Cancellable elements and unique decomposition for connected sums of closed manifolds in dimensions at least 4

For $n \ge 4$, let $\mathcal{M}^{\mathrm{Cat}}_n$ be the commutative monoid of closed, oriented, connected $n$-manifolds in the category Cat, where Cat is Top (topological), PL (piecewise-linear), or Diff (smooth), with the operation of connected sum $\#$. Two questions posed by Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasprowski, Markus Land and Johnny Nicholson (2021) remain open. First, does there exist a non-trivial cancellable element in at least one of these monoids for $n \ge 4$ — that is, a manifold $M$, not a unit, such that $M \# N \cong M \# N'$ always implies $N \cong N'$ — or in the simply connected submonoids for $n \ge 6$? All known results point the other way: $\mathbb{CP}^2$, $S^2 \times S^{n-2}$ for $n \ge 4$, and $S^5 \times S^{n-5}$ for $n \ge 17$ (in the simply connected submonoid) are all non-cancellable. Second, is the submonoid generated by closed aspherical manifolds (manifolds whose universal cover is contractible) a unique-decomposition monoid for $n \ge 4$ in each of the three categories? Assuming the Borel conjecture the answer in Top is yes; unconditionally, and in PL/Diff, nothing is known, and exotic tori (manifolds homeomorphic, but not isomorphic in Cat, to the standard torus) have been suggested as a possible source of failure.

Background

The connected sum $M \# N$ of two closed connected $n$-manifolds is formed by deleting an open $n$-ball from each and gluing the results along the resulting boundary $(n-1)$-spheres; up to the standard caveats about orientation, this is a well-defined commutative, associative operation, and the closed oriented connected $n$-manifolds in a fixed category Cat $\in \{$Top, PL, Diff$\}$ form a commutative monoid $\mathcal{M}^{\mathrm{Cat}}_n$ whose identity is the sphere $S^n$. The units of this monoid are the homotopy spheres (in Diff they form the Kervaire–Milnor group $\Theta_n$ of diffeomorphism classes of homotopy $n$-spheres under connected sum; in Top they are trivial by the Poincaré conjecture, and in PL trivial except possibly in dimension 4). An element $M$ is called cancellable if $M \# N \cong M \# N'$ implies $N \cong N'$ for all $N, N'$; it is called irreducible if $M \cong N \# N'$ forces $N$ or $N'$ to be a unit. In dimension 3 the Kneser–Milnor prime decomposition makes $\mathcal{M}_3$ a unique-factorisation monoid, and this underpins 3-manifold topology. In dimensions $n \ge 4$ the structure is far wilder: $\mathbb{CP}^2$ is not cancellable in $\mathcal{M}_4$, $S^2 \times S^{n-2}$ is not cancellable for $n \ge 4$, and $S^5 \times S^{n-5}$ is not cancellable (within the simply connected submonoid) for $n \ge 17$; these double-decomposition phenomena are the source paper's Theorems 1.3 and 1.4. On the positive side, the Smale–Wall classification of highly connected manifolds implies that in Diff the submonoid of highly connected manifolds ($\pi_i = 0$ for $i \le \lfloor n/2 \rfloor - 1$) is a unique-decomposition monoid in every even dimension $n = 2k$ with $k \equiv 3, 5, 7 \pmod 8$ and $k \notin \{15, 31, 63\}$. A closed manifold is aspherical if its universal cover is contractible; equivalently it is a $K(\pi_1, 1)$-space, i.e. its only non-trivial homotopy group in positive degree is its fundamental group. The Borel conjecture asserts that any homotopy equivalence between closed aspherical manifolds is homotopic to a homeomorphism; it is one of the central open problems in rigidity and remains open in general. The fundamental group of a closed aspherical manifold can never be a nontrivial free product, which is why the aspherical submonoid behaves better under connected sum than the full monoid: the Grushko–Neumann and Kurosh subgroup theorems give uniqueness of free-product decompositions on the group side, and the Borel conjecture promotes the resulting homotopy equivalences to homeomorphisms.

Problem Statement

Two linked questions about the monoid $\mathcal{M}^{\mathrm{Cat}}_n$ of closed, oriented, connected $n$-manifolds under connected sum (Cat $\in \{$Top, PL, Diff$\}$). (1) Cancellable elements: for $n \ge 4$, does there exist a non-trivial cancellable element in at least one of the monoids $\mathcal{M}^{\mathrm{Top}}_n$, $\mathcal{M}^{\mathrm{PL}}_n$, $\mathcal{M}^{\mathrm{Diff}}_n$ — i.e. a manifold $M$ that is not a unit such that $M \# N \cong M \# N'$ implies $N \cong N'$ for all closed $n$-manifolds $N, N'$? And for $n \ge 6$, does there exist such an element in the simply connected submonoids $\mathcal{M}^{\mathrm{Top,sc}}_n$, $\mathcal{M}^{\mathrm{PL,sc}}_n$, $\mathcal{M}^{\mathrm{Diff,sc}}_n$ — that is, a simply connected $M$ that is not a unit, such that $M \# N \cong M \# N'$ implies $N \cong N'$ for all simply connected closed $n$-manifolds $N, N'$? (2) Unique decomposition of aspherical sums: in each of the three categories Top, PL and Diff, is the monoid generated by closed aspherical manifolds a unique-decomposition monoid for $n \ge 4$ — i.e. whenever $N_1 \# \cdots \# N_k \cong M_1 \# \cdots \# M_l$ with all $N_i$ and $M_j$ closed aspherical $n$-manifolds, must $k = l$ and the factors agree up to isomorphism in the category, after reordering? A positive answer to (2) in Top is known conditional on the Borel conjecture; the unconditional question is open in every category, and (1) is open in full.

References

1. Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasprowski, Markus Land, Johnny Nicholson, 'Connected sum decompositions of high-dimensional manifolds', in: 2019–20 MATRIX Annals, MATRIX Book Series, Springer, 2021. DOI: 10.1007/978-3-030-62497-2_1, https://doi.org/10.1007/978-3-030-62497-2_1; arXiv:1909.02628, https://arxiv.org/abs/1909.02628.
2. Esma Dirican Erdal, 'Multiplicativity of Reidemister-Franz Torsion for Even Manifolds', arXiv:2511.01786 (2025), https://arxiv.org/abs/2511.01786.
3. Johnny Nicholson, 'Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction', International Mathematics Research Notices 2024(16), 11832–11864 (2024). DOI: 10.1093/imrn/rnae141, https://doi.org/10.1093/imrn/rnae141; preprint arXiv:2005.01664 (2020), https://arxiv.org/abs/2005.01664.