= U-adic completion of Heegaard Floer homology
{c}
{title2=$\mathbf{HF}^-(Y)$}
Completing the minus <chain complex> in powers of $U$ replaces $\mathbb Z[U]$ by the <formal power series ring> $\mathbb Z[[U]]$; its homology is written $\mathbf{HF}^-$. This permits infinite sums of <Heegaard Floer cobordism maps> whose exponents tend to infinity, as occur in surgery formulas. A series with constant coefficient $\pm1$ is a unit: its inverse is found recursively, or by the geometric series for its positive-order part. Such a series need not have a polynomial inverse, which is why exact triangles involving sums over all Spin-c extensions require a completion in the minus theory. Individual homogeneous cobordism maps are still defined on the ordinary minus groups.
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