Chern's conjecture for closed affine manifolds and the Hopf sign conjecture: the surviving open core of the Hirsch–Thurston Euler-characteristic questions

Two long-standing open problems ask how a geometric structure or curvature hypothesis constrains the Euler characteristic of a closed manifold (compact, without boundary), and both were addressed by the same foliated-bundle and content machinery in Hirsch and Thurston's 1975 Annals paper 'Foliated Bundles, Invariant Measures and Flat Manifolds'. The first is Chern's conjecture: every closed affine manifold has zero Euler characteristic. Open since around 1955, it is known for complete affine manifolds, for class C holonomy, for manifolds locally a product of surfaces, and for special affine manifolds, and remains open in general. The torsion-allowed version of this vanishing statement — under Hirsch–Thurston's terminology, which allows torsion in the connection — is false: Smillie (1977) constructed closed flat-connection manifolds with nonzero Euler characteristic. The second is the classical Hopf conjecture: for a closed $2n$-manifold of strictly negative sectional curvature, $(-1)^n\chi(M)>0$. It is proven only in dimensions 2 and 4 and open in dimensions 6 and above. The class C corollary that Hirsch–Thurston flagged as possibly provable directly has since been proved directly (Chen, Gusevskij, de Carvalho Júnior).

Background

Two of the oldest open problems about the Euler characteristic ask how a geometric structure or curvature hypothesis constrains it on a closed manifold (compact, without boundary).

An affine structure on a smooth $n$-manifold $M$ is an atlas whose transition maps are restrictions of affine maps $x\mapsto Ax+b$ of $\mathbb{R}^n$; equivalently, $M$ carries a flat (zero-curvature) affine connection that is torsion-free, meaning the connection has no antisymmetric part. Parallel transport around loops yields the holonomy representation $\pi_1(M)\to\mathrm{GL}(n,\mathbb{R})$. A complete affine manifold is one whose developing map — the canonical local diffeomorphism from the universal cover to $\mathbb{R}^n$ obtained by analytically continuing a chart along paths — is a homeomorphism; a special affine manifold is one whose holonomy lies in $\mathrm{SL}(n,\mathbb{R})$, i.e. which carries a parallel volume form. These must be distinguished from manifolds merely carrying a flat linear connection with torsion allowed, a strictly weaker structure.

Chern's conjecture (c. 1955) asserts that every closed affine manifold has vanishing Euler characteristic $\chi(M)=0$. It is known only under additional hypotheses: Kostant and Sullivan (1975) proved it for complete affine manifolds; Hirsch and Thurston (1975) for holonomy in class C (defined below) or admitting a content; Bucher and Gelander (2011) for manifolds locally a product of surfaces; Klingler (2017) for special affine manifolds. The survey of Löh and Raptis (2023) confirms that the general case is open. Hirsch and Thurston use 'flat manifold' to allow torsion in the connection, and in that generality the vanishing statement is false: Smillie (1977) constructed closed flat-connection manifolds with nonzero Euler characteristic. Since Smillie's examples are non-aspherical (their universal cover is not contractible), Bucher and Gelander formulate the modern generalized Chern conjecture for aspherical closed manifolds.

The Hirsch–Thurston proof works through foliated bundles: a fibre bundle $E\to B$ whose total space carries a foliation (a decomposition into immersed submanifolds, the leaves, locally modeled on parallel slices) transverse to the fibres; such a bundle is determined by a holonomy homomorphism $\pi_1(B)\to\mathrm{Homeo}(F)$ of the fibre. They isolated two weakenings of 'preserves an invariant probability measure': class C, the smallest class of groups containing all amenable groups (groups admitting a finitely additive left-invariant probability measure on their bounded functions, e.g. all solvable groups) and closed under free products and finite extensions; and a content on the fibre, a linear functional on continuous functions that fixes the constants. They showed that foliated bundles with content-preserving or class C holonomy have torsion primary obstruction (the obstruction class to sectioning an associated sphere bundle), and deduced the vanishing statements above, including for compact negatively curved manifolds $M$ with $\pi_1(M)$ in class C or with content-preserving stable-foliation holonomy, the stable foliation of the geodesic flow being the partition into strong stable manifolds (the sets of tangent vectors that converge exponentially fast toward a given vector under the flow), which is in general only Hölder continuous (power-law continuous) in the transverse direction (Hasselblatt 1994; Hasselblatt and Wilkinson 1997).

On the negative-curvature side, the classical Hopf conjecture, which goes back to Hopf's 1932 survey 'Differentialgeometrie und topologische Gestalt', predicts that a closed $2n$-manifold $M$ with a Riemannian metric of strictly negative sectional curvature satisfies $(-1)^n\chi(M)>0$. It holds in dimension 2 by the Gauss–Bonnet theorem and in dimension 4 because the Chern–Gauss–Bonnet integrand (the curvature form whose integral equals $\chi(M)$) has a pointwise sign there — an observation of Milnor recorded in Chern's 1955 paper, as documented by Connell, Ruan, and Wang (2023), who also record that Geroch (1976) showed this pointwise strategy fails in dimension 6 and higher.

Problem Statement

This problem concerns two open conjectures, both of which ask how a geometric structure or curvature hypothesis forces the vanishing or sign of the Euler characteristic of a closed manifold:

(1) Chern's conjecture (the surviving core of the flat-manifold question): Does every closed affine manifold $M$ — a compact manifold without boundary carrying a flat, torsion-free affine connection, equivalently an atlas with affine transition maps — have Euler characteristic $\chi(M)=0$?

(2) The Hopf sign conjecture (the surviving core of the negative-curvature question): If $M$ is a closed manifold of dimension $2n$ with $n\geq 3$ admitting a Riemannian metric of strictly negative sectional curvature, is $(-1)^n\chi(M)>0$?

Both conjectures were addressed by the same foliated-bundle and content machinery in Hirsch and Thurston's paper, and both concern how flatness or negative curvature constrains the Euler characteristic. Two sub-questions raised alongside them in the source have been settled and are not part of the open core: the torsion-allowed version of (1), refuted by Smillie's 1977 examples; and the class C corollary of (2) (that the fundamental group of a compact negatively curved manifold lies outside class C), proved directly by Chen, Gusevskij, and de Carvalho Júnior. Each of (1) and (2) is independently refutable: resolving (1) settles Chern's conjecture, and resolving (2) settles the Hopf sign conjecture in the open range of dimensions six and above. The objective of this problem is to settle both (1) and (2).

References

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3. Shiing-Shen Chern, On Curvature and Characteristic Classes of a Riemann Manifold, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 20 (1955), 117-126. DOI: 10.1007/BF02960745, https://doi.org/10.1007/BF02960745.
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