Heegaard Floer homology ($H{F}^{\circ }(Y,\mathfrak{s})$)

Heegaard Floer homology assigns groups $H{F}^{\circ }(Y,\mathfrak{s})$ to a closed oriented three-manifold with a Spin-c structure. A pointed Heegaard diagram of genus $g$ gives product Lagrangian tori ${\mathbb{T}}_{\alpha },{\mathbb{T}}_{\beta }\subset {\mathrm{Sym}}^g(\Sigma )$; the chain complex is generated by their intersection points and its differential counts Whitney disk classes represented by holomorphic disks. The minus, infinity, plus and hat versions use different powers or quotients of the variable $U$. The Heegaard Floer cobordism maps and the Mixed Heegaard Floer invariant relate these groups to smooth four-dimensional topology.

For a closed smooth oriented four-manifold $M$ with $b_2^{+}(M)>1$, remove two balls to obtain $W:S^3\to S^3$, and choose an Admissible cut for a Heegaard Floer mixed map. Positivity and naturality show that the minus map of $W_1$ lands in $H{F}_{\mathrm{red}}^{-}$, while the plus map of $W_2$ descends from $H{F}_{\mathrm{red}}^{+}$. ThusThe coefficient of the bottom plus generator in this map, applied to $U^r{\Theta }^{-}\otimes \zeta$, defines ${\Phi }_{M,\mathfrak{s}}(U^r\otimes \zeta )\in \mathbb{Z}$, up to the orientation sign, with $\zeta \in {\Lambda }^{\ast }(H_1(M)/\mathrm{torsion})$. It vanishes unless $2r+\mathrm{deg}\zeta =(c_1(\mathfrak{s}{)}^2-2\chi (M)-3\sigma (M))/4$. The construction is independent of the admissible cut by the cobordism composition and associativity laws. Two positive pieces explain the condition $b_2^{+}>1$.

If ${\Phi }_{M,\mathfrak{s}}\ne 0$ and a connected closed oriented embedded surface $S$ has positive genus and $S^2\ge 0$, thenBlow up $S^2$ points on $S$ and choose the $\pm 1$ exceptional determinant classes which increase the Chern pairing on its proper transform by $S^2$. The Blowup formula for the mixed Heegaard Floer invariant preserves nonvanishing, while the proper transform has square zero. A cut disjoint from it allows the relevant cobordism map to pass through $H{F}^{\pm }(S\times S^1,\mathfrak{s}{|}_{S\times S^1})$. These groups vanish when the Chern pairing exceeds $2g-2$, so the mixed map would then vanish, a contradiction. Reverse the surface orientation for the absolute value. The positive-genus condition is essential: a small nullhomologous sphere has square zero and zero pairing and would violate the displayed inequality.

Let $\tilde{M}=M\#{\overline{\mathbb{CP}}}^{,2}$, and let $E$ be the exceptional sphere. Extend a Spin-c structure by $c_1(\tilde{\mathfrak{s}})=c_1(\mathfrak{s})\pm \mathrm{PD}[E]$. Then ${\Phi }_{\tilde{M},\tilde{\mathfrak{s}}}={\Phi }_{M,\mathfrak{s}}$. More generally, an evaluation $⟨c_1(\tilde{\mathfrak{s}}),E⟩=\pm (2\ell +1)$ introduces the factor $U^{\ell (\ell +1)/2}$. This follows by putting the blowup on one side of an admissible cut: the punctured exceptional piece is negative definite, its minus and plus tower maps are multiplication by this power of $U$, and the cobordism composition law gives the formula. In the $\pm 1$ case the exponent is zero, so nonvanishing is preserved.

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.

A smooth oriented cobordism $W:Y_1\to Y_2$ with a Spin-c structure induces $U$-equivariant maps $F_{W,\mathfrak{s}}^{\circ }$ on the corresponding Heegaard Floer homology groups. A handle decomposition defines the maps using generator selection for one- and three-handles and holomorphic triangle counts for two-handles. Composition sums over extensions of the prescribed Spin-c structures. When the boundary determinant classes are torsion, the absolute grading shift isThe square is defined using the rational relative lift. If $b_2^{+}(W)>0$, the infinity map vanishes. Between rational homology spheres, when $b_1(W)=b_2^{+}(W)=0$, the infinity map is an isomorphism of Laurent towers.

An admissible cut splits a four-dimensional cobordism as $W=W_1{\cup }_N{W}_2$, with $b_2^{+}(W_1),b_2^{+}(W_2)>0$, and with the Mayer-Vietoris connecting map $H^1(N;\mathbb{Z})\to H^2(W;\mathbb{Z})$ equal to zero. The last condition removes the ambiguity in gluing Spin-c structures on the two pieces. Positivity makes both infinity cobordism maps vanish, allowing passage through the Reduced Heegaard Floer homology of $N$. A useful cut is the boundary of a neighborhood of a positive-square embedded surface, chosen so that its complement still has a positive direction. The nonzero normal Euler number makes $H^1$ of the neighborhood surject onto $H^1$ of its boundary, proving the connecting-map condition.

