We’ll assume everything is smooth and oriented.

Note that \({\mathbb{Z}}_2\) can be replaced with \({\mathbb{Z}}\), but it’s technical and we won’t discuss it here. For the first half of the course, we’ll just discuss \(\widehat{HF}\), and we’ll discuss the latter 3 in the second half.

These invariants can be used to compute the Thurston seminorm of a 3-manifold:

Note that this can’t be a norm, since if \({\mathbb{S}}^2, {\mathbb{T}}^2 \in [S] \implies {\left\lVert {\alpha } \right\rVert}= 0\).

The Thurston norm \({\left\lVert {a} \right\rVert}\) can be computed from this data by considering a perturbed version of \(\widehat{HF}\), denoted \(\underline{\widehat{HF}}\), in the following way: taking the first Chern class \(c_1(\mathfrak{s}) \in H^2(M)\) (which can be associated to every 2-dimensional vector bundle), we have \begin{align*} {\left\lVert { \alpha} \right\rVert} = \max_{\underline{\widehat{HF}}(M, \mathfrak{s} ) \neq 0 } {\left\lvert {{\left\langle { c_1(\mathfrak{s}) },~{ \alpha } \right\rangle} } \right\rvert} .\end{align*}

This gives obstructions for two of the following important properties of contact structures:

• Being overtwisted, or
• Being Stein fillable.

We’ll also discuss similar invariants for knots that were created after these invariants for manifolds.

There are better lower bounds for \(u(K)\) defined using \(\widehat{HFK}\) which are not lower bounds for the slice genus. There are also other lower bounds for the slice genus with different names (see Jen Hom’s survey), some of which are stronger than \(\tau\).

Another application of having these lower bounds is that we can construct exotic (or fake) \({\mathbb{R}}^4\)s, i.e. 4-manifolds \(X\) homeomorphic to \({\mathbb{R}}^4\) but not diffeomorphic to \({\mathbb{R}}^4\).

All of these invariants work nicely in a \((3+1){\hbox{-}}\)TQFT: we have invariants of 3-manifolds \(M_i\) and knots in them, so we can talk about cobordisms between them: \(W^4\) a compact oriented 4-manifold with \({{\partial}}W^4 = -M_1 {\coprod}M_2\).

Osvath-Szabó define a map \begin{align*} F_{W, t}^{-}: HF^{-}(M_1, { \left.{{t}} \right|_{{M_1}} } ) \to HF^{-}(M_2, { \left.{{t}} \right|_{{M_2}} }) \end{align*} using \(t\) coming from the splitting of spin\(^c\) structure which yields an invariant of closed 4-manifolds referred to as mixed invariants.

Similarly, if we have knots in 3-manifolds we can define a cobordism \((M_1, K_1) \to (M_2, K_2)\) as \((W^4, F)\) where \(W^4\) is a cobordism \(M_1\to M_2\) and \(F\hookrightarrow W\) is a smoothly embedded surface that is a cobordism from \(K_1\to K_2\) with \(F \cap M_i = K_i\) and \({{\partial}}F = -K_1 {\coprod}K_2\).

This similarly yields a map

\begin{align*} F_{W, F t}^{-}: HF^{-}(M_1, K_1, { \left.{{t}} \right|_{{M_1}} } ) \to HF^{-}(M_2, K_2, { \left.{{t}} \right|_{{M_2}} }) \end{align*}

This smoothly embedded surface in the middle can be used to study other smoothly embedded surfaces in 4-manifolds, which has been done recently.

For Morse Theory, there are some good exercises in Audin’s book – essentially anything other than the existence questions. The first 8 look good on p. 18.

Today:

1. Overview of the construction of HF, and

2. A discussion of Morse Theory.

First goal: discuss how the name “Heegard” fits in.

This is essentially an infinite-dimensional version of Morse homology.

