Affine structures on nontrivial connected sums of closed manifolds

Whether a nontrivial connected sum of closed manifolds can carry a flat affine structure — equivalently, whether every closed affine manifold of dimension at least three is prime under connected sum — is a basic open structural question in flat affine geometry. It is the stated motivating question of Wu's study of embedded spheres in affine manifolds (arXiv:1110.3541) and a recognized standing problem of the field, recorded as Goldman's Problem 6 in the open-problems list of the 1996 Brazil–USA workshop; Wu sharpens it in dimension three as the conjecture that all closed orientable affine 3-manifolds are prime. Every known closed affine manifold is prime; existing obstructions (the dimension-two classification, Smillie's free-product holonomy obstruction, radiance-theoretic nonexistence on certain Seifert fibered spaces, Cooper–Goldman's exclusion of real projective structures on $\mathbb{RP}^3 \# \mathbb{RP}^3$ , and Wu's rigidity of embedded spheres) cover only special classes, and the Euler characteristic gives no constraint in odd dimensions or for zero-characteristic summands such as $T^n \# T^n$ . Choi's convex–concave decomposition theory reduces the three-dimensional question, outside the affine Hopf class, to whether toral $\pi$ -submanifolds can occur — but his attempted full proof of primeness was withdrawn, and the reduction is conditional. The question remains unresolved in every dimension $n \ge 3$ .

Background

A flat affine structure on a smooth $n$ -manifold $M$ is a maximal atlas of charts with values in the affine space $\mathbb{A}^n$ (Euclidean space equipped with its affine geometry, i.e. the group $x \mapsto Ax + b$ of transformations rather than the full projective group) whose transition maps are restrictions of such affine transformations; equivalently, it is a torsion-free flat connection on the tangent bundle of $M$ . The structure is encoded by a developing map $\mathrm{dev}: \tilde{M} \to \mathbb{A}^n$ , a local diffeomorphism from the universal cover that is equivariant with respect to the holonomy representation $h: \pi_1(M) \to \mathrm{Aff}(n)$ describing how loops act on charts. The structure is complete when $\mathrm{dev}$ is a global diffeomorphism, so that $M$ is a quotient $\mathbb{A}^n/\Gamma$ ; it is radiant when $h$ is conjugate into the linear group $\mathrm{GL}(n)$ (fixes a point of $\mathbb{A}^n$ ). Which closed manifolds admit flat affine structures is a classical and largely open problem: unlike Euclidean (metric-flat) manifolds, classified by Bieberbach's theorem, affine manifolds are not classified, and incomplete structures resist the available tools.

The connected sum $M_1 \# M_2$ of two closed $n$ -manifolds — delete an open $n$ -ball from each and glue along the resulting boundary spheres — is the basic operation for building manifolds; in dimension three every closed manifold decomposes into prime pieces, where a manifold is prime if it cannot be written as a connected sum of two manifolds neither of which is the sphere $S^n$ . Whether the affine condition is compatible with any nontrivial connected-sum decomposition was posed as the "basic topological question" of Wu's study of embedded spheres: "Can a non-trivial connected sum of closed manifolds admit an affine structure?" In dimension two a classical 1955 theorem (see the account in Klingler 2017) leaves only the torus and the Klein bottle, both prime, so the question concerns $n \ge 3$ , where Wu sharpens it as Conjecture 3.3: all closed orientable affine 3-manifolds are prime. The question long predates Wu's preprint: Choi records it as Goldman's Problem 6 in the open-problems list of the 1996 Brazil–USA workshop in affine geometry.

