Summary: Starting from a Lie group \(G\) whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional \(BF\) theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah’s axioms to manifolds equipped with principal \(G\)-bundle. The case \(G= \text{GL} (4,\mathbb{R})\) is especially interesting because every 4-manifold is then naturally equipped with a principal \(G\)-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.