The spin bordism group for the classifying space $BG$ of group $G$ is denoted as $\Omega^{Spin}_d(BG)$. $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, E. H. Brown, Jr. and F. P. Peterson, The structure of the Spin cobordism ring, Ann. of Math. 86, 271-298 (1967).).
Would you be able to illuminate the following statement, in the similar context: If a bordism group contains a free part, its Pontryagin dual has a U(1) factor. If we can ignore such continuous parameters, this is equivalent to only considering the torsion subgroup of $\Omega^{Spin}_d(pt)$. They are the elements of the Pontryagin dual of the torsion subgroup of $\Omega^{Spin}_d(pt)$.
Can one work on these statements explicitly to see what has been gone through and what has been thrown away, by reducing "$\Omega^{Spin}_d(pt)$" to "the Pontryagin dual of the torsion subgroup of $\Omega^{Spin}_d(pt)$"?