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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).

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I believe the bordism groups are nonzero in every dimension after some relatively small finite dimension, just by looking at the Poincaré polynomial in Anderson-Brown-Peterson's earlier paper "Spin Cobordism".

However, from their splitting result, all of these groups come from suspensions of Eilenberg-MacLane spectra. In particular, the image of the Hurewicz map $\Omega^{fr}_* \rightarrow \Omega^{spin}_*$ is zero in dimensions 4k-1 because that's true for connective covers of KO and nonzero suspensions of EM-spectra. Atiyah et. al. are only ever using that framed manifolds in that dimension are bounded by Spin-manifolds, so their arguments still work, despite the mistaken claim about Spin bordism groups in general.

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    $\begingroup$ Thanks, that makes sense, although it's a puzzling oversight from that era. I'll have to think about whether the manifolds in my situation are indeed framed. $\endgroup$
    – Danny Ruberman
    Commented Jan 31, 2017 at 14:30
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    $\begingroup$ I once saw a math sci net review of a different paper that included the line “... $\Omega^{Spin}_{4k-1}$, which the authors seem unaware is nonzero in general”. $\endgroup$
    – Theo Johnson-Freyd
    Commented Mar 1, 2019 at 15:32
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The spin cobordism groups $\Omega^{spin}_n$ have been computed for $n \leq 127$; see section 10 of Secondary Invariants for String Bordism and tmf by Bunke and Naumann. They use MAPLE together with the decomposition of the 2-completion of $MSpin$ found by Anderson, Brown, and Peterson in their paper Spin Cobordism.

In particular, one sees that $\Omega^{spin}_{4k-1}$ is zero for $1 \leq k \leq 9$, but $\Omega^{spin}_{39} \cong \mathbb{Z}_2\oplus\mathbb{Z}_2 \neq 0$. In fact, $\Omega^{spin}_{4k-1} \neq 0$ for $10 \leq k \leq 32$.

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