For the purposes of a remark in a paper in preparation, I would like to know if anyone can confirm that $\Omega^{spin}_{4k-1} = 0$.

In the Atiyah-Patodi-Singer paper, Spectral asymmetry and Riemannian geometry: II (Math. Proc. Camb. Phil. Soc., 78, (1975), 405–432) they give an analytical interpretation of the Adams $e$-invariant; see page 422. This starts with the statement that the spin cobordism group $\Omega^{spin}_{4k-1} = 0$, with a reference to Stong's book on cobordism. I didn't find this statement there, and so went back to the original Anderson-Brown-Peterson papers ([1] The Structure of the Spin Cobordism Ring, Annals 86, No. 2 (Sep., 1967), pp. 271-298 and research announcement [2] Spin Cobordism, Bull. AMS (1966), 256-260) on which Stong's treatment is based. Extracting the answer from those papers seems fairly strenuous, so I hope someone more expert than I can help.

One possibly confusing item (pointed out to me by Vitaly Lorman) is in Theorem 1.7 of [2], which reports some computer calculations of the image of $\Omega^{spin}_n$ in unoriented cobordism. This image is reported to be non-trivial for $n = 39, 43$, which seems to contradict the APS statement. Perhaps those calculations are wrong, although the authors say that they confirm the main results of the paper.

By the way, the same vanishing statement also appears in the paper of Atiyah-Smith (Compact lie groups and the stable homotopy of spheres, Topology 13, pp. 135-142, 1974).