moving the C variables to the front introduces only the fixed dimensional sign (-1)^{nd}, and eliminating their identity block leaves the differential of the pulled-back \mathcal V-y_0 on Z. Thus the intersection signs agree with its zero signs up to one fixed sign, independent of the point, sheet, and component. Division by the positive number R does not change them. Their nonzero sum is therefore \deg(G,\Omega,0) up to that sign. Doubling the parameter may cancel a global intersection count between its two copies; it does not change the local degree over the first copy used here.

It remains to check the derivative bound. By Lemma 3.7 and Equation (3.10),

In the cutoff transition region, \norm{\mathcal V-y_0}_\infty<4T_0. Writing E_\eta=\sup\abs{\eta'}, differentiation of Equation (3.16) gives for every row

The estimate also holds where the cutoff is constant. Together with Equation (3.9), it proves Equation (3.2) with J=\max\{J_1,K_0(1+12E_\eta),1\}, increasing R_0 if necessary. Finally G is a smooth compactly supported map on the lifted product geometry of the cover Z\to M\times A; each of its covariant derivatives therefore has finite supremum for this fixed R.□

Proof. Write E=N^{-1}\partial_z and give the graph the upward unit normal \nu=-U+vE. Our convention for the second fundamental form is II(X,Y)=\ip{\nabla_X\nu,Y}, so its trace is

The vertical Killing field \partial_z has normal speed Nv along the graph, and its tangential part projects to NvU on the base. Vertical translation preserves the graph’s mean curvature as a function of that base point. The normal and tangential variation terms must therefore cancel, giving

Here a normal variation of speed \psi changes H by the negative of the Jacobi operator applied to \psi, while a tangential variation differentiates H in its tangential direction. This fixes the sign on the right of (4.7).

The classical warped-product formulas [2, Section 7], together with the Gauss equation, give, respectively,

Expanding \Delta_{\rm gr}(Nv) in (4.7) therefore yields

The formula for P in (4.5) follows directly by inverting the rank-one addition g+N^2df^2.

Put b_0=d\log N. For a function on the base, \Delta_{\rm gr}N is its ambient Hessian traced over the graph, minus H\nu(N). Since \Hess_{\rm amb}N(E,E)=N\abs{b_0}^2, this gives

Substitute H=D-b_0(U) in (4.8). The Hessian terms cancel because

The result is

When N\equiv1, all terms involving b_0 vanish. This proves (4.4) without any condition on D other than its definition.

For the remaining calculation use the graph-orthonormal frame T=ve_1+rE,e_2,\ldots,e_{\dim B}. For base vector fields lifted independently of z, the ambient connection satisfies

Differentiating \nu=-U+vE gives

Moreover,

For the last equality, symmetry of \Hess f gives \nabla_{e_1}e_1=\sum_{j>1}A(e_j)e_j.

After substitution in (4.10), the C terms give r^2\abs{C}^2. The remaining terms are squares of the component of A along e_1 and of its components orthogonal to e_1. After division by r^2, the coefficient of A(e_1)^2 is

For every j>1, the corresponding coefficient is

Since e_1r=rv^2(1+\mathfrak p)A(e_1), these are precisely (4.3). Also

which proves the positivity assertion. Finally, \det(g+N^2df^2)=W^2\det g and vW=1 prove the volume identity.□

Proof. It suffices to prove the assertion when \dim B\ge2. If d=\dim B<2, put T_0=\T^{2-d} with its flat product metric and consider \overline L=L-\Delta_{T_0} on B\times T_0, with right-hand side pulled back from B. Its principal matrix is uniformly positive and its zeroth-order coefficient is still at least c_0. The result in dimension two gives a unique solution; translation invariance of the data and uniqueness make that solution independent of T_0. It therefore descends to the required solution on B. Restriction to a fixed torus point gives the same uniform estimates, and lifting two possible solutions proves uniqueness on B.

