A new mathematical preprint announces an infinite supply of closed, simply connected spin four-manifolds with positive signature, each carrying a smooth structure that is not diffeomorphic to the others. In plain terms, the construction produces endlessly many topologically constrained four-dimensional spaces that are smoothly distinct.
The construction feeds into bounds for a quantity called Λ_s. The paper states an asymptotic upper bound whose leading term is 8s, while a lower bound with the same leading term follows only if the symplectic version of the Bogomolov–Miyaoka–Yau inequality is assumed.
How the family is built
The construction starts with a prime-indexed family. For every prime p at least 5, the paper uses the index n = 12p and shows that the corresponding complex surfaces X₁₂p are simply connected and spin. These surfaces are then used in the central four-manifold construction.
The paper says this approach bypasses the difficulty of finding suitable self-intersection-zero tori by using a symplectic normal sum with homotopy elliptic surfaces. In the central construction, the pairs (X_n, Σ_n) and (E(r)_K, Σ_K) are combined. The selected curve Σ_n has self-intersection 4p.
The resulting family, called W_n, is reported to be symplectic, simply connected, spin and irreducible, and to have the infinity-two property. Along the prime-indexed family, its Chern-to-Euler ratio approaches 3 as p grows and stays below the BMY line.
A tighter asymptotic picture
The immediate technical estimate applies only in the theorem’s stated range: for an admissible even s no larger than one eighth of the signature of W_n, the paper gives an upper bound of -10s - 1 plus one quarter of c21(W_n). That formula connects the constructed manifolds to the broader Λ_s problem.
After accounting for gaps between consecutive primes, the paper states the asymptotic result Λ_s ≤ 8s + O(s^(4/5)d(s)). In ordinary language, the dominant growth is 8s, with an additional term whose size depends on the prime-gap quantity d(s).
Under Cramér’s conjecture about prime gaps, the formula becomes Λ_s ≤ 8s + O(s^(4/5)(log s)^2). This is a conditional refinement of the upper bound, not an unconditional estimate.
What remains conditional
The paper’s lower bound rests on a separate assumption. If the symplectic Bogomolov–Miyaoka–Yau inequality holds for every simply connected spin symplectic four-manifold, then the authors obtain Λ_s ≥ 8s - 1 for every even s at least 0. Under that same assumption, Λ_s divided by s tends to 8 as s becomes large.
Taken together, the upper and lower statements place both sides at the leading scale 8s. They do not produce an unconditional exact value: the lower result depends on the BMY assumption, while the sharper upper result depends on Cramér’s conjecture.
The preprint goes further by proposing the exact formula Λ_s = |8s - 1| for every even s at least 0. It labels this as a conjecture and states that the formula remains open for all s, so the expression is a target for future work rather than a theorem.
The document is identified as an arXiv preprint, and its reported results are presented through constructions and invariant relations for the W_n family. The third author reports partial support from an NSERC Discovery grant.
Paper data and sources
Original title: Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds
Authors: Shintaro Fushida-Hardy, Robert Harris, B. Doug Park
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text