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The 4-dimensional oriented cobordism group of closed manifolds is $\Omega^{SO}_4=Z$, and we know that it cannot be generated by a mapping torus, since the Pontryagin for $p_1$ is zero for any mapping torus. (A mapping torus is a fiber bundle over $S^1$.)

In question Is $SU(3)/SO(3)$ cobordant with a mapping torus? it is shown that the 5-dimensional oriented cobordism group $\Omega^{SO}_5=Z_2$ is generated by a mapping torus.

My question is that is there any oriented cobordism group that has a torsion part which cannot be generated by any mapping torus?


I found a related paper: http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0339224-X/ It shows that at dimension $n=12$ mod 16, the torsion part of a version of Lashof cobordism group $\Omega_n(v)$ is fully generated by mapping tori of the diffeomorphism of $RP^{n-1}$. In particular Tor$\Omega_{12}(v)=Z_2$ and Tor$\Omega_{28}(v)=6Z_2$. ($\Omega_n(v)$ is denoted as $\Omega^G_n$ in wiki article http://en.wikipedia.org/wiki/Cobordism .)

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