research/AMR-010-0205.md

metadata

id: AMR-010-0205
classification: SOLVED-IN-LITERATURE
wording_corrected: 'no'

AMR-010-0205 — Branched covers of S×S along the diagonal admit no smooth NPC metric

Problem (corrected statement if needed)

The dataset transcription matches Bestvina's source list verbatim; no correction was needed. Original wording (Bestvina, Questions in Geometric Group Theory, updated July 2004, Q 2.5, https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf):

(Exercise in [BGS85, p. 2]) Take a closed surface S of genus ≥ 2. Let V = S × S and let Σ ⊂ V denote the diagonal. Let Ṽ be a nontrivially ramified finite cover of V along Σ. Then Ṽ has a natural piecewise hyperbolic CAT(0) metric. Show that Ṽ admits no C²-smooth Riemannian metric with curvature K ≤ 0.

Here [BGS85] = W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Progress in Mathematics 61, Birkhäuser, 1985 (DOI 10.1007/978-1-4684-9159-3), where this is the first exercise of the book.

Setup and why the premises hold:

• The diagonal Σ is totally geodesic in the product of hyperbolic metrics on S × S, of codimension 2. Ramified covers of an NPC Riemannian manifold along a totally geodesic codimension-2 submanifold carry a natural locally CAT(0) (here piecewise hyperbolic) length metric — an observation of Gromov; see also R. Charney, M. Davis, "Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds", Amer. J. Math. 115(5) (1993), 929–1009 (DOI 10.2307/2375063; verified via Crossref).
• Nontrivially ramified finite covers exist: S × S ∖ Σ is the configuration space of two ordered points on S, and a Mayer–Vietoris computation (the normal bundle of Σ is TΣ, of Euler number 2 − 2g) shows the meridian of Σ is torsion of order dividing 2g − 2 in H₁(S × S ∖ Σ; ℤ); since 2g − 2 is even, at least a double branched cover exists for every g ≥ 2. (My own check of the BGS premise; I verified the torsion bound, not the exact order.)

Status / Literature

Solved. The exercise was carried out by Stephan Stadler:

• S. Stadler, "An obstruction to the smoothability of singular nonpositively curved metrics on 4-manifolds by patterns of incompressible tori", Geom. Funct. Anal. 25(5) (2015), 1575–1587. DOI 10.1007/s00039-015-0341-8 (verified via Crossref); arXiv:1312.2198 (Dec 2013; verified via the arXiv API). Also Chapter/result of his LMU dissertation New obstructions to smooth nonpositively curved metrics in dimension 4 (advisor B. Leeb, defended 16 July 2014; verified at https://edoc.ub.uni-muenchen.de/19614/).

Stadler's Theorem 1 (stated as "Exercise 1 in [BGS85]"): Let V be a closed 4-manifold which admits a non-trivial finite branched covering β: V → Σ × Σ over the product of a hyperbolic surface Σ with itself with branching locus the diagonal ΔΣ. Then V admits no smooth Riemannian metric of nonpositive sectional curvature. Since "no smooth NPC metric" is stronger than "no C² NPC metric", this settles Q 2.5 completely. The paper's abstract states it is "answering affirmatively a question of Gromov" and the introduction says "The purpose of this note is to do this exercise."

Prior related milestone (different examples, first of their kind): M. Davis, T. Januszkiewicz, J.-F. Lafont, "4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings", Duke Math. J. 161(1) (2012), 1–28 (DOI 10.1215/00127094-1507259; verified via Crossref) — smooth 4-manifolds with isolated ℤ²'s whose invariant flats are "knotted at infinity", impossible in smooth Hadamard 4-manifolds. Stadler's approach is complementary: the branched covers have plenty of ℤ²'s, forcing an over-dense pattern of flat tori.

