= Integer-surgery exact triangle in Heegaard Floer homology
{c}
For a knot $K$ in an integral homology three-sphere and $n>0$, fix a <Spin-c structure> $\mathfrak t_k$ on the $n$-surgery. Write $\mathfrak s_i$ for structures on zero-surgery with $\langle c_1(\mathfrak s_i),[\widehat F]\rangle=2i$, using a capped Seifert surface. In compatible labels the completed-minus triangle is
$$
\cdots\longrightarrow\mathbf{HF}^-(Y)\longrightarrow\bigoplus_{i\equiv k\ (\mathrm{mod}\ n)}\mathbf{HF}^-(Y_0,\mathfrak s_i)\longrightarrow\mathbf{HF}^-(Y_n,\mathfrak t_k)\longrightarrow\mathbf{HF}^-(Y)\longrightarrow\cdots.
$$
One can equally use the plus theory without completing. The first map is the congruence-restricted zero-surgery trace map; the last is the sum of reverse, orientation-reversed $n$-surgery trace maps. The middle map uses a Heegaard triple with an auxiliary lens-space generator. For $n=1$ that lens space is $S^3$, giving the usual three two-ended cobordism maps. For $n>1$ it is important to retain the auxiliary boundary and its fixed Spin-c structure. The <U-adic completion of Heegaard Floer homology> makes the sums of maps well-defined.
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