Take a smooth compact exhaustion and solve the zero-boundary Dirichlet problem on each member. Bounded-domain elliptic Dirichlet theory applies since the principal matrix is uniformly positive and the zeroth-order term has the indicated sign; see, for example, the linear Schauder theory in [10, Theorems 6.2 and 6.14]. The maximum principle gives, for right-hand side F,

independently of the exhaustion domain. On every smaller reference chart whose enlargement is contained in that domain, the interior estimate gives

with one constant for all charts. Local compactness and a diagonal subsequence produce a solution on B. Each fixed enlarged chart eventually lies in the exhaustion, so the same estimate holds at every point of the limit. Taking the supremum over charts proves that the solution belongs to C_b^{2,\alpha} with a bounded norm. No estimate at the moving exhaustion boundary is needed. If B is compact, one instead uses the closed-manifold version of the same theory.

For uniqueness, suppose L\xi=0 and S=\sup_B\xi>0. Center uniform charts at a sequence of points where \xi\to S. The pulled-back coefficients have a convergent local C^{0,\beta} subsequence, and the functions a convergent C^{2,\beta} subsequence, for 0<\beta<\alpha. At the center of the limiting chart the limiting function attains the maximum S. There its gradient vanishes and its Hessian is negative semidefinite, so

a contradiction. Apply the same argument to -\xi. This also proves the global maximum principle used in (5.1) for any solution in the uniform class. Higher-order interior estimates give the final assertion.□

Proof. We may assume \dim B\ge2. Once the theorem is proved in those dimensions, the lower-dimensional cases follow by taking the product with T_0=\T^{2-\dim B}, using the product metric and pulling back \ell. The solution on B\times T_0 is invariant under T_0 translations by uniqueness, and hence is pulled back from a function on B. For such a function the flux is (U,0) and its product divergence is \diver_g U, so the descended function solves exactly (5.2). Restriction to a fixed torus point gives the uniform and parameter bounds, and lifting two possible solutions gives uniqueness. The graph metric claims then follow from the same final bounds below.

Replace \ell by \tau\ell, 0\le\tau\le1, and start at f=0 for \tau=0. We use C_b^{2,\alpha}(B), for a fixed 0<\alpha<1, as the solution space.

Linearization and openness. As a function of the covector df, the flux U has transverse eigenvalue r(t)/t and radial eigenvalue

Both extend positively and smoothly through t=0, since N is constant there. Hence the equation is uniformly elliptic on every bounded gradient range.

On the open set of C_b^{2,\alpha} functions with a positive uniform margin in the tangent band, the equation is a smooth map into C_b^{0,\alpha}. The apparent nonsmoothness of r=\abs{U} at zero causes no problem: \lambda is a smooth function of r^2, and the flux is smooth in df. At an individual solution, moving the right-hand side to the left gives the linearization

Here a_f is the derivative of the flux. After expansion, the coefficients are uniformly Hölder, and the zeroth-order coefficient is at least \eps B_0s^\beta/w>0. The individual C_b^{2,\alpha} bound supplies a bounded gradient range, so Lemma 5.1 makes the linearization invertible. The Banach implicit function theorem proves openness in \tau.

Comparison and individual regularity. Two solutions in this class agree. To see this, take a positive supremum of their difference, if one exists, and pass to a limiting uniform chart as in Lemma 5.1. At the limiting maximum their gradients agree and their Hessians are ordered. At that common gradient the right-hand side of the equation is strictly decreasing in height, with derivative at most -\eps B_0s^\beta/w. The two equations are therefore incompatible. The same argument compares a solution with a strict upper or lower barrier.

Each individual solution has uniform bounds in every higher derivative before we seek bounds independent of \tau. Indeed its given C_b^{2,\alpha} norm and band margin make the equation uniformly elliptic with smooth nonlinearities on a bounded range. In a chart, write its nondivergence form as

For z=\partial_k f, differentiation, initially by difference quotients, gives

The coefficients and source are uniformly C^{0,\alpha} using only the individual C^{2,\alpha} bound. The difference quotients have bounded sup norms, so interior Schauder estimates on smaller charts give f\in C_b^{3,\alpha}. Iterating gives C_b^\infty. These constants may at this stage depend on the individual solution.