Use the convention that $C{F}^{-}$ is generated by $[\mathbf{x},i]$ with $i<0$, and $U[\mathbf{x},i]=[\mathbf{x},i-1]$. The chain complexes areThe last complex is generated by $[\mathbf{x},0]$. Equivalently it is $C{F}^{-}/UC{F}^{-}$ with its absolute grading shifted up by two. The short exact sequence $0\to C{F}^{-}\to C{F}^{\infty }\to C{F}^{+}\to 0$ gives the fundamental long exact sequence. For a rational homology sphere in each Spin-c structure, $H{F}^{\infty }\cong \mathbb{Z}[U,U^{-1}]$. The plus tower is ${\mathcal{T}}_d^{+}=\mathbb{Z}[U,U^{-1}]/U\mathbb{Z}[U]$, with bottom degree $d$; its corresponding minus tower is a free $\mathbb{Z}[U]$ module with top degree $d-2$.

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.

The reduced minus and plus groups areThe connecting homomorphism of the fundamental long exact sequence induces an isomorphism $\delta :H{F}_{\mathrm{red}}^{+}\to H{F}_{\mathrm{red}}^{-}$ of degree $-1$. This follows immediately from exactness: its image is the indicated kernel, and its kernel is precisely the image quotiented out on the plus side. These groups isolate the information lost by inverting $U$.

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.

For an admissible pointed Heegaard diagram, $C{F}^{-}$ is a free $\mathbb{Z}[U]$ chain complex with one generator for each point of ${\mathbb{T}}_{\alpha }\cap {\mathbb{T}}_{\beta }$. Each generator is an unordered selection containing exactly one intersection on each alpha circle and exactly one on each beta circle. After choosing coherent orientations,Here $\hat{\mathcal{M}}$ is the moduli space modulo strip translation, $\mu$ is the Maslov index, and $n_z$ is multiplicity at the basepoint. Holomorphic representatives have nonnegative domain multiplicities. The differential lowers grading by one and $U$ lowers it by two. Boundary strata of one-dimensional moduli spaces give the cancellation proving ${\partial }^2=0$. Its homology is $H{F}^{-}$.

Over $\mathbb{Z}$, holomorphic disk moduli spaces in a Heegaard Floer chain complex must be oriented compatibly with gluing: the orientation of a concatenated disk class is the product determinant-line orientation of its two factors. Boundary orientations of compactified one-dimensional moduli spaces then make broken disks cancel, yielding ${\partial }^2=0$. Isolated regular disks contribute $+1$ or $-1$. Changing the chosen orientation at an intersection generator changes the corresponding row and column signs of the differential and yields an isomorphic integral chain complex. Cobordism triangle counts require compatible orientation choices as well; an overall orientation ambiguity accounts for the usual sign ambiguity of the closed invariant.

An embedded positive domain which is a bigon or rectangle, has convex alternating alpha and beta corners, and contains no other selected coordinate in its interior, has Maslov index one. A bigon has Euler measure $1/2$ and two corner averages $1/4$; a rectangle has Euler measure zero and four corner averages $1/4$. For the usual regular complex structure, its holomorphic disk count modulo translation is $\pm 1$. For a bigon the Riemann map to the strip is unique up to translation. For a rectangle, the tautological correspondence represents the disk in the symmetric product by a branched double cover of the strip onto the rectangular domain: its conformal modulus fixes the branch parameter, and the only remaining freedom is translation. The resulting isolated regular disk gives one point in the oriented moduli space. The differential coefficient is $\pm U^{n_z}$, so a basepoint-free domain contributes a unit.

For generators in the same Spin-c structure and a Whitney disk class $\psi$ joining them,The ambiguity is the divisibility of $c_1(\mathfrak{s})$ on $H_2(Y)$; a torsion determinant class gives an integer relative grading. Adding a sphere class changes $\mu$ by two and $n_z$ by one, so it leaves the difference unchanged. For a domain $D$, the index can be calculated as $\mu (D)=e(D)+n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D)$, where $e$ is Euler measure and each $n_{\mathbf{x}}$ is the sum of averages of the four adjacent region multiplicities at the selected intersections.

For a torsion Spin-c structure, the relative grading of Heegaard Floer homology lifts canonically to a rational absolute grading, normalized by the bottom plus generator of $S^3$ having degree zero. Multiplication by $U$ has degree $-2$, and a Heegaard Floer cobordism map has degree $(c_1^2-2\chi -3\sigma )/4$. The correction term $d(Y,\mathfrak{s})$ is the lowest grading of the image of $H{F}^{\infty }$ in $H{F}^{+}$ for a rational homology sphere. A pure tower then has bottom plus degree $d$ and top minus degree $d-2$ in the $i<0$ minus convention. Keeping that shift explicit prevents errors of two in surgery and mixed-invariant calculations.

For intersection generators $\mathbf{x},\mathbf{y}$, a Whitney disk is a map of a disk into the symmetric product, with its two boundary semicircles mapped to the two product tori and its endpoints mapped to $\mathbf{x},\mathbf{y}$. Its homotopy class belongs to ${\pi }_2(\mathbf{x},\mathbf{y})$. Projecting it to the Heegaard surface gives an integral domain. With the surface oriented, the convention is ${\partial }_{\alpha }D=\mathbf{y}-\mathbf{x}$. The obstruction to a disk is the class $ϵ(\mathbf{x},\mathbf{y})\in H_1(Y)$ obtained by joining the points along alpha arcs and returning along beta arcs. In the Heegaard Floer setup, its vanishing is equivalent to equality of the generators' Spin-c structures; for the rational homology sphere diagrams this is equivalent to ${\pi }_2(\mathbf{x},\mathbf{y})\ne \varnothing$.