The evidence is one-sided. The complete affine 3-manifolds are classified by Fried and Goldman — precisely the closed 3-manifolds finitely covered by a 2-torus bundle over the circle, all prime — and the radiant ones are likewise classified (surveyed in Wu); every known closed orientable affine 3-manifold is prime. On the obstruction side, the Euler characteristic gives no information in odd dimensions or for zero-characteristic summands, since $\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - \chi(S^n)$ : the summand arrangement $T^n \# T^n$ is excluded by no known characteristic-class obstruction. Smillie proved that the holonomy of a compact affine manifold cannot factor through a free product of finite groups, which excludes connected sums of lens spaces (quotients of $S^3$ by free linear cyclic actions) and related decompositions; the related radiance obstruction of Goldman and Hirsch shows that certain Seifert fibered 3-manifolds — 3-manifolds foliated by circles over a 2-dimensional base — admit no affine structure at all (Carrière, Dal'bo and Meigniez). Cooper and Goldman proved the sharpest single nonexistence result of this type: $\mathbb{RP}^3 \# \mathbb{RP}^3$ , the simplest nontrivial connected sum of a closed 3-manifold with itself, admits no real projective structure at all, and an affine structure would induce one, so this sum is excluded. Wu's rigidity theorem shows that for $n \ge 3$ , under the hypotheses that a developing map embeds a lift of the boundary sphere in $\mathbb{A}^n$ and maps a neighborhood of the lift into the closure of the bounded component of the complement (strict convexity of the embedded image enters only through the proof's reduction, not the statement), the embedded sphere bounds only the standard ball; consequently, a developing map is never injective on a lift of a nontrivial separating sphere of a closed affine manifold. This is the strongest general constraint known, and its hypotheses are assumptions that hold for no a priori reason.

The most substantial recent progress is Choi's convex–concave decomposition theory. A closed affine 3-manifold that is not an affine Hopf manifold (a quotient of $\mathbb{A}^3$ with a point removed by a contraction, the standard non-complete examples) decomposes along projectively embedded toral $\pi$ -submanifolds — embedded 2-tori whose fundamental groups inject into that of the manifold — into irreducible pieces; a closed affine 3-manifold without such a submanifold is irreducible or affine Hopf, hence prime, and one covered by a domain in affine space is likewise irreducible or affine Hopf. Primeness outside the affine Hopf class therefore reduces to whether toral $\pi$ -submanifolds occur at all, which Choi poses as an explicit open question; his attempted full proof of primeness was withdrawn after a central step failed, and Gordon's Dehn-surgery examples show that gluings of the residual type could a priori produce reducible manifolds.

The question is intertwined with three famous open conjectures that Wu restates as context. The Chern conjecture (around 1955) asserts that the Euler class of a closed affine manifold vanishes; the connected-sum question is its natural counterexample side, since Smillie constructed closed manifolds of nonzero Euler characteristic, described in Wu's paper as arising from connected sums, whose tangent bundles admit flat connections, and Wu notes that such manifolds "might potentially be counter-examples" to the Chern conjecture — whether they in fact carry torsion-free affine structures is open — making connected sums the natural hunting ground for potential counterexamples. The Markus conjecture (1960) asserts that a closed affine manifold is complete if and only if its holonomy preserves a volume form. The Auslander conjecture (1964) asserts that the fundamental group of a closed complete affine manifold is virtually polycyclic (has a finite-index subgroup built by successive extensions of cyclic groups). None of the three is resolved, and none would decide the connected-sum question in odd dimensions or for zero-characteristic summands.

Problem Statement

Does there exist, for some $n \ge 3$ , a closed $n$ -manifold that admits a flat affine structure (an $(\mathbb{A}^n, \mathrm{Aff}(n))$ -structure, equivalently a torsion-free flat connection on its tangent bundle) and that decomposes as a nontrivial connected sum $M_1 \# M_2$ with neither $M_1$ nor $M_2$ homeomorphic to $S^n$ ? Equivalently: is every closed affine manifold of dimension at least three prime with respect to connected sum? The general question and its three-dimensional instance — Wu's Conjecture 3.3, that every closed orientable affine 3-manifold is prime — both bear on one objective, deciding whether the affine condition is compatible with any nontrivial connected-sum decomposition, and are not equivalent: the three-dimensional instance is the special case of the general question. An example in any single dimension settles the whole problem, since one reducible closed affine manifold refutes primeness in general, and an orientable three-dimensional example simultaneously refutes Conjecture 3.3, whereas the non-existence of such a decomposition is a claim about every closed affine manifold of dimension at least three. Conjecture 3.3 is the orientable three-dimensional case and leaves dimensions $n \ge 4$ open.

References

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