Uniform tangent-band barriers. For \delta>0 consider

Their gradient and divergence terms are bounded uniformly in \tau and in small \delta. At f_+ the tangent term is negative and diverges uniformly to -\infty as \delta\downarrow0; at f_- it diverges uniformly to +\infty. This uses \lambda\ge B_0s^\beta>0. Choose a sufficiently small fixed \delta so that these are strict upper and lower barriers. Comparison gives, for every solution along the homotopy,

In particular the heights and the band margins are now controlled independently of \tau.

A uniform gradient bound. Fix one solution along the continuation path. If \sup_B t\le1, there is nothing to prove. Otherwise choose points where t approaches its supremum. The individual higher bounds just established allow the metric, data, and solution to converge in uniform charts, in enough derivatives for the graph equation and its Jacobi identity. The limiting slope attains its maximum at the center. This passage uses finite bounds for that one solution; the estimate obtained at the maximum will depend only on the fixed data and the uniform band margin.

At this maximum, t>1, so N=1 on a neighborhood. The graph is locally an ordinary product graph, v=(1+t^2)^{-1/2} has a minimum, and dr=0. Put

Differentiating the right-hand side of the equation along U, and using dr=0 and vU(f)=r^2, the vertical translation identity becomes

The band bound implies fixed constants 0<h_{\min}\le h_e\le h_{\max}<\infty. The fixed uniformly controlled base has \Ric_g\ge-C_{\rm Ric}g. At the minimum of v, the left-hand side of (5.7) is at least -C_{\rm Ric}v. Since r^2>1/2 there and \abs{U(\ell)}\le\norm{d\ell}_{\infty}, it follows that

This bounds v below by a positive constant and hence bounds t above, uniformly in \tau. One may add 1 to the denominator when writing an explicit positive bound. All constants in this final estimate depend only on the fixed data and the already uniform band margin; the individual higher norms used to attain the limiting maximum have disappeared.

Uniform higher estimates and closedness. After the gradient estimate, the flux is uniformly elliptic on a fixed bounded range. Absorbing the metric volume density, write

Every coordinate derivative z=\partial_k f satisfies

The drift coefficient and the nondivergence source are uniformly bounded; the last term is the divergence of a uniformly bounded vector field. Interior Hölder regularity for scalar divergence-form equations gives a uniform C^{0,\alpha'} bound for z on smaller charts, for some \alpha'>0; we use [10, Theorem 8.24]. In that theorem’s notation take b^i=d=0, retain the bounded drift, and treat \mathcal H_fz+\mathcal H_{x_k} as scalar data and \mathcal A^i_{x_k} as divergence data, with the signs determined by (5.8). On a fixed chart these bounded data satisfy its L^{r/2} and L^r hypotheses for any finite r>\dim B. The estimate uses the local L^2 norm of z, already controlled by the gradient bound; it requires no bound for the derivatives of z. Thus its constants depend only on ellipticity, the fixed chart size, the displayed data bounds, and the gradient bound. We obtain a uniform C^{1,\alpha'} bound for f.

The original equation now has uniformly C^{0,\alpha'} nondivergence coefficients and right-hand side. Interior Schauder estimates give C^{2,\alpha'} bounds, and (5.5) gives higher uniform bounds by iteration. In particular these imply uniform C^{2,\alpha} bounds for the originally chosen \alpha<1, by using a bound on one further derivative. Given a sequence of solvable parameters tending to \tau_*, local compactness on an exhaustion produces a solution at \tau_* with the same uniform bounds and band margin. The solvable set is therefore closed as well as open, and contains all of [0,1].

Symmetries and parameter dependence. An isometry preserving the data takes a solution to a solution; uniqueness gives invariance. More generally, an isometry carrying one set of data to another carries their solutions to one another by pullback. The uniform solution class is preserved: covariant Hölder norms for the uniformly controlled metric are equivalent to the reference-chart norms, and these intrinsic norms are unchanged under an isometry. In particular all old torus translations preserve a solution when they preserve the input data.

