Standardness of the extended Cappell–Shaneson homotopy 4-spheres and inequivalent smooth structures on the associated nonorientable 4-manifolds (Torres, 2026, Questions D–F)

Torres (arXiv:2607.15062, 2026) extends the classical Cappell–Shaneson construction by assembling $k$ mapping tori of the 3-torus with Cappell–Shaneson monodromies along their sections and performing surgery, producing pairs of homotopy 4-spheres $\Sigma^\epsilon_{A_1\cdots A_k}$ for every collection of Cappell–Shaneson matrices, and, on the nonorientable side, sets $\{Q_r\}$ of homotopy $\mathbb{RP}^4$s and $\{X_r\}$ of manifolds homeomorphic to a given closed nonorientable $X$ containing a suitable $\mathbb{RP}^2$. The $\eta$-invariant of Pin$^+$ structures settles the stable (connect-sum-with-$S^2\times S^2$) classification of these families but says nothing about the unstabilized diffeomorphism types. The paper leaves open whether the new homotopy 4-spheres are diffeomorphic to $S^4$, and whether the construction yields genuinely inequivalent smooth structures: pairwise non-diffeomorphism within $\{X_r\}$ or $\{Q_r\}$ (Question D), behavior under a single $\mathbb{CP}^2$ stabilization (Question E), and exoticness relative to $\mathbb{RP}^4\#\mathbb{CP}^2$ or $\mathbb{RP}^2\times S^2$ (Question F). These questions extend the open Smooth 4D Poincaré Conjecture and the search for closed nonorientable 4-manifolds with more than two inequivalent smooth structures.

Background

The smooth 4-dimensional Poincaré conjecture (SPC4) asserts that every smooth 4-manifold homotopy equivalent to $S^4$ is diffeomorphic to $S^4$. By Freedman's 1982 classification of simply connected topological 4-manifolds, every such homotopy 4-sphere is at least homeomorphic to $S^4$, so SPC4 is purely a question about smooth structures. The oldest and most studied proposed source of counterexamples is due to Cappell and Shaneson (1976). A CS matrix is an integer matrix $A\in SL(3,\mathbb{Z})$ with $\det(A-I)=1$; the mapping torus $M_A$ of a self-diffeomorphism of the 3-torus $T^3$ inducing $A$ on homology is a $T^3$-bundle over $S^1$ (the diffeomorphism is called the monodromy). Removing a tubular neighborhood of a section — a loop meeting each fiber once — and regluing $D^2\times S^2$ with one of two framings $\epsilon\in\mathbb{Z}/2$ yields a pair of homotopy 4-spheres $\Sigma^\epsilon_A$. Decades of work (Aitchison–Rubinstein, Akbulut, Akbulut–Kirby, Gompf) proved large subfamilies of these diffeomorphic to $S^4$, most recently infinite families by Iwaki (2025), but no result covers all CS matrices, and the general question remains a standard test case for SPC4.

The same 1976 paper produced the first inequivalent smooth structure on the real projective 4-space $\mathbb{RP}^4$: gluing $\mathbb{RP}^4$ minus a neighborhood of its generator loop to the mapping torus of an orientation-reversing torus monodromy (an NCS matrix, i.e. $B\in GL(3,\mathbb{Z})$ with $\det B=-1$ and $\det(B^2-I)=1$) gives a manifold $Q$ homeomorphic to $\mathbb{RP}^4$ but not diffeomorphic to it, even after connect sum (denoted $\#$) with any number of copies of $S^2\times S^2$. The key tool is due to Stolz (1988). A Pin$^+$ structure is a lift of the tangent bundle's structure group appropriate to nonorientable manifolds with vanishing second Stiefel–Whitney class $w_2$; its $\eta$-invariant, a spectral invariant of the associated Dirac operator, descends to the Pin$^+$ bordism group $\Omega_4^{\mathrm{Pin}+}\cong\mathbb{Z}/16$, is additive under disjoint union and invariant under bordism, and is unchanged by diffeomorphism, so unequal values obstruct the existence of a diffeomorphism. Two 4-manifolds are stably diffeomorphic when they become diffeomorphic after connect summing both with $N$ copies of $S^2\times S^2$ for some $N$; Stolz showed that the Euler characteristic and the $\eta$-invariant determine the stable diffeomorphism class of closed nonorientable 4-manifolds with fundamental group of order two. Ruberman (1984) showed there are exactly two smooth $s$-cobordism classes of manifolds homeomorphic to $\mathbb{RP}^4$, and Hambleton–Kreck–Teichner (1994) gave the topological classification that identifies such homeomorphism types. No closed nonorientable 4-manifold is currently known to admit more than two inequivalent smooth structures.

