Standardness of Cappell-Shaneson homotopy 4-spheres: Gompf's conjecture on Cappell-Shaneson matrices

Cappell-Shaneson homotopy 4-spheres, constructed in 1976 from matrices $A\in SL(3;\mathbb{Z})$ with $\det(A-I)=1$ and a framing $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$, are the most-studied potential counterexamples to the smooth 4-dimensional Poincare conjecture (SPC4). Gompf (2010) conjectured that every such matrix is Gompf equivalent - connected by matrix similarity and the trace-shift move $(c,d,n)\mapsto(c,d,n+kd)$ - to the special matrix $A_{0}$, which would imply every Cappell-Shaneson sphere is diffeomorphic to the standard $S^{4}$. The conjecture is verified only for traces $-64\le n\le 69$ and a few exceptional values (Kim-Yamada), plus 146 infinite arithmetic-progression families of matrices (Iwaki 2024), organized through ideal class monoids of the simplest cubic orders $\mathbb{Z}[\theta_{n}]$. As of August 2026, the general conjecture, and in particular the existence or non-existence of an exotic Cappell-Shaneson sphere, remains open.

Background

A homotopy 4-sphere is a smooth closed 4-manifold homotopy equivalent to $S^{4}$. The smooth 4-dimensional Poincare conjecture (SPC4) asserts that every homotopy 4-sphere is diffeomorphic to $S^{4}$; it is the last open case of the Poincare problem. Freedman (1982) proved the topological counterpart - every homotopy 4-sphere is homeomorphic to $S^{4}$ - so a counterexample to SPC4 would be an exotic smooth structure on the topological $S^{4}$, invisible to homotopy and to every currently known smooth invariant. In all dimensions other than 4 the classification of homotopy spheres is essentially understood (Kervaire-Milnor), which is why dimension 4 is exceptional.

Cappell and Shaneson (1976) introduced what became the most intensively studied family of potential counterexamples. Let $T^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$ be the 3-torus. A Cappell-Shaneson (CS) matrix is a matrix $A\in SL(3;\mathbb{Z})$ with $\det(A-I)=1$; it induces a diffeomorphism $f_{A}\colon T^{3}\to T^{3}$, and the mapping torus $W_{A}=T^{3}\times\mathbb{R}/(x,t)\sim(f_{A}(x),t-1)$, surgered along the circle $\{y\}\times\mathbb{R}$ with one of the two possible framings $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$ (a choice of how a tubular neighborhood of the circle is reglued), yields a homotopy 4-sphere $\Sigma_{A}^{\varepsilon}$, a Cappell-Shaneson sphere.

Every CS matrix is similar to a standard CS matrix $X_{c,d,n}$, indexed by the integer triples $\mathcal{CS}=\{(c,d,n)\in\mathbb{Z}^{3}: f_{n}(c)\equiv 0 \pmod{d},\ d\neq 0\}$, where $f_{n}(x)=x^{3}-nx^{2}+(n-1)x-1$ and

$$X_{c,d,n}=\begin{bmatrix}0&a&b\\ 0&c&d\\ 1&0&n-c\end{bmatrix},$$

with $a,b$ uniquely determined by the conditions $\det(X_{c,d,n})=1$ and $\det(X_{c,d,n}-I)=1$, so that $X_{c,d,n}$ has trace $n$ (Aitchison-Rubinstein; Kim-Yamada, Definition 2.5). Write $A_{n}=X_{1,1,n+2}$. The subfamily $\Sigma_{A_{n}}^{\varepsilon}$ was long considered the simplest: Aitchison and Rubinstein (1984) proved $\Sigma_{A_{n}}^{0}$ standard for all $n$; Gompf (1991) proved $\Sigma_{A_{0}}^{1}$ standard; Akbulut (2010) completed the case $\Sigma_{A_{n}}^{1}$ for all $n$. These proofs use Kirby calculus - the manipulation of handle decompositions of 4-manifolds through handle slides and cancellations - and took over thirty years to cover even this subfamily.

Gompf (2010) then isolated the common mechanism. He proved that multiplying a standard CS matrix on either side by powers of an explicit matrix $\Delta$ produces new CS matrices whose spheres are diffeomorphic to the original, and that this operation corresponds exactly to the trace shift $(c,d,n)\mapsto(c,d,n+kd)$, $k\in\mathbb{Z}$. The equivalence relation $\sim$ on $\mathcal{CS}$ generated by matrix similarity together with these shifts is called Gompf equivalence, and Gompf-equivalent matrices have diffeomorphic spheres. Since the spheres of $A_{0}=X_{1,1,2}$ are standard, any CS matrix Gompf equivalent to $A_{0}$ yields a standard sphere. Note that $A_{n}$ itself is Gompf equivalent to $A_{0}$ (take $d=1$, $k=-n$). Gompf conjectured (Conjecture 3.6 of that paper) that every CS matrix is Gompf equivalent to $A_{0}$; this implies the folklore conjecture that every CS sphere is diffeomorphic to $S^{4}$, itself a special case of SPC4. Gompf verified his conjecture for traces $-6\le n\le 9$ and $n=11$.

