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My question regards the map defined in Atiyah,Bott,Shapiro "Clifford modules", which equals the index of the Clifford-linear Dirac operator: $$\alpha:\Omega^\mathrm{Spin}_\ast(M)\longrightarrow KO^{-n}.$$ Is $\alpha$ injective in dimension $n\equiv 0,1,2,4\mod 8$ (otherwise it's the 0 map)? I think this is true up to dimension $n=8$ by direct check on the generators, what about $n\geq 9$?

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  • $\begingroup$ This fails already in dimension 8. The quaternionic projective plane generates an infinite cyclic summand in $\Omega_8^{\operatorname{spin}}$ whose $\alpha$-invariant is trivial. $\endgroup$
    – archipelago
    Commented Mar 7, 2018 at 3:30

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No. See for example the work of Anderson, Brown, and Peterson

https://projecteuclid.org/euclid.bams/1183527786

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