The Thom Obstruction

Thom spectra arise from maps of loop spaces to the classifying space of the stable unitary group — they’re the homotopy-theoretic generalization of cobordism theories. Many important spectra in chromatic homotopy theory (MU, BP) are Thom spectra. Truncated Brown-Peterson spectra BP⟨n⟩ — which capture chromatic information at height ≤ n — are natural candidates to be Thom spectra as well.

They’re not, at least for n ≥ 2 at the prime 2 (arXiv:2603.11440). The proof uses topological Hochschild homology with specific coefficient systems, computed via a new variant of the Brun spectral sequence. The THH computation detects an obstruction: if BP⟨n⟩ were a Thom spectrum with the expected E_3-MU-algebra structure, its THH would have a specific form. It doesn’t.

The structural insight: the Thom spectrum construction is a “geometric” origin for a spectrum — it means the spectrum comes from geometry (cobordism, bundles, classifying spaces). The obstruction says BP⟨n⟩ for n ≥ 2 doesn’t come from geometry in this sense. Its existence is algebraic, not geometric. THH — which computes a form of “free loop space homology” — distinguishes between spectra with geometric origins and those without. The spectra look similar from the outside (both have ring structures, both fit into the chromatic picture), but THH sees inside and finds that the internal structure is different. The tool detects the difference between geometric and algebraic provenance.