For parameter dependence it is essential to use the stated mixed bounds. If h(x,a) is any input coefficient in a local parameter chart, bounds on its mixed derivatives imply smoothness of a\mapsto h(\cdot,a) in every C_b^{j,\alpha} norm. Explicitly, Taylor’s formula in a, applied also to x derivatives through order j+1, gives

The additional x derivative controls the Hölder seminorm on the uniformly bounded charts. Applying the same argument to every parameter derivative proves the claimed Banach-space smoothness.

The equation is therefore a smooth Banach map in a as well as f. The fixed reference charts supply the Banach spaces, while the common metric equivalence on V' supplies locally uniform ellipticity on bounded gradient ranges. At each parameter, the invertible linearization gives a locally smooth solution by the implicit function theorem. Applying the higher-order part of Lemma 5.1 gives the same conclusion in every higher norm, and uniqueness identifies all the local branches. Compactness of A supplies uniform mixed bounds and a common band margin using finitely many such neighborhoods.

For gluing, suppose a fixed reference deck transformation \psi carries one local data set to another, so that g_a^{(2)}=\psi^*g_a^{(1)} and \ell_a^{(2)}=\psi^*\ell_a^{(1)}. It is then an isometry between those data sets. Naturality and uniqueness give f_a^{(2)}=\psi^*f_a^{(1)}. Extending \psi by the identity on old circle factors gives the compatibility needed for the flat bundles below.

Finally, df is bounded, N is positive and bounded on its attained range, and v is bounded below. Thus g^+ is uniformly equivalent to the reference product metric and has bounded coefficient derivatives of every order. Its completeness follows from that equivalence to a complete reference metric.□

Proof. Set

Choose a smooth nondecreasing cutoff \phi which is zero for t\le1/4, one for t\ge1/2, and satisfies 0\le\phi'\le10. Let

Writing \mathfrak p=tN'/N, we have

The expression vanishes outside [1/4,1/2], which justifies the displayed bound.

For one coordinate, denote the original function by \ell and the current full fibre metric by g. Solve (5.2) with these choices and replace

Theorem 5.2 applies on \widetilde M\times\T^b. Its uniqueness gives torus invariance and compatibility with the parameter trivializations. Consequently df vanishes on the circle directions, so the metric retains the form (6.1). Its block on E_M has only increased. Since

(the inequality is immediate for t<1/2, and N=1 for t\ge1/2), the new coordinate has derivative norm at most L/4. Subsequent steps can only improve this bound. The tangent band gives the height estimate in (6.2).

It remains to estimate the scalar loss in a way that can be summed. At t>0, put z=e_1r. Differentiating \lambda(r)=B_0(r^2+s^2)^{\beta/2} along U=re_1 gives

The part of -2U(D) contributed by this derivative is -2\gamma zD. Hence

By (6.6), the final term of (4.3) is nonnegative.

Measure the vector U in the metric at the start of the pass, and set

Since the current metric on E_M is at least g_*, one has b_U\le r. The input coordinate still satisfies \abs{d\ell}_{g_*}\le L, so U(\ell)\le Lb_U, while U(f)=rt. Differentiating the tangent in the equation therefore contributes an adverse term bounded above by

Since 0\le r<1 and \beta<1,

Thus the D^2 part of (6.8) is at most

which is absorbed by (6.7).

For the remaining part, first suppose r<s. Then

Here s=q^{-3} and \beta>0. If instead r\ge s, then t\ge1/4: for t<1/4 one has N=s and r< s/4. On the adverse region where Lb_U-\eps rt>0, it follows that

Moreover,

Consequently, on that region,

Combining these estimates and substituting B_0 from (6.4) gives the one-coordinate inequality

For example C_0=128\sqrt2 e^3 is sufficient. The inequality extends by continuity to the boundary of \{t>0\}. On the interior of \{t=0\} the graph is constant and the update adds a constant circle, so it holds there as well.

