Friedl–Hambleton–Melvin–Teichner conjecture on equivariant intersection forms for π1 ≅ Z

Establish whether, for every closed smooth oriented 4‑manifold X with fundamental group isomorphic to the infinite cyclic group Z, the equivariant intersection form \widetilde{I}_X is extended from the integers; that is, determine whether there exists an integral unimodular form Q such that \widetilde{I}_X = Q \otimes_{\mathbb{Z}} \mathbb{Z}[\pi_1(X)].

Background

The paper develops an algorithm to compute equivariant intersection forms for 4‑manifolds obtained via torus surgeries and shows these forms are extended from the integers in their constructions. This provides supporting evidence for a broader conjectural picture concerning 4‑manifolds with infinite cyclic fundamental group.

Known results establish extendedness in certain stable ranges (e.g., via Hambleton–Teichner), but the authors’ examples lie outside these ranges, making the conjecture particularly relevant. The classification results of Freedman–Quinn, Stong–Wang, and Hambleton–Kreck–Teichner further underscore the importance of understanding equivariant intersection forms in determining homeomorphism types.