Explicit diffeomorphism types, multiplicities, and distinguishing invariants in the De Michelis-Freedman family of exotic $\mathbb{R}^4$'s

The 1992 paper of De Michelis and Freedman constructed, inside standard $\mathbb{R}^4$, a one-parameter family $\{R_t\}_{t\in[0,1]}$ of open subsets each homeomorphic to $\mathbb{R}^4$, and proved (in ZFC set theory) that the family realizes continuum many pairwise nondiffeomorphic smooth structures. The proof is indirect: it combines a gauge-theoretic invariant (Kotschick's $\Phi$-invariant, shown to commute with passage to end-periodic geometric limits) with a counting argument. It never exhibits an explicit pair of parameters $t\neq t'$ with $R_t\not\cong R_{t'}$, never determines how often each diffeomorphism type recurs in the family, and leaves open whether the construction's simplest 'model case' (a two-component ribbon link with the simplest unramified Casson handles) is exotic at all. The problem, flagged by the authors themselves as an uncarried-out project, is to make the family explicit: determining which choices of ribbon disks and Casson handles produce which smooth structures and with which multiplicities, which choices suffice for exoticness, and how to construct invariants that distinguish family members directly.

Background

A smooth 4-manifold is called an exotic $\mathbb{R}^4$ if it is homeomorphic but not diffeomorphic to the standard $\mathbb{R}^4$; such manifolds exist only in dimension four and embody the failure of the smooth h-cobordism theorem. An h-cobordism is a cobordism between two manifolds each of which is a deformation retract of it; in dimensions at least five the h-cobordism theorem makes every smooth h-cobordism a product, but in dimension four Donaldson's gauge theory provides smooth non-product h-cobordisms between homeomorphic simply connected 4-manifolds, while Freedman's topological theory makes every topological h-cobordism of such manifolds a product. This gap between the topological and smooth categories is exactly where exotic $\mathbb{R}^4$'s arise.

The basic smooth substitute for a 2-handle in 4-manifold topology is a Casson handle: an infinite tower of 2-handles, each one self-plumbed (that is, made to intersect itself transversely, in points called kinks) along the double-point loops of the previous stage. Freedman proved that every Casson handle is homeomorphic to the open 2-handle $D^2\times\mathbb{R}^2$, while gauge theory implies that most are not diffeomorphic to it. The simplest Casson handle, denoted $CH^+$, has exactly one kink at each stage, all positive; an unramified Casson handle is one whose tower has no branching. A ribbon disk is a disk smoothly embedded in the 4-ball $B^4$ whose radial height function has no interior local maxima. Deleting tubular neighborhoods of a collection of ribbon disks from $B^4$, gluing Casson handles along the meridians of the deleted disks instead of standard 2-handles, and then removing the remaining boundary produces a ribbon $\mathbb{R}^4$ — the class of potentially exotic $\mathbb{R}^4$'s that embed in the standard one. Freedman's theorem makes every such manifold homeomorphic to $\mathbb{R}^4$; the question is always smooth exoticness.

De Michelis and Freedman built their family from a smooth non-product h-cobordism $W^5$ between nondiffeomorphic closed simply connected 4-manifolds: the Barlow surface $B$ (a simply connected complex surface) and the rational surface $Q=\mathbb{CP}^2\#8\overline{\mathbb{CP}}^2$ (connected sum of the complex projective plane, with orientation reversed on eight summands, denoted $\overline{\mathbb{CP}}^2$). The two are distinguished by Kotschick's $\Phi$-invariant, a signed count of irreducible anti-self-dual $SO(3)$ connections (connections whose curvature component of one self-duality type vanishes). Their key analytic theorem is that $\Phi$ commutes with the passage between a compact manifold and an end-periodic geometric limit — a noncompact manifold whose end is an infinite repetition of a fixed compact piece — the Fredholm theory for which is due to Taubes. Inside such a cobordism they exhibit a proper, topologically trivial h-cobordism between two copies of a ribbon $\mathbb{R}^4$, and varying the Casson handles attached to a fixed ribbon complement produces the family $\{R_t\}$, $t\in[0,1]$, the parameter being a 'topological radius' measured from the end. For $t$ in the standard middle-thirds Cantor set the members admit a nearly explicit description $R_t\cong\mathbb{R}^4\setminus X_t$, where $X_t$ is a Cantor set of 'wild arcs' (arcs not locally flat at any point) whose geometry depends on $t$. A diagonal argument partitions the parameter interval into continuum many equivalence classes of the relation '$R_t\cong R_{t'}$', each of which is countable, so the family contains continuum many diffeomorphism types — but nonconstructively: no explicit pair of distinct types and no multiplicity is identified.