The conjecture also has an arithmetic reformulation introduced by Aitchison-Rubinstein and developed by Kim-Yamada. An order in a number field is a subring that is a free $\mathbb{Z}$-module of maximal rank (here $\mathbb{Z}[\theta_{n}]$, where $\theta_{n}$ is a root of $f_{n}$, a simplest cubic field of Shanks type); its ideal class monoid $C(\mathbb{Z}[\theta_{n}])$ is the set of nonzero ideals modulo the relation $I\approx J$ iff $\alpha I=\beta J$ for some nonzero $\alpha,\beta\in\mathbb{Z}[\theta_{n}]$, a monoid under ideal multiplication that generalizes the ideal class group and is a group only for integrally closed (maximal) orders. By the Latimer-MacDuffee-Taussky correspondence, similarity classes of trace-$n$ CS matrices correspond bijectively to elements of $C(\mathbb{Z}[\theta_{n}])$; non-invertible ideal classes correspond to matrices not similar to any $A_{n}$. Kim-Yamada discovered a hidden symmetry - Gompf's conjecture holds for trace $n$ iff it holds for trace $5-n$ - and verified it for $-64\le n\le 69$ and $n=-69,-66,71,74$, using computer-algebra computation of ideal class monoids. Iwaki (2024, journal version 2025) extended this to 146 infinite families: whenever a triple $(c,p,n_{0})$ satisfies the simultaneous congruences $(2c-1)n_{0}\equiv 3c^{2}-1 \pmod{p}$ and $(c^{2}-c)n_{0}\equiv c^{3}-c-1 \pmod{p^{2}}$, the matrices $X_{c,p,p^{2}k+n_{0}}$ are not similar to any $A_{n}$ and their spheres are standard, provided the residue class of $n_{0}$ modulo $p$ meets a previously verified trace. Beyond such finite intervals and arithmetic progressions nothing is known, and no available invariant can test an individual candidate sphere for exoticness.

Problem Statement

Question (Gompf's conjecture on Cappell-Shaneson matrices; Conjecture 3.6 of Gompf 2010, Conjecture 1.3 of Iwaki 2024): is every Cappell-Shaneson matrix $A\in SL(3;\mathbb{Z})$ with $\det(A-I)=1$ Gompf equivalent to $A_{0}=X_{1,1,2}$? Explicitly, since every CS matrix is similar to a standard one $X_{c,d,n}$ with $(c,d,n)\in\mathcal{CS}=\{(c,d,n)\in\mathbb{Z}^{3}: f_{n}(c)\equiv 0 \pmod{d},\ d\neq 0\}$, $f_{n}(x)=x^{3}-nx^{2}+(n-1)x-1$:

$$(c,d,n)\sim(1,1,2)\ \ \text{for every }(c,d,n)\in\mathcal{CS},$$

where $\sim$ is the equivalence relation generated by similarity of standard CS matrices and the moves $(c,d,n)\mapsto(c,d,n+kd)$ for $k\in\mathbb{Z}$. Equivalently, in the arithmetic formulation of Kim-Yamada and Iwaki, transporting Gompf equivalence through the Latimer-MacDuffee-Taussky correspondence (which identifies similarity classes of trace-$n$ CS matrices with elements of $C(\mathbb{Z}[\theta_{n}])$): for every integer $n$, is every element of the ideal class monoid $C(\mathbb{Z}[\theta_{n}])$ Gompf equivalent to the principal class?

An affirmative answer implies that every Cappell-Shaneson homotopy 4-sphere $\Sigma_{A}^{\varepsilon}$, for both framings $\varepsilon\in\mathbb{Z}/2\mathbb{Z}$, is diffeomorphic to the standard $S^{4}$ (Iwaki's Conjecture 1.2 / the folklore conjecture), a special case of the smooth 4-dimensional Poincare conjecture. A negative answer would produce an explicit CS matrix (and corresponding non-invertible ideal class) that is not Gompf equivalent to $A_{0}$, hence a concrete candidate whose sphere is not certified standard by this method; because Gompf equivalence is only a sufficient condition for standardness, a negative answer would not by itself refute SPC4 - it would identify a candidate exotic sphere whose exoticness would have to be established by other means. The conjecture is currently verified for traces $-64\le n\le 69$ and $n=-69,-66,71,74$, and for 146 infinite arithmetic-progression families of matrices; the general case - including the existence or non-existence of an exotic Cappell-Shaneson sphere - is open.

References

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