Weak Poincaré transversality for atomic four-dimensional surgery targets (Freedman's homotopy-theoretic reformulation of the four-dimensional topological surgery conjecture)

The four-dimensional topological surgery conjecture predicts that a degree-one normal map $f\colon (M,\partial M)\to (X,\partial X)$ from a compact 4-manifold to a 4-dimensional Poincaré duality pair (the homotopy-theoretic analogue of a manifold with boundary), which is a $\mathbb{Z}[\pi_1 X]$-homology equivalence over the boundary and has vanishing Wall surgery obstruction, can be corrected by normal bordism rel boundary to a homotopy equivalence; it is known only for fundamental groups of subexponential growth. In 1988 Freedman recast the conjecture as a question in homotopy theory. For an atomic target — the mapping cylinder $X = M(N^3 \to_\alpha \vee S^1)$ of a map from a closed 3-manifold to a wedge of circles inducing a surjection on $\pi_1$ with perfect kernel — the question is whether $X$ admits, rel boundary, a decomposition into a 4-dimensional Lefschetz duality piece $H$ glued to products $Y_i \times I$ along the faces $Y_i \times\{0,1\}$, each $(Y_i,\Sigma_i)$ a 3-dimensional duality pair whose boundary surface $\Sigma_i \subset N^3$ realizes the inverse image of a circle wedge point. The paper proves that this weak transversality property yields surgery kernels represented by immersed 2-spheres with null-homotopic double-point loops (condition-$\pi$), that condition-$\pi$ in turn implies solvability of the corresponding surgery problem, and that the surgery conjecture is equivalent to weak transversality for all atomic targets. Both directions remain open: the equivalent link-slicing formulations are under active attack, Freedman has conjectured that the conjecture fails for non-abelian free groups, and no counterexample is known. The problem is to prove or refute weak Poincaré transversality for every atomic target.

Background

Wall's surgery theory asks whether a degree-one normal map $f\colon (M,\partial M)\to (X,\partial X)$ — a map of degree one with compatible stable normal data from a compact topological $n$-manifold to an $n$-dimensional Poincaré duality pair, a finite pair of spaces whose homology satisfies Poincaré–Lefschetz duality with local coefficients, the homotopy-theoretic analogue of a manifold with boundary — can be modified into a homotopy equivalence. The allowed modification is normal bordism rel boundary: $f$ and $f'$ are joined by a cobordism that restricts to the product on the boundary. For $n \ge 5$ the answer is governed by an algebraic obstruction $\theta(f)$ in a Wall $L$-group, a Witt group of quadratic forms over the group ring $\mathbb{Z}[\pi_1 X]$, whose vanishing is necessary and sufficient [22]. In dimension four this statement for arbitrary fundamental groups is the topological surgery conjecture, the direct extension of Wall's program to the last open dimension. Freedman settled the simply connected case, culminating in the classification of simply connected topological 4-manifolds, on the strength of the disk embedding theorem (immersed disks in a 4-manifold with transverse dual spheres can be replaced by locally flat — locally linearly modeled — embedded ones, provided $\pi_1$ conditions hold) [4], and the conjecture is known for the good groups, those for which the disk embedding theorem is available: Freedman–Teichner and Krushkal–Quinn [6,7,8] showed that every group of subexponential growth is good, a class containing all virtually nilpotent groups (groups with a nilpotent subgroup of finite index).

Freedman's 1988 paper [1] attacks the conjecture through the atomic surgery problems introduced by Casson–Freedman [3]: targets $X = M(\alpha)$, the mapping cylinder of a map $\alpha\colon N^3 \to \vee S^1$ from a closed 3-manifold to a finite wedge of circles with $\alpha_*$ surjective and $\ker\alpha_*$ perfect (equal to its own commutator subgroup); such an $X$ is a compact 4-dimensional Poincaré duality pair with boundary $N^3$, free fundamental group, and $\pi_2 X = 0$. The Freedman–Quinn book reduces the general conjecture to these targets, the canonical problems [5, Chapter 12]. The paper's Theorem 1 proves surgery succeeds under condition-$\pi$: after a preliminary normal bordism rel boundary, the surgery kernel $\ker(\pi_2 f)$ is represented by an immersion of a disjoint union of $S^2 \vee S^2$'s, hyperbolic in the surgery sense, in which every loop in the image — in particular every double-point loop, a loop passing through a self-intersection — is null-homotopic in the source 4-manifold (the image is $\pi_1$-null). Theorem 2 derives condition-$\pi$ for atomic targets from a weak Poincaré transversality hypothesis on $\alpha$, using the codimension-two placement theory of Cappell–Shaneson [2] — an obstruction theory, through groups denoted $\Gamma$, for replacing submanifolds by homology-equivalent ones — to cut the target into duality pieces and reassemble it from integral homology equivalences. The weak transversality property asks that $(X, N^3)$ be homotopy equivalent rel boundary to a pair decomposing as $H \cup (\coprod_i Y_i \times I)$, where $(H,\partial H)$ is a 4-dimensional $\mathbb{Z}$-Lefschetz duality pair (a pair carrying a fundamental class inducing chain-level duality isomorphisms), each $(Y_i,\Sigma_i)$ is a 3-dimensional $\mathbb{Z}$-Lefschetz duality pair with $\Sigma_i$ an orientable closed surface, and the $\Sigma_i \subset N^3$ realize the inverse images of the distinguished points of the wedge under a map homotopic to $\alpha$. The paper's conclusion is that the four-dimensional topological surgery conjecture is equivalent to the solvability of this weak Poincaré transversality problem for all atomic targets.

