I'm trying to find a reference that discusses the homotopy groups of $\mathrm{MSPL}_k$ and $\mathrm{MSTop}_k$, the Thom spectra for $k$-dimensional oriented PL and Top manifolds. I just need the fact that the homotopy groups have odd torsion. It might be buried in Chapter 5 of Madsen-Milgram, but I cannot quite unearth it.
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$\begingroup$ What do you mean by the Thom spectra for $k$-dimensional manifolds? Madsen-Milgram in Ch. 5 analyze the Thom spectra $\mathrm{MSPL}$ and $\mathrm{MSTop}$ corresponding to PL and topological cobordism in all dimensions. They analyze these spectra after inverting $2$, which may be what you're looking for. $\endgroup$– Arun DebrayCommented Dec 28, 2016 at 15:09
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$\begingroup$ Yes I mean MSPL and MSTop corresponding to PL and topological cobordism, as you say. I have looked at Madsen-Milgram but I am not sure if it implies that the homotopy groups of these spectra have odd torsion. $\endgroup$– Stanley ChangCommented Dec 29, 2016 at 19:21
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