Gauging fermion parity and summing over spin structures are subtly distinct operations. We introduce ‘bosonisation cohomology’ groups \( {H}_B^{d+2}(X) \) to capture this difference, for theories in spacetime dimension d equipped with maps to some X. Non-trivial classes in \( {H}_B^{d+2}(X) \) contain theories for which (−1)F is anomaly-free, but spin structure summation is anomalous. We formulate a sequence of cobordism groups whose failure to be exact is measured by \( {H}_B^{d+2}(X) \) , and from here we compute it for X = pt. The result is non-trivial only in dimensions d ∈ 4ℤ + 2, being due to the presence of gravitational anomalies. The first few are \( {H}_B^4={\mathbb{Z}}_2 \) , probed by a theory of 8 Majorana-Weyl fermions in d = 2, then \( {H}_B^8={\mathbb{Z}}_8 \) , \( {H}_B^{12}={\mathbb{Z}}_{16}\times {\mathbb{Z}}_2 \) . We rigorously derive a general formula extending this to every spacetime dimension. Along the way, we compile many general facts about (fermionic and bosonic) anomaly polynomials, and about spin and pin− (co)bordism generators, that we hope might serve as a useful reference for physicists working with these objects. We briefly discuss some physics applications, including how the \( {H}_B^{12} \) class is trivialised in supergravity. Despite the name, and notation, we make no claim that \( {H}_B^{\bullet }(X) \) actually defines a cohomology theory (in the Eilenberg-Steenrod sense).
