Adjunction inequalities and the Davis hyperbolic four-manifold
Francesco Lin, Bruno Martelli
TL;DR
This work proves that all Seiberg–Witten invariants of the Davis hyperbolic 4‑manifold D vanish by exploiting adjunction inequalities on a large collection (864) of genus‑2 totally geodesic surfaces. The authors translate the adjunction constraints into linear inequalities on the lattice spanned by a chosen basis of H2(D;Z) and combine this with the moduli space dimension formula dim = (c1(s)^2 − 2χ − 3σ)/4 = (c1(s)^2 − 52)/4 (since χ=26 and σ=0) to show c1(s)^2 < 52 for all spin^c structures satisfying the adjunction relations, hence vanishing invariants. The core novelty is the explicit computation of intersection data for the 864 surfaces, the reduction to a 72‑dimensional lattice, and a computer‑assisted inequality check (with extensive symmetry pruning) that bounds |c1(s)^2| by 32. This provides a concrete demonstration of vanishing SW invariants for a non‑almost‑complex hyperbolic 4‑manifold and showcases a computational approach that could extend to other hyperbolic examples via higher adjunction refinements.
Abstract
The Davis hyperbolic four-manifold $\mathcal{D}$ is not almost-complex, so that its Seiberg-Witten invariants corresponding to zero-dimensional moduli spaces are vanishing by definition. In this paper, we show that all the Seiberg-Witten invariants involving higher-dimensional moduli spaces also vanish. Our proof involves the adjunction inequalities corresponding to 864 genus two totally geodesic surfaces embedded inside $\mathcal{D}$.