On ordinary cohomology in homotopy type theory, specifically on integral cohomology:
Guillaume Brunerie, Axel Ljungström, Anders Mörtberg, Synthetic Integral Cohomology in Cubical Agda, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) 216 (2022) doi:10.4230/LIPIcs.CSL.2022.11
On homotopy groups of spheres in homotopy type theory, specifically on the first stable homotopy group of spheres:
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