Stein nonfillability of a torus prequantization

Definitions and mathematical statement

Let $\omega$ be an integral symplectic form on $T^4$, with the integral normalization understood so that $[\omega]\in H^2(T^4;{\mathbb{Z}})$. Let $\pi:P^5\to T^4$ be the principal circle bundle with $c_1(P)=[\omega]$, equipped with its prequantization contact structure $\xi=\ker\alpha$, where the connection curvature is the pullback of the appropriately normalized $\omega$. The source's Question $1.17$ conjectures that $(P,\xi)$ has no Stein filling: no compact complex three-dimensional Stein domain has this contact boundary.

The aid reports a stronger topological statement: the total space of every principal circle bundle over $T^4$ admits no six-dimensional Stein or Weinstein filling, regardless of contact structure on that total space. Its stated obstruction uses that $P$ is a closed oriented aspherical five-manifold, so $P\simeq B\pi_1(P)$ and its classifying map sends $[P]$ nontrivially to group homology. A six-dimensional Stein/Weinstein domain has handles of index at most three; viewed relative to its entire boundary $P$, this implies $\pi_1(P)\to\pi_1(W)$ is an isomorphism. A classifying map then extends across $W$, forcing the boundary fundamental class to map to zero, a contradiction. The filling has boundary exactly $P$, with the usual orientations. This is the aid's reported resolution, not independent mathematical review of its topology inputs.

Its additional integral-cohomology proposition concerns the symplectic Euler class $e=du$, where $d\geq1$, $u\in H^2(T^4;{\mathbb{Z}})$ is primitive, and \[ Q=\langle u\smile u,[T^4]\rangle\ne0. \] For $P_e$ define $a=\pi^*u\in H^2(P_e;{\mathbb{Z}})$. It generates the torsion summand ${\mathbb{Z}}/d{\mathbb{Z}}$, trivial when $d=1$, and the aid asserts \[ a^2\ne0\quad\Longleftrightarrow\quad d\nmid Q. \] In the integral alternating-form normal form, choosing an oriented basis $x_1,\ldots,x_4$ of $H^1(T^4;{\mathbb{Z}})$, \[ u=x_1\smile x_2+m\,x_3\smile x_4,\qquad Q=2m, \] so the test is $d\nmid2m$. This cup-square refinement is distinct from the fundamental-class nonfillability obstruction, which does not require that square to be nonzero.

Short English statement

The aid reports that torus circle-bundle total spaces are aspherical and cannot bound six-dimensional Stein or Weinstein domains, so the prequantization conjecture follows. It also gives a separate divisibility test for the square of a torsion cohomology class; nonfillability does not depend on that square being nonzero.

Sources

• Local UnsolvedMath registry, permanent ID $20002075$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[412], zero-based index $412$.
• AIM workshop "Holomorphic curves in contact geometry," Question $1.17$: https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The source specifies a contact circle bundle over $T^4$ with first Chern class the symplectic class. No PDF was opened in this session.
• Supplied research attempt $1$, result full_solution, run status "solved," rigor label "proved," novelty confidence "low." Registry status "solved," classification SOLVED-IN-LITERATURE, updated 14 August 2026. Its candidate contribution is the torsion cup-square proposition in addition to the reported topological resolution.
• No full authoritative references for the Stein handle theorem, classifying-space obstruction or integral normal-form calculation are supplied. Original remarks and literature fields are empty, with no proposer or year.