= Non-torsion Heegaard Floer homology of the sphere-circle product
{c}
{title2=$HF^-(S^1\times S^2,\mathfrak s_k)$}
Let $\mathfrak s_k$ on $S^1\times S^2$ have $\langle c_1(\mathfrak s_k),[S^2]\rangle=2k\ne0$. The ordinary groups are
$$
HF^-(S^1\times S^2,\mathfrak s_k)=\mathbb Z[U]/(1-U^{|k|}),\qquad HF^\infty(S^1\times S^2,\mathfrak s_k)=\mathbb Z[U,U^{-1}]/(1-U^{|k|}),\qquad HF^+=\widehat{HF}=0.
$$
A 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.
Back to article page