Torres (arXiv:2607.15062, July 2026) extends the construction. Assembling $k$ mapping tori $M_{A_i}$ of CS matrices along disjoint sections produces a bundle over $S^1$ whose fiber is the connected sum $k\cdot T^3$ of $k$ three-tori, and surgery on a section with framing $\epsilon\in\mathbb{Z}/2$ yields Theorem A: a pair of homotopy 4-spheres $\Sigma^\epsilon_{A_1\cdots A_k}$ for every $k\ge 1$ and every collection of CS matrices ($k=1$ recovers the classical family). Theorem B assembles $r$ NCS-matrix mapping tori the same way and glues the result to $\mathbb{RP}^4$ minus a neighborhood of its generator loop, producing a set $\{Q_r:r\in\mathbb{N}\}$ of homotopy $\mathbb{RP}^4$s: for odd $r$, $\eta$-invariant values obstruct any diffeomorphism $Q_r\#(n-1)\cdot(S^2\times S^2)\to\mathbb{RP}^4\#(n-1)\cdot(S^2\times S^2)$ for any $n$, while for even $r$ the manifolds are stably diffeomorphic to $\mathbb{RP}^4$. Theorem C generalizes the mechanism: if a closed nonorientable 4-manifold $X$ contains a smoothly embedded $\mathbb{RP}^2$ whose tubular neighborhood is the twisted 2-disk bundle $D^2\widetilde{\times}\mathbb{RP}^2$, replacing that neighborhood by the corresponding piece of $Q_r$ gives a manifold $X_r$ homeomorphic to $X$, and under the hypotheses $H^1(X;\mathbb{Z}/2)=\mathbb{Z}/2$, $w_2(X)=0$, and an $\eta$-invariant condition, odd $r$ again gives a stable non-diffeomorphism. Thus the stable behavior of all these families is settled by the $\eta$-invariant, and the universal covers of the $Q_r$ lie among the Theorem A spheres — but nothing is known about the unstabilized diffeomorphism types, and this gap is precisely what the open questions target. The source records the historical motivation: Akbulut showed that applying the Cappell–Shaneson method to $\mathbb{RP}^2\times S^2$ produces no inequivalent smooth structure and that $Q\#\mathbb{CP}^2\cong\mathbb{RP}^4\#\mathbb{CP}^2$, and Gompf showed that any two closed homeomorphic nonorientable 4-manifolds become diffeomorphic after connect summing with sufficiently many copies of $\mathbb{CP}^2$.

Problem Statement

Determine the unstabilized smooth structure types of the manifolds produced by Torres's extension of the Cappell–Shaneson construction (arXiv:2607.15062). The common objective is to decide whether this construction yields genuinely inequivalent smooth structures — its stable classification being already settled by the $\eta$-invariant. The following questions make this precise:

(1) Are the homotopy 4-spheres $\Sigma^\epsilon_{A_1\cdots A_k}$ of Theorem A diffeomorphic to the standard $S^4$, for arbitrary collections $\{A_1,\dots,A_k\}$ of CS matrices ($A\in SL(3,\mathbb{Z})$, $\det(A-I)=1$; $k\ge 1$) and both framings $\epsilon\in\mathbb{Z}/2$? An affirmative answer for all would remove the largest explicit proposed family of counterexamples to the smooth 4-dimensional Poincaré conjecture; a single negative answer would refute it.

(2) (Question D of the source, with the Introduction's variant.) Is there a closed nonorientable 4-manifold $X$, containing a smoothly embedded $\mathbb{RP}^2$ with tubular neighborhood the twisted 2-disk bundle $D^2\widetilde{\times}\mathbb{RP}^2$, for which the set $\{X_r:r\in\mathbb{N}\}$ consists of pairwise non-diffeomorphic elements? The sharpest instance is $X=\mathbb{RP}^4$, namely the homotopy $\mathbb{RP}^4$s $Q_r$ of Theorem B: are the $Q_r$ pairwise non-diffeomorphic — which would yield the first known closed nonorientable 4-manifold with at least three inequivalent smooth structures — or, more weakly, does $\{Q_r\}$ contain more than two pairwise non-diffeomorphic elements? A positive answer to the pairwise $Q_r$ question implies a positive answer both to the general question and to the weaker $Q_r$ question.

(3) (Question E.) Is $X_r\#\mathbb{CP}^2$ diffeomorphic to $X\#\mathbb{CP}^2$ for every $r\in\mathbb{N}$ — that is, does a single connect sum with the complex projective plane erase the inequivalence produced by the construction?