Finally sum (6.11) over the q coordinates. At a fixed point of Z, the inverse metrics of the E_M blocks all act on the same n-dimensional space E_M^*. If U_i is the vector used at the ith step, (4.5) gives the telescoping identity

Taking its g_*-trace gives \sum_i b_{U_i}^2\le n. The q terms 1/q sum to 1, proving (6.3). In particular the growing number of old circle directions does not appear in this trace. Uniform control and smooth parameter dependence at the new stage follow from Theorem 5.2 applied successively a finite number of times.□

Proof. In a parameter chart about a_* write a=a_*+P^{-1}y. The parameter block becomes g_A(a_*+P^{-1}y), and the other metric coefficients are the corresponding fibre coefficients evaluated at a_*+P^{-1}y. All parameter derivatives through order two tend uniformly to zero. Their limits, at y=0, are the two-jets of the fibre metric with a=a_* frozen, times the constant Euclidean parameter block g_A(a_*). Inverses remain uniformly bounded for this fixed stage. The scalar-curvature formula is a continuous expression in these metric two-jets and their inverses, so its limit is uniformly the fibre scalar curvature. The finitely many parameter charts and the assumed bounds make the convergence uniform over all of Z.

The parameter differential of H acquires the factor P^{-1}, proving the second assertion. The reference product on Z is the pullback of a metric on the closed manifold M\times A and is complete. The assembled metric, with its finite number of circle blocks, is uniformly equivalent to this complete reference product and has bounded coefficient derivatives.□

Proof. Start with the lifted metric of M and no circles, and perform passes indexed by j=0,\ldots,k-1 with

Lemma 6.1 applies inductively because every coordinate derivative is divided by at least four in each pass. Every stage is finite, so the uniform control required for its successor follows from the last assertion of that Lemma. All local constructions match under the flat parameter transitions by uniqueness of the graph solution.

For any number of passes, the total scalar loss is at most

Choose C_n=(400/3)C_0(n+1). Under (6.13), all the fibre scalar curvatures are at least 3\kappa/4. Each coordinate is modified exactly once per pass, so

No coordinate count enters this estimate. The fibrewise Euclidean operator norm is at most \sqrt q\,L_k, since each of the q rows has norm at most L_k=4^{-k}L_0.

At this fixed stage apply Lemma 6.2, making the parameter scale so large that the scalar-curvature loss from assembly is at most \kappa/4 and the parameter part of dG^{(k)} has operator norm at most L_k. The domain’s fibre and parameter blocks are orthogonal. Cauchy–Schwarz therefore bounds the full differential by \sqrt{qL_k^2+L_k^2}, proving (6.14). The resulting base metric on Z is h_k; all circle warps remain positive and independent of circle variables. The completeness and finite-stage bounds follow from the same assembly Lemma.□

Proof. The left-hand side of (6.13) is at most

Choose a sufficiently large finite R once. In particular q, A, and the original comparison map are now fixed while k remains arbitrary. Apply Theorem 6.3 at this fixed scale.

Choose a smooth map Q_q:\R^q\to S^q which is constant outside the Euclidean unit ball and has degree one on the one-point compactification. It can be chosen with the north pole as a regular value having the sole preimage 0. For example, in polar coordinates use

where \rho(r) is a positive linear function near zero, increases to \pi, and is smoothly constant at \pi in a collar of r=1 and beyond; choose the orientation, or compose with a reflection, to give degree one. The expression is smooth at zero because \rho is linear there. Let D_q<\infty bound its differential.

By the comparison theorem there is a precompact open \Omega\subset Z with \deg(G,\Omega,0)\ne0 and \norm{G}_{\infty}>3 on \partial\Omega. The homotopy

stays outside the unit ball on \partial\Omega, by (6.14). Define

The boundary is compact and its image is separated from the unit ball. Thus Q_q\circ G^{(k)} is constant on a neighborhood of the boundary, and this definition is smooth even if \partial\Omega is not smooth. The map is constant outside the compact set \overline\Omega.

Homotopy invariance of local degree gives

The degree of F_k is this same integer: count preimages of a regular target sufficiently near the north pole, where Q_q is an orientation-preserving local diffeomorphism, or equivalently compose the relative degree classes. Hence F_k has nonzero degree. Its differential is bounded by D_q\sqrt{q+1}\,4^{-k}J/R, establishing (6.16) with C(q)=D_q\sqrt{q+1}.