If \(L_1 \cap L_2\) is finitely many points, case we can define a chain complex \begin{align*} CF(M^{2n}, L_1, L_2) \mathrel{\vcenter{:}}={\mathbb{Z}}_2[L_1 \cap L_2] ,\end{align*} the \({\mathbb{Z}}_2{\hbox{-}}\)vector space generated by the intersection points of the Lagrangian submanifolds. We’ll define a differential by essentially counting discs between intersection points:

We’ll want to write \({{\partial}}x = c_y y + \cdots\) where \(c_y\) is some coefficient. How do we compute it? In this case, we have half of the boundary on \(L_1\) and half is on \(L_2\)

So we can the number of holomorphic discs from \(x\) to \(y\). We’ll get \({\partial}^2 = 0 \iff \operatorname{im}{\partial}\subset \ker {\partial}\), and \(HF\) will be kernels modulo images. In more detail, we’ll have \begin{align*} {{\partial}}x = \sum_y \sum_{\mu(\varphi) = 1} \# \widehat{\mathcal{M}} (\varphi)y && \widehat{\mathcal{M}}(\varphi) = \mathcal{M}(\varphi) / {\mathbb{R}} \end{align*} where \(\widehat{\mathcal{M}}\) will (in good cases) be a 1-dimensional manifold with finitely many points. Note that it’s not necessarily true that \(CF\) has a grading!

Given a 3-manifold \(M^3\), we’ll associate a Heegard diagram \(\Sigma, \alpha, \beta\). Note the \(g{\hbox{-}}\)element symmetric group acts on \(\prod_{i=1}^g \Sigma\) by permuting the \(g\) coordinates, so we can define \(\operatorname{Sym}^g(\Sigma) \mathrel{\vcenter{:}}=\prod_{i=1}^g \Sigma / S_g\).

Write \({\mathbb{T}}_{\alpha} \mathrel{\vcenter{:}}=\prod_{i=1}^g \alpha_i \subseteq \prod_{i=1}^g \Sigma\) for a \(g{\hbox{-}}\)dimensional torus; this admits a quotient map to \(\operatorname{Sym}^g(\Sigma)\). We can repeat this to obtain \({\mathbb{T}}_{\beta}\). Then \(HF^{-}(M)\) will be a variation of Lagrangian Floer Homology for \((\operatorname{Sym}^g(\Sigma), {\mathbb{T}}_{\alpha}, {\mathbb{T}}_{\beta} )\).

Note that the diffeomorphism is entirely determined by the image of the curve \(\alpha\) . The Knot Floer chain complex of \(K\) will allow us to compute any flavor \(HF^{-}M _{\varphi} (K)\) of Floer homology. Why is this important: any closed 3-manifold is surgery on a link in \(S^3\). However there are many more computational tools available here and not in the other theories: combinatorial approaches to compute, exact sequences, bordered Floer homology.

Next time: we’ll talk about “integer surgeries.”

Let \(M^n\) be a smooth closed manifold, then the goal is to study the topology of \(M\) by studying smooth functions \(f \in C^ \infty (M, {\mathbb{R}})\). We’ll need \(f\) to be generic in a sense we’ll discuss later.

This is only well-defined at critical points (check!). Note that we need \(\tilde w\) so that \(\tilde w \cdot f\) is again a function (and not a number) which can be differentiated again. You can also take e.g. \(\tilde v \cdot (\tilde w \cdot f)\), differentiating with respect to the vector field instead of just the vector \(v\), but we’re plugging in \(p\) in either case.

