Let |$\{{\mathcal{T}}_n\}$| be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran et al. It is known that for each |$n\ne 1$| the group |${\mathcal{T}}_n/{\mathcal{T}}_{n+1}$| has infinite rank and |${\mathcal{T}}_1/{\mathcal{T}}_2$| has positive rank. In this paper, we show that |${\mathcal{T}}_1/{\mathcal{T}}_2$| also has infinite rank. Moreover, we prove that there exist infinitely many Alexander polynomials |$p(t)$| such that there exist infinitely many knots in |${\mathcal{T}}_1$| with Alexander polynomial |$p(t)$| whose nontrivial linear combinations are not concordant to any knot with Alexander polynomial coprime to |$p(t)$| , even modulo |${\mathcal{T}}_2$|. This extends the recent result of Cha on the primary decomposition of |${\mathcal{T}}_n/{\mathcal{T}}_{n+1}$| for all |$n\ge 2$| to the case |$n=1$|. To prove our theorem, we show that the surgery manifolds of satellite links of |$\nu ^+$| -equivalent knots with the same pattern link have the same Ozsváth–Szabó |$d$| -invariants, which is of independent interest. As another application, for each |$g\ge 1$| , we give a topologically slice knot of concordance genus |$g$| that is |$\nu ^+$| -equivalent to the unknot. [ABSTRACT FROM AUTHOR]