Non-torsion Heegaard Floer homology of the sphere-circle product ($H{F}^{-}(S^1\times S^2,{\mathfrak{s}}_k)$)

Let ${\mathfrak{s}}_k$ on $S^1\times S^2$ have $⟨c_1({\mathfrak{s}}_k),[S^2]⟩=2k\ne 0$. The ordinary groups areA sufficiently wound genus-one diagram gives a two-generator summand whose chain differential is multiplication by $1-U^{|k|}$: the two relevant disks have opposite coherent signs and basepoint multiplicities differing by $|k|$. The map is injective over $\mathbb{Z}[U]$, so its homology is the displayed quotient. Multiplication by $U$ is invertible in the quotient, showing that minus-to-infinity is an isomorphism and that the plus and hat groups vanish. In the U-adic completion of Heegaard Floer homology, $1-U^{|k|}$ is a unit, so both completed minus and infinity groups vanish. This example shows why genus bounds for the completed-minus or plus groups must not be transferred unchanged to the ordinary minus theory.