If you fix a coordinate chart in a neighborhood of \(p\), then the bilinear form is represented by a matrix given by \begin{align*} (d^2 f)_p = H_p = \qty{ {\frac{\partial ^2}{\partial x_j {\partial}x_i}\,}(p)}_{ij} .\end{align*}

In local coordinates, we can write \(v = \sum_{i=1}^n a_i {\frac{\partial }{\partial x_i}\,}\) and \(w = \sum_{i=1}^n b_i {\frac{\partial }{\partial x_i}\,}\), and thus \begin{align*} (d^2 f)_p(v, w) = \mathbf{b}^t H_p \mathbf{a} = \sum_{1 \leq i,j \leq n} a_i b_j {\frac{\partial ^2 f}{\partial x_i {\partial}x_j}\,}(p) .\end{align*}

We’ll see that almost every smooth function is Morse, and these are preferable since they have a simple and predictable structure near critical points and don’t do anything interesting elsewhere.

We have \begin{align*} H_p = \begin{bmatrix} -2&&&&&&\\ &\ddots&&&&&\\ &&-2&&&&\\ &&&2&&&\\ &&&&\ddots&&\\ &&&&&2&\\ &&&&&&2 \end{bmatrix} = -2 I_{\lambda} \oplus 2 I_{n- \lambda} .\end{align*}

If \(\lambda=n\)??

??

Define \(M_a \mathrel{\vcenter{:}}= f ^{-1} ((- \infty , a])\); we then want to consider how \(M_a\) changes as \(a\) changes:

We have \begin{align*} df( \nabla f) = g(\nabla f, \nabla f) = {\left\lVert {\nabla f} \right\rVert}^2 .\end{align*}

Goal: we want to use Morse functions (smooth, nondegenerate critical points) to study the topology of \(M\). Recall that the torus had 4 critical points,

We defined the index as the number of negative eigenvalues of the Hessian matrix. Here the highest index will be the dimension of the manifold, and by the Morse lemma the two intermediate critical points will be index 1.

We want to use the Morse function to decompose the manifold, so we consider \(M_a \mathrel{\vcenter{:}}= f ^{-1} ((- \infty , a ])\). If \(f ^{-1} [a, b]\) does not contain a critical point, then \(M_a \cong M_b\) and \(f ^{-1} (a) \cong f ^{-1} (b)\). So taking \(M_{1/2}\) and \(M_{3/4}\) here both yield discs:

Passing through critical points does change the manifold, though:

There is a deformation retract \(M_b \to M_a \cup C_ \lambda\), where \(C_ \lambda\) is a \(\lambda{\hbox{-}}\)cell given by \(D_ \lambda \times\left\{{0}\right\}\). For example:

We’ll use this to define a chain complex \(C_*(f, g)\) where \(g\) is a chosen metric, define a differential, and use this to define a homology theory. For notation, we’ll write \(\operatorname{crit}(f)\) as the set of critical points of \(f\), and given \(p, q\in \operatorname{crit}(f)\) with \(\gamma\) a trajectory running from \(p\) to \(q\), we have \begin{align*} W^u(p) \cap W^s(q) = \left\{{ \gamma(t) {~\mathrel{\Big|}~} \gamma(t) \overset{t\to -\infty }\to p,\, \gamma(t) \overset{t\to +\infty }\to q }\right\} .\end{align*}

This means that metrics can be perturbed to become Morse-Smale.

If \(\operatorname{ind}(p) - \operatorname{ind}(q) = 1\), then \(\dim \mathcal{M}(p, q) = \operatorname{ind}(p) - \operatorname{ind}(q) - 1 = 0\), making \(\mathcal{M}(p, q)\) a compact 0-dimensional manifold, which is thus finitely many points, meaning there are only finitely many trajectories connecting \(p\to q\) and it becomes possible to define a Morse complex.

Next time we will work on proving this.

Last time: defined the Morse complex. Assumed \((f, g)\) was a Morse-Smale pair, where \(f\) is a Morse function and \(g\) is a Riemannian metric, and this guarantees that if \(p, q\in \operatorname{crit}(f)\) with \(\operatorname{ind}(p) - \operatorname{ind}(q) = 1\), then (among other things) there are finitely many gradient trajectories \(p\leadsto q\). We denoted this \(\mathcal{M}(p, q)\). The chain complex was defined by \(C_i(f, g) \mathrel{\vcenter{:}}=\bigoplus_{\operatorname{ind}(p) = i} {\mathbb{Z}}_2 \left\langle{ p }\right\rangle\) with differential \({{\partial}}_i: C_i \to C_{i-1}\) was defined by sending an index \(i\) critical point \(p\) to \(\sum_{\operatorname{ind}(q) = i-1} \# \mathcal{M}(p, q) q \pmod 2\).