Later work has mostly pursued equivalent link-theoretic reformulations: the conjecture holds if and only if all good boundary links — links $L$ admitting a homomorphism from $\pi_1(S^3 \setminus L)$ onto a free group taking meridians to free generators, with perfect kernel — are freely slice, i.e. bound disjoint locally flat disks in $B^4$ whose complement has free fundamental group [5,16]; and if and only if the universal surgery problems with trivial Lagrangian (a Lagrangian being a half-basis submodule of the surgery kernel on which the intersection form vanishes) are solvable [14], with the controlled $1/2$-$\pi_1$-null relaxation of condition-$\pi$ — requiring the double-point loops to be trivial only in a weaker, commutation-controlled sense — supplying further universal models [15]. Klein's recent Poincaré surgery theory [21], despite the name, concerns simply connected Poincaré duality spaces in dimensions at least five and does not address the four-dimensional question. The question is stated in Freedman's homotopy-transversality formulation; by the 1988 equivalence, resolving it in either direction resolves the four-dimensional topological surgery conjecture itself, the famous open problem of which this is a reformulation.

Problem Statement

Let $\vee S^1$ be a finite wedge of circles with basepoint, and in each circle a distinguished point $P_i$ distinct from the basepoint; let $N^3$ be a closed 3-manifold; and let $\alpha\colon N^3 \to \vee S^1$ be a map such that $\alpha_*\colon \pi_1(N^3)\to\pi_1(\vee S^1)$ is surjective and $\ker\alpha_*$ is perfect (equal to its own commutator subgroup). Let $X = M(\alpha)$ be the mapping cylinder, the quotient of $N^3 \times I \sqcup \vee S^1$ obtained by identifying $(n,1)$ with $\alpha(n)$; $X$ is a compact 4-dimensional Poincaré duality pair (a finite pair of spaces whose homology satisfies Poincaré–Lefschetz duality with local coefficients, the homotopy-theoretic analogue of a manifold with boundary) with boundary $N^3$, free fundamental group, and $\pi_2 X = 0$ [1,3,5].

Following [1], call $\alpha$ weakly Poincaré transverse if there is a pair $(X',N^3)$ homotopy equivalent to $(X,N^3)$ rel boundary such that $X' = H \cup (\coprod_i Y_i \times I)$, glued along $\coprod_i Y_i \times\{0,1\}$, where: (i) $(H,\partial H)$ is a 4-dimensional $\mathbb{Z}$-Lefschetz duality pair — $\pi_1 H$ finitely generated, $\partial H$ a 3-dimensional $\mathbb{Z}$-Poincaré duality space, and a class $\mu\in H_4(H,\partial H;\mathbb{Z})$ inducing the duality isomorphisms; (ii) each $(Y_i,\Sigma_i)$ is a 3-dimensional $\mathbb{Z}$-Lefschetz duality pair — $Y_i$ has finitely generated $\pi_1$, $\Sigma_i$ is an orientable closed surface, and a class $\mu\in H_3(Y_i,\Sigma_i;\mathbb{Z})$ induces the duality isomorphisms $H^*(Y_i,\Sigma_i;\mathbb{Z})\to H_{3-*}(Y_i;\mathbb{Z})$ and $H^*(Y_i;\mathbb{Z})\to H_{3-*}(Y_i,\Sigma_i;\mathbb{Z})$; and (iii) the surface $\Sigma_i \subset \partial X' = N^3$ is the inverse image of the distinguished point $P_i$ under a map homotopic to $\alpha$.

Question: is every such $\alpha$ weakly Poincaré transverse? Prove this, or else exhibit a specific pair $(N^3,\alpha)$ satisfying the hypotheses for which no such decomposition exists.

By [1] (Theorems 1 and 2 and the concluding equivalence), together with the reduction of general four-dimensional surgery problems to atomic ones [3,5, Chapter 12], an affirmative answer is equivalent to the four-dimensional topological surgery conjecture for arbitrary fundamental groups — every degree-one normal map to a 4-dimensional Poincaré duality pair that is a $\mathbb{Z}[\pi_1]$-homology equivalence over the boundary with vanishing Wall obstruction is normally bordant rel boundary to a homotopy equivalence — while a single counterexample would refute that conjecture. In the geometrically equivalent form, the question is whether the corresponding atomic surgery problems satisfy condition-$\pi$: after normal bordism rel boundary, the surgery kernel $\ker(\pi_2 f)$ is represented by an immersion of a disjoint union of $S^2 \vee S^2$'s, hyperbolic in the surgery sense (the kernel form being a direct sum of hyperbolic planes), in which every loop in the image — in particular every double-point loop — is null-homotopic in the source. By the 1988 equivalence, the question simultaneously settles the surgery conjecture, the famous open problem of which this is Freedman's derived reformulation.

References

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