For the last assertion, multiply each circle warp by a sufficiently large positive constant. Because all coefficients are independent of the circle variables and the circle blocks are mutually orthogonal, this is locally a constant change of each circle coordinate. It preserves scalar curvature. The base metric and the norm of dF_k are unchanged. At the fixed stage all warps have positive uniform lower bounds, so their constant multipliers can make every inverse circle length at most 4^{-k}J/R. The operator norm of the projection to the product flat torus is the maximum of these inverse lengths, with no factor depending on the number of circles.□

Proof of Theorem 7.2. Fix d\ge2 and \chi>0. Suppose that the data in the theorem exist, and write b for the finite number of circles. Set D=d+b. Thus Y^d carries the stabilized metric

of scalar curvature at least \chi, and the nonzero-degree map F:Y\to S^d has differential bounded by a number \delta>0 in the base metric g. We derive a positive lower bound for \delta which is independent of b.

Choose a fixed two-sided embedded \T^{d-1} in S^d away from the exterior value of F, with a closed tubular neighborhood parametrized by \T^{d-1}\times[-2,2]. Such embeddings can be built inductively in a Euclidean chart: add one ambient dimension and take the boundary of a small trivial two-dimensional normal disk bundle around the preceding torus. After a rotation, the target tube and all its projection bounds depend only on d.

Choose regular interval levels s_-\in(-5/4,-3/4) and s_+\in(3/4,5/4) and pull back the strip between them. This gives a compact smooth band W\subset Y, with a map to that strip of nonzero relative degree. Compactness follows because the closed target tube avoids the exterior value. Regularity of the two levels gives a smooth boundary; nonzero relative degree follows by counting preimages of a regular interior target of F.

Write the target strip as \T^{d-1}\times[s_-,s_+]. Double W with the opposite orientation on the second copy. On both copies retain the \T^{d-1} component of the target map, and send the interval component monotonically along complementary semicircles of S^1. To make this map smooth across the seams, first choose disjoint source boundary collars and precompose the entire base map, including its torus component, with a smooth collar reparametrization that is constant in the normal variable near the boundary. On the first copy, choose the circle angle \alpha(t) to be \pi/2+\pi t/2 for |t|\le1/2, extend it monotonically to 0 and \pi at the two walls, and make it flat near the endpoints. On the second copy use the complementary semicircle, with the same endpoint values in S^1. The two torus components then agree and are constant in the normal variable at each seam, while the circle components agree there as well. Hence they define a smooth map

with the identity on the final torus.

All collar changes can be confined near the walls, away from |t|\le1/2. Choose the open target arc

Its full preimage in the first copy is precisely |t|<1/4, where the map is unchanged and its interval slope is the fixed number \pi/2. The second copy maps to the complementary semicircle and has no preimage over J. Counting preimages of a regular target over J therefore recovers the nonzero relative degree of the original band map. The map f has nonzero degree, on at least one connected component if the double is disconnected.

Each m_\nu has a positive minimum on the compact band. Enlarge its circle length by a constant factor until the joint projection to the standard \T^b has operator norm at most \delta on W. This changes no local scalar curvature: a constant change of a circle coordinate is a local isometry between the two warped metrics. The operator norm of the joint projection is the maximum inverse circle length, rather than a sum over b.

Give the closed double any smooth metric agreeing with the enlarged stabilized metric throughout the chosen middle substrip and a neighborhood of its closure. On the preimage of J\times\T^{d+b-1} it then has scalar curvature at least \chi. The base part of f has differential at most C_d\delta, because only the fixed tube projection and a fixed middle-arc parametrization are used there. Its torus part has differential at most \delta. The orthogonal block structure therefore gives

independently of b. No metric or curvature control elsewhere on the double is required by the cited theorem.

Applying (7.2) gives

The fixed tube and middle-arc choices depend only on d. In particular we may take

A map with differential at most this number contradicts the preceding lower bound. The threshold depends only on d and \chi, independently of the finite circle count b.□

Proof of Theorem 1.1. Suppose, to the contrary, that M^n, n\ge2, has a positive-scalar-curvature metric and that (c_M)_*[M]\ne0 in H_n(B\Gamma;\Q). Fix the metric and choose \kappa>0 with \Scal_M\ge\kappa.