Work done

1. Confirmed the dataset wording against Bestvina's PDF (fetched directly): Q 2.5 is transcribed verbatim; the only artifacts are typographical (˜V, C2, ≤0).
2. Verified the resolution and every citation above against Crossref/arXiv (DOIs and the arXiv abstract page 1312.2198), and read the argument in the arXiv HTML version.
3. Checked why "cheap" obstructions cannot do the exercise, as my own sanity analysis:
   • Ṽ is aspherical (its universal cover with the pulled-back metric is CAT(0), hence contractible), so one cannot argue via contractibility.
   • For a k-fold branched cover, χ(Ṽ) = kχ(V) − (k−1)χ(Σ) = (2g−2)(k(2g−2) + k − 1) > 0. This is consistent with the sign of χ for NPC 4-manifolds (the 4-dimensional Hopf sign question has an affirmative answer), so Euler characteristic gives no obstruction.
   • For a double cover, Hirzebruch's branched-cover signature formula gives σ(Ṽ) = 2σ(V) − ½[Σ]² = g − 1 ≠ 0, but nonzero signature is also no obstruction to NPC in general (compact complex-hyperbolic surfaces have σ ≠ 0 and K < 0).
   • Hence the obstruction is genuinely geometric, not characteristic-class or fundamental-group-theoretic — consistent with the problem being open from 1985 to 2013.
4. Summary of Stadler's proof (from the arXiv version): The universal cover X of Ṽ with the singular CAT(0) metric contains two rigid convex product subsets interacting badly: (a) lifts of "product blocks" Σ⁺ × Σ̄⁻ disjoint from the diagonal, convex subsets Y₁ × Y₂ preserved by a product F × F of free subgroups (product rigidity à la Monod/Schroeder), and (b) a product Z × ℝ whose cross-section Z contains an ideal triangle, whose three flats come from flat half-planes in c × c ⊂ Σ × Σ orthogonal to the diagonal along a nonperiodic simple geodesic — such flats branch along the singular locus π⁻¹(ΔΣ) and are shown (Lemma 4) to be pointed Hausdorff limits of Γ-periodic flats, hence quasi-isometry invariant (via Kleiner and Lang–Schroeder). One defines a "coarse intersection" relation between flats that is a quasi-isometry invariant and, in smooth Hadamard manifolds, coincides with transverse point intersection. The configuration (conditions (i)–(vii)) therefore transfers to any CAT(0) space with a geometric π₁(Ṽ)-action; but in a smooth Hadamard manifold it forces either two flats to share a quadrant and coincide, or Tits-distance < π between antipodal ideal points — a contradiction (Claims 1–2). Hence π₁(Ṽ) acts geometrically on no Hadamard 4-manifold, i.e. Ṽ carries no smooth NPC metric.

Result

SOLVED-IN-LITERATURE. Q 2.5 is an exercise from BGS85 (1985) that stood for ~28 years and was proved by Stadler (arXiv:1312.2198, 2013; GAFA 25 (2015) 1575–1587; LMU thesis 2014): any closed 4-manifold finitely covering S × S with nontrivial ramification along the diagonal admits no smooth (a fortiori no C²) Riemannian metric of nonpositive sectional curvature, despite carrying a natural piecewise-hyperbolic locally CAT(0) metric. No new proof by me; my contribution is verification of the source wording, of all citations, and a triage of why elementary obstructions (asphericity, χ, σ, π₁) provably cannot settle it.

What remains

Nothing for the problem as stated — it is fully resolved with the stronger conclusion ("smooth" in place of "C²"). Open directions in the vicinity (not part of the assigned problem): the general smoothability question for singular locally CAT(0) metrics on closed manifolds (e.g. which Davis–Januszkiewicz–Lafont-type or Charney–Davis hyperbolization manifolds admit smooth NPC metrics — positive smoothing results exist in other settings, e.g. Ontaneda's Riemannian hyperbolization, not verified here); and whether π₁(Ṽ)-type groups can act geometrically on CAT(0) 4-manifolds of lower regularity (e.g. C¹ or topological Hadamard manifolds), which Stadler's theorem does not address.