In this paper we study the bordism groups of manifolds with semi-free
$S^{1}$-actions, denoted by $SF_{n}\left(S^{1}\right)$. We study
the multiplicative structure by using a J -homomorphism map. We also
study the construction K, which gives a set of multiplicative generators,
presenting an algebraic interpretation of this geometric construction.
As an application, we analyze the homomorphisms $r_{p}:SF_{*}\left(S^{1}\right)\rightarrow SF_{*}\left(\mathbb{Z_{\textrm{p}}}\right)$
from the bordism group of semi-free $S^{1}$-actions on the bordism
group of $\mathbb{Z_{\textrm{p}}}$ -actions induced by the restriction
functors.
