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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 $⟨c_1({\mathfrak{s}}_i),[\hat{F}]⟩=2i$, using a capped Seifert surface. In compatible labels the completed-minus triangle isOne 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.

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.

Use ordinary integer surgery coefficients, so the positive trace has intersection form $(n)$. Let ${\hat{F}}_0\subset K_0$ be the capped Seifert surface. Label the Spin-c structures byOn the trace $X_n:S^3\to K_n$, let ${\hat{F}}_n^2=n$ and label its structures ${\mathfrak{r}}_j$ by $⟨c_1({\mathfrak{r}}_j),[{\hat{F}}_n]⟩=2j-n$. Restriction identifies ${\mathfrak{t}}_k$ with $j\equiv k(\mathrm{mod}n)$. In these compatible labels the Integer-surgery exact triangle in Heegaard Floer homology isHere boldface denotes the U-adic completion of Heegaard Floer homology. Only finitely many zero-surgery structures contribute, by the genus bound on a capped Seifert surface, so the middle direct sum causes no difficulty. The infinite sums of trace maps do require completion. The printed minus notation should be understood in this completed sense; with the polynomial version, a sum over every extension need not even have values in $\mathbb{Z}[U]$. The plus version has the same ordering and needs no completion.

The two ordinary surgery traces give $A_k$ and $C_k$. For $X_0:S^3\to K_0$, $H^2(X_0)=\mathbb{Z}$, and its Spin-c structures ${\mathfrak{q}}_i$ have determinant evaluations $2i$ on the capped Seifert surface. Their restrictions are the unique structure on $S^3$ and ${\mathfrak{s}}_i$ on $K_0$. Thus the component of $A_k$ in the $i$th summand is $F_{X_0,{\mathfrak{q}}_i}^{-}$ for $i\equiv k$. For the orientation-reversed and turned-around trace $V_n:K_n\to S^3$, $H^2(V_n)=\mathbb{Z}$ and ${\hat{F}}_n^2=-n$. Its structures ${\mathfrak{r}}_j$ still have characteristic evaluations $2j-n$, restrict to ${\mathfrak{t}}_{j\mathrm{mod}n}$ on $K_n$ and to the unique structure on $S^3$, andCharacteristic parity explains the allowed evaluations; changing a structure by the generator of $H^2$ changes its evaluation by two, and the boundary restriction reduces the label modulo $n$.

For $n>1$, $B_k$ has an auxiliary lens-space input. More precisely, take the three-boundary surgery cobordism, or equivalently a two-ended cobordism $Z_n:K_0\#L_n\to K_n$, where $L_n$ is positive $n$-surgery on the unknot. Its incoming lens-space factor is fixed to the distinguished structure ${\mathfrak{u}}_0$ and its canonical top Floer generator ${\Theta }_{L_n}$. ThenThis describes the boundary restrictions exactly, including the boundary that must not be suppressed. A concrete Kirby diagram for $Z_n$ starts with the split incoming surgeries of coefficients $0$ and $n$, then attaches a zero-framed circle linking each incoming component once. Its full linking matrix is $\left(\begin{array}{ccc}0& 0& 1\\ 0& n& 1\\ 1& 1& 0\end{array}\right)$. Quotienting cohomology by the two incoming surgery columns gives $H^2(Z_n)=\mathbb{Z}\oplus \mathbb{Z}/n$. Accordingly its Spin-c structures may be labeled $(i,q)$ with incoming restrictions $({\mathfrak{s}}_i,{\mathfrak{u}}_q)$ and outgoing restriction ${\mathfrak{t}}_{i-q}$. In particular, fixing $q=0$ gives exactly one extension for each $i$, and its outgoing class is $i\mathrm{mod}n$. Thus $B_k$ is a Heegaard Floer cobordism map with the fixed auxiliary generator, rather than an unqualified map of a trace from $K_0$ to $K_n$. When $n=1$, $L_n=S^3$ and all three arrows become the familiar sums of ordinary two-ended Heegaard Floer cobordism maps.