Heegaard Floer plus and infinity groups and absolute grading for rational homology spheres

A rational homology sphere is a closed connected oriented three-manifold $Y$ with $H_{\ast }(Y;\mathbb{Q})\cong H_{\ast }(S^3;\mathbb{Q})$. Let $\mathfrak{t}$ be a Spin${}^c$ structure on $Y$, meaning a principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c(3)$ lift of the oriented frame bundle. A pointed Heegaard diagram for $(Y,\mathfrak{t})$ consists of a closed oriented surface $\mathrm{\Sigma }$ of genus $g$, two systems $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_g)$ and $\mathit{\beta }=({\beta }_1,\dots ,{\beta }_g)$ of pairwise disjoint simple closed curves which bound complete disk systems in the two genus-$g$ handlebodies of a Heegaard splitting of $Y$, with every ${\alpha }_i$ transverse to every ${\beta }_j$, and a basepoint $z\in \mathrm{\Sigma }\setminus (\mathit{\alpha }\cup \mathit{\beta })$; the diagram is restricted to the generators assigned to $\mathfrak{t}$. Compactness of $\mathrm{\Sigma }$ makes the transverse intersection set finite. A periodic domain is an integral two-chain in the complementary regions whose boundary is an integral sum of attaching curves. The diagram is strongly admissible for $\mathfrak{t}$ when every nonzero periodic domain $P$ with $n_z(P)=0$ and $⟨c_1(\mathfrak{t}),[P]⟩\ge 0$ has at least one positive and one negative region coefficient.

For such a diagram, put ${\mathbb{T}}_{\alpha }={\alpha }_1\times \cdots \times {\alpha }_g$ and ${\mathbb{T}}_{\beta }={\beta }_1\times \cdots \times {\beta }_g$ in ${\mathrm{Sym}}^g(\mathrm{\Sigma })={\mathrm{\Sigma }}^g\mathrm{/}{S}_g$. A generator is an intersection point $\mathbf{x}\in {\mathbb{T}}_{\alpha }\cap {\mathbb{T}}_{\beta }$. A Whitney disk class $\varphi \in {\pi }_2(\mathbf{x},\mathbf{y})$ is a relative homotopy class of disks in ${\mathrm{Sym}}^g(\mathrm{\Sigma })$ with boundary arcs on the two product tori and endpoints $\mathbf{x},\mathbf{y}$; its Maslov index is $\mu (\varphi )$ and its local multiplicity at $z$ is $n_z(\varphi )$. For a regular almost-complex datum for which every relevant count is finite and the disk-count operator squares to zero, define $C{F}^{\mathrm{\infty }}(Y,\mathfrak{t})$ over ${\mathbb{F}}_2[U,U^{-1}]$ on generators $[\mathbf{x},i]$, $i\in \mathbb{Z}$, with differential

$${\mathrm{\partial }}^{\mathrm{\infty }}[\mathbf{x},i]=\sum _{\mathbf{y}}\sum _{\begin{array}{c}\varphi \in {\pi }_2(\mathbf{x},\mathbf{y})\\ \mu (\varphi )=1,n_z(\varphi )\ge 0\end{array}}{\mathrm{\#}}_2(\mathcal{M}(\varphi )\mathrm{/}\mathbb{R})[\mathbf{y},i-n_z(\varphi )]\mathrm{.}$$

Here ${\mathbb{F}}_2$ is the field with two elements, $U[\mathbf{x},i]=[\mathbf{x},i-1]$, and ${\mathrm{\#}}_2$ is the finite count modulo two. The submodule generated by $[\mathbf{x},i]$ with $i<0$ is $C{F}^{-}(Y,\mathfrak{t})$, $C{F}^{+}(Y,\mathfrak{t})=C{F}^{\mathrm{\infty }}\mathrm{/}C{F}^{-}$, and $H{F}^{\mathrm{\infty }}$ and $H{F}^{+}$ denote their homology when the differential is established. Relative Maslov gradings are fixed by the Fredholm index and the shift under $U$ is $-2$.

When the standard absolute rational Maslov grading is available, normalize it so that the generator of $H{F}^{\mathrm{\infty }}(S^3)$ has degree zero. For a smooth Spin${}^c$ cobordism $(W,\mathfrak{s})$ between rational homology spheres, the induced map is assigned degree shift

$$\frac{⟨c_1(\mathfrak{s}{)}^2⟩-2\chi (W)-3\sigma (W)}{4},$$

where $c_1(\mathfrak{s})$ is the first Chern class, $\chi (W)$ is Euler characteristic, $\sigma (W)$ is the signature, and its square is evaluated using the rational intersection form of $W$. This is a convention for a Floer package when constructed; it asserts neither existence of generic regular data nor that the disk-count differential squares to zero. Heegaard invariance, the absolute grading on general $(Y,\mathfrak{t})$, and cobordism-map properties are separate claims.

Spin-c structures as lifts of the oriented frame bundle

Ozsvath and Szabo, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Sections 2–4

Hom, Lecture notes on Heegaard Floer homology, Sections 2.1–2.2