The complexity of (generalized) Casson handles is measured by their capped groups, and compared using the reembedding theorem of Freedman's disk-embedding program, which reembeds any Casson handle into one of prescribed complexity. De Michelis and Freedman state that the actual multiplicities 'could, in principle, be worked out' by studying an explicit h-cobordism and the reembedding proof, bounding 'the increase in capped-group complexity required to achieve reembedding,' but that 'this interesting project has not been carried out.' They also isolate a model case for their construction — the two-component ribbon link of their Figures 3.2/3.7 with the simplest unramified Casson handle on each meridian — as the simplest instance, 'not known to be associated with any nontrivial h-cobordisms and consequently not known to be exotic.' They further remark that the $\Gamma$-invariant, or 'polynomial invariant', is in principle suitable for this discussion, though their analysis is carried out in detail only for Kotschick's 'simpler' $\Phi$-invariant.

The surrounding toolkit was subsequently developed: Gompf showed the model-case ribbon link does arise from the simplest known nontrivial h-cobordism (due to Akbulut), so exotic ribbon $\mathbb{R}^4$'s are constructible from it with sufficiently complex handles; Bižaca and Gompf proved that the simplest Casson handle $CH^+$ suffices for their two-disk pretzel-link example; Gadgil's end Floer homology (a direct limit of the Heegaard Floer homology groups — a Floer-theoretic invariant of 3-manifolds — taken over a compact exhaustion of the manifold) detects exoticness for general slice-disk complements; and in 2026 Eli, Hom and Lidman distinguished infinitely many slice $\mathbb{R}^4$'s built with the same $CH^+$ using knot Floer homology. What remains missing is the explicit smooth-structure accounting for the original De Michelis-Freedman family.

Problem Statement

Make the De Michelis-Freedman family explicit. The overarching objective is an explicit smooth-structure accounting of the family $\{R_t\}_{t\in[0,1]}$ of ribbon $\mathbb{R}^4$'s (a fixed ribbon complement with varying attached Casson handles), replacing the nonconstructive cardinality argument by constructive mathematics. Three jointly supporting targets, in the source's own terms:

(1) Multiplicities and explicit distinctness. Determine, for explicit pairs of parameters $t\neq t'$, whether $R_t$ and $R_{t'}$ are diffeomorphic; compute the multiplicity with which each diffeomorphism type occurs in the family (the counting argument shows each type occurs at most countably often); and identify which explicit Casson handles — which signed trees, equivalently which capped-group complexities — realize which diffeomorphism types.

(2) Minimal complexity. Determine which collections of ribbon disks in $B^4$ and Casson handles yield exotic $\mathbb{R}^4$'s. In particular: is the model case of the construction (the two-component ribbon link of De Michelis-Freedman Figures 3.2/3.7, with the simplest unramified Casson handle on each meridian) exotic? More generally, does the simplest Casson handle $CH^+$ suffice for every ribbon complement arising from a nontrivial h-cobordism?

(3) Distinguishing invariants. Construct computable invariants that separate family members directly.

All three targets are jointly required, and within (2) the general $CH^+$ question subsumes its model case.

References

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