Suppose \(L_0^n, L_1^n \subset M^{2n}\) are compact with \(L_0 \pitchfork L_1\), so the intersection is finitely many points.

We can do Morse theory on the space of paths between them: \begin{align*} \mathcal{P}(L_0, L_1) \mathrel{\vcenter{:}}=\left\{{ \gamma: I\to M {~\mathrel{\Big|}~}\gamma(0) \in L_0, \gamma(1) \in L_1}\right\} .\end{align*}

We’ll find analogs of Morse functions on \(P(L_0, L_1)\) such that the critical points are constant paths, i.e. \(L_0 \cap L_1\). The Morse inequalities then gives bounds on the number of intersection points between \(L_0\) and \(L_1\).

The general setup for next time: we’ll have \((M^{2n}, \omega)\) a symplectic manifold, a pair \(L_0, L_1 \subset M\) such that \(L_0 \pitchfork L_1\), and we want to do Morse Homology on \(\mathcal{P}(L_0, L_1)\).

We’re working with a symplectic manifold, i.e. a pair \((M^{2n}, \omega)\) where \(\omega \in \Omega^2\) is closed, i.e. \(d \omega = 0\), and nondegenerate, i.e. \(\bigvee^n \omega \neq 0\). We were also consider \(L^n_0, L^n_1 \subset M\) Lagrangian submanifolds, i.e. \({ \left.{{ \omega }} \right|_{{L_i}} } = 0\). The goal is to do something like Morse homology on \(\mathcal{P}(L_0, L_1)\) where the critical points corresponds to intersection points \(L_0 \cap L_1\), where we’ll assume \(L_0 \pitchfork L_1\).

The functional \(f\) is defined on the universal cover \(\overline{\mathcal{P}}(L_0, L_1) \to {\mathbb{R}}\). We can get around knowing much about \(f\) because we only ever need derivatives \(df\) and a metric \(g\) on the path space to talk about the gradient \(\nabla_g f\). We’ll define a 1-form \(\alpha: T \mathcal{P}(L_0, L_1) \to {\mathbb{R}}\), where we can define this tangent space as \(T_{\gamma} \mathcal{P}(L_0, L_1)\) where \(\gamma(s): I \to M\). Set \(u(s, t)\) to be a path from \(\gamma\) to \(\gamma'\) where \(u(s, 0) = \gamma\) and \(u(s, 1) = \gamma'\) and \({\frac{\partial u}{\partial t}\,}\Big|_{t=0}\), which is a tangent vector to \(\gamma\) and thus \({\frac{\partial u}{\partial t}\,}(s, 0) \in T_{\gamma(s)}M\).

Upshot: tangent vectors in \(T_{\gamma} \mathcal{P}(L_0, L_1)\) are given by \(\xi(s) \in T_{\gamma(s)}M\) for every \(s \in I\), i.e. a way to push the path off of itself to obtain a new path.

We can thus define \begin{align*} \alpha: T\mathcal{P}(L_0, L_1) &\to {\mathbb{R}}\\ ( \gamma, \xi \in T_{\gamma} \mathcal{P}) &\mapsto \alpha_{\gamma}{\xi} \mathrel{\vcenter{:}}=\int_0^1 \omega( \dot{\gamma}(s), \xi(s)) ds .\end{align*} Does this have the property we want? I.e. is it zero when \(\gamma\) is the constant path?