Theorem 3.1 supplies comparison data at arbitrarily large finite scales R, with a constant J independent of R. Choose one sufficiently large finite R for Corollary 6.4, and keep all resulting comparison data A,Z,q,G,\Omega fixed. For every finite integer k\ge1, that corollary gives a smooth metric g_k on Z and positive smooth warps m_{k,\nu} such that

It also gives a smooth map F_k:Z\to S^q, constant outside a compact set and of nonzero degree, satisfying

The metric coefficients and the parameter enlargement may depend on the finite pass count k.

The sphere map uses the full base Z, with its n fibre directions and \dim A parameter directions. The relevant instance of Theorem 7.2 has d=q=n+\dim A, b=kq, and scalar lower bound \chi=\kappa/2. Its threshold \delta(q,\kappa/2) is independent of b. Since q is fixed, choose a finite k so large that

The resulting stabilized metric and nonzero-degree map contradict the obstruction. This proves rational inessentiality.□

Proof of Corollary 1.2. Let M be one connected component of a closed smooth aspherical manifold. Scalar curvature is identically zero in dimensions zero and one, so suppose n\ge2. If M is oriented, its classifying map is a homotopy equivalence because M is aspherical. It therefore sends the nonzero rational fundamental class to a nonzero class, contradicting Theorem 1.1 if M admits positive scalar curvature.

If M is not orientable, its orientation double cover is a closed connected oriented smooth manifold with the same contractible universal cover. A positive-scalar-curvature metric on M lifts to such a metric on this aspherical cover, contradicting the oriented case.□

Proof of Corollary 1.3. Work on one connected component of M. In dimensions zero and one the full Riemann curvature tensor vanishes, so suppose that \dim M\ge2. Write (\widehat M,\widehat g)=(M,g) if M is orientable. Otherwise, let \widehat M\to M be its connected orientation double cover and let \widehat g be the pullback of g. In either case \widehat M is closed, oriented and aspherical, and \widehat g has nonnegative scalar curvature.

By Lemma 2.2, either \widehat M admits a metric of strictly positive scalar curvature or the lifted given metric \widehat g is Ricci-flat. Corollary 1.2 excludes the first alternative, so \Ric_{\widehat g}=0. The compact-manifold consequence of the Cheeger–Gromoll splitting theorem [7] gives an isometric splitting of the universal Riemannian cover

where N is a compact connected manifold without boundary and \widetilde g is the lift of the original metric g. Since M is aspherical, \widetilde M is contractible. The factor N is a deformation retract of \R^k\times N, hence is contractible as well. If m=\dim N>0, then H_m(N;\Z/2\Z)\ne0 by the mod-two fundamental class, a contradiction. Thus N is a point, \widetilde g is Euclidean, and g is flat because the covering maps are local isometries.□

Proof of Corollary 1.4. Choose compatible basepoints and put \phi=f_*:\pi_1(M)\to\pi_1(N). The K(\pi,1) mapping property gives

see [19, Theorem 1B.8 and Proposition 1B.9]. Since N is aspherical, c_N is a homotopy equivalence. Naturality and the degree identity therefore give

Thus M is rationally essential, and Theorem 1.1 rules out the stated metric. No injectivity or surjectivity condition on \phi is used.

For the connected sum, the pinch N\#P\to N\vee P followed by projection to N has degree +1: a point on the unchanged N-side has one orientation-preserving local preimage. This construction is valid for every n\ge2, including surfaces, so the first assertion applies.□

Proof of Corollary 1.5. Simplicial volume is the sum of the simplicial volumes of the finitely many connected components, so it suffices to treat one component. The circle case was proved in Section 2. In dimensions at least two, Lemma 2.2 either supplies a positive-scalar-curvature metric or gives zero simplicial volume directly in the Ricci-flat case. In the former alternative, Theorem 1.1 and Lemma 2.1 give the same conclusion.□