U-adic completion of Heegaard Floer homology (${\mathbf{HF}}^{-}(Y)$)

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 $\pm 1$ 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.