(4) (Question F.) Is there an $X_{r_0}\in\{X_r\}$ that is not diffeomorphic to $X$ for $X=\mathbb{RP}^4\#\mathbb{CP}^2$ or $X=\mathbb{RP}^2\times S^2$?

Questions (1)-(4) are independent.

Here $Q_r$ and $X_r$ are the smooth 4-manifolds defined by the constructions of the source (its equations (5.17) and (6.2)); all diffeomorphism questions are in the smooth category, with no restriction on $k$, $r$, the matrices, the framing, or the gluing choices.

References

1. Rafael Torres, A novel construction of homotopy 4-spheres and real projective 4-spaces à la Cappell-Shaneson, arXiv preprint (2026). DOI: 10.48550/arxiv.2607.15062. URL: https://arxiv.org/abs/2607.15062
2. Sylvain E. Cappell and Julius L. Shaneson, Some new four-manifolds, Annals of Mathematics (2) 104 (1976), 61-72. DOI: 10.2307/1971056. URL: https://doi.org/10.2307/1971056
3. Selman Akbulut, A fake 4-manifold, Contemporary Mathematics 35 (1984), 75-141. DOI: 10.1090/conm/035/780576. URL: https://doi.org/10.1090/conm/035/780576
4. Selman Akbulut, Cappell-Shaneson homotopy spheres are standard, Annals of Mathematics (2) 171 (2010), 2171-2175. DOI: 10.4007/annals.2010.171.2171. URL: https://doi.org/10.4007/annals.2010.171.2171
5. Robert E. Gompf, More Cappell-Shaneson spheres are standard, Algebraic & Geometric Topology 10 (2010), 1665-1681. DOI: 10.2140/agt.2010.10.1665. URL: https://doi.org/10.2140/agt.2010.10.1665
6. Kazunori Iwaki, Infinite families of standard Cappell-Shaneson homotopy 4-spheres, Topology and its Applications (2025), article 109293. DOI: 10.1016/j.topol.2025.109293. URL: https://doi.org/10.1016/j.topol.2025.109293
7. Ronald Fintushel and Ronald J. Stern, An exotic free involution on S^4, Annals of Mathematics (2) 113 (1981), 357-365. DOI: 10.2307/2006987. URL: https://doi.org/10.2307/2006987
8. Stephan Stolz, Exotic structures on 4-manifolds detected by spectral invariants, Inventiones Mathematicae 94 (1988), 147-162. DOI: 10.1007/bf01394348. URL: https://doi.org/10.1007/bf01394348
9. Daniel Ruberman, Invariant knots of free involutions of S^4, Topology and its Applications 18 (1984), 217-224. DOI: 10.1016/0166-8641(84)90011-7. URL: https://doi.org/10.1016/0166-8641(84)90011-7
10. Ian Hambleton, Matthias Kreck, and Peter Teichner, Nonorientable 4-manifolds with fundamental group of order 2, Transactions of the American Mathematical Society 344 (1994), 649-665. DOI: 10.2307/2154500. URL: https://doi.org/10.2307/2154500
11. Michael Hartley Freedman, The topology of four-dimensional manifolds, Journal of Differential Geometry 17 (1982), 357-453. DOI: 10.4310/jdg/1214437136. URL: https://doi.org/10.4310/jdg/1214437136
12. Daniel Nash, New homotopy 4-spheres, Pacific Journal of Mathematics 256 (2012), 165-176. DOI: 10.2140/pjm.2012.256.165. URL: https://doi.org/10.2140/pjm.2012.256.165
13. Robert E. Gompf, Stable diffeomorphism of compact 4-manifolds, Topology and its Applications 18 (1984), 115-120. DOI: 10.1016/0166-8641(84)90004-X. URL: https://doi.org/10.1016/0166-8641(84)90004-X
14. Jae Choon Cha and Min Hoon Kim, Calegari's homotopy 4-spheres from fibered knots are standard, arXiv preprint (2024). DOI: 10.48550/arxiv.2411.10051. URL: https://arxiv.org/abs/2411.10051
15. Valentina Bais and Rafael Torres, Smooth structures on non-orientable 4-manifolds via twisting operations, arXiv preprint (2023), to appear in Bulletin of the London Mathematical Society. DOI: 10.48550/arxiv.2308.04227. URL: https://arxiv.org/abs/2308.04227
16. Hisaaki Endo, Kazunori Iwaki, and Andrei Pajitnov, Cappell-Shaneson polynomials, arXiv preprint (2025). DOI: 10.48550/arxiv.2507.10885. URL: https://arxiv.org/abs/2507.10885