Recall that we had a symplectic manifold \((M^{2n}, \omega)\) with \(L_0, L_1 \subset M\) two Lagrangians. We wanted to do something like Morse theory on \(\mathcal{P}(L_0, L_1)\).

What ingredients do we need?

• Something to replace \(-df\): \(\alpha\)

• Something to replace the vector field \(-\nabla\): we defined a metric \(g^\mathcal{P}\) using \(\alpha\)

To define \(\alpha\) we needed to look at \begin{align*} T_{ \gamma} \mathcal{P}= \left\{{ \xi: I\to TM {~\mathrel{\Big|}~}\xi(s) \in T_{\gamma(s)}M }\right\} ,\end{align*} which is like a collection of tangent vectors along \(\gamma\) giving a way to deform the path. Since \(\alpha\in \Omega^1(\mathcal{P})\), for any \(\gamma\) it induces a map \begin{align*} T_{ \gamma} M &\xrightarrow{\alpha} {\mathbb{R}}\\ \xi &\mapsto \alpha_{ \gamma} (\xi) \mathrel{\vcenter{:}}=\int_0^1 \omega( \dot{ \gamma}, \xi)\,ds .\end{align*}

We wanted to define the gradient, for which we needed a metric on \(\mathcal{P}\). We did this by lifting a metric from \(M\). Pick an almost complex structure \(J\) compatible with \(\omega\), then this yields a Riemannian metric defined by \(g(v, w) = \omega(v, Jw)\). Then we can define \begin{align*} g^\mathcal{P}_{\gamma}(\xi, \eta) \mathrel{\vcenter{:}}=\int_0^1 g( \xi(s), \eta(s) )\,ds .\end{align*} We used this to compute the vector field \(-\operatorname{grad}_{\gamma} J \cdot{\gamma}(s)\). What are its trajectories? These are paths of paths \(u(s, t) \mathrel{\vcenter{:}}= u_t(s)\) such that \({\frac{\partial }{\partial t}\,} u_t(s) = J {\frac{\partial }{\partial s}\,} u_t\). We thus get an equation \begin{align*} {\frac{\partial u}{\partial t}\,}(s, t) = J {\frac{\partial u}{\partial s}\,}(s, t) .\end{align*}

For \(x, y \in L_0 \cap L_1\) trajectories connecting \(x\) to \(y\), we’ll write this as \begin{align*} \mathcal{M}(x, y) \mathrel{\vcenter{:}}=\left\{{ u(s, t) :[0,1] \times{\mathbb{R}}\to M \substack{ u(0, t) \in L_0 \\ u(1, t) \in L_1 \\ u(s, t) \overset{t\to - \infty }\to x \\ u(s, t) \overset{t\to \infty }\to y \\ {\frac{\partial u}{\partial t}\,} = J {\frac{\partial u}{\partial s}\,} } }\right\} .\end{align*}

We can modify this PDE to make things look familiar: multiply both sides with \(J\) to obtain \begin{align*} J {\frac{\partial u}{\partial t}\,} = J^2 {\frac{\partial u}{\partial s}\,} \implies J {\frac{\partial u}{\partial t}\,} = - {\frac{\partial u}{\partial s}\,} \implies {\frac{\partial u}{\partial s}\,} + J {\frac{\partial u}{\partial t}\,} = 0 ,\end{align*} which is the Cauchy-Riemann equation.

Schematically, the situation is the following:

Using the Riemann mapping theorem, the strip on the left-hand side is biholomorphic to \({\mathbb{D}}\subseteq {\mathbb{C}}\) with \(\pm i\) removed:

Due to the limit conditions at infinity in the strip, we can extend \(u\) to a \(J{\hbox{-}}\)holomorphic map from the entire disc by sending \(i\mapsto y\) and \(-i\mapsto x\).

In Morse homology, we have an \({\mathbb{R}}\) action on the moduli space of trajectories, and that also shows up here. Here \({\mathbb{R}}\curvearrowright\mathcal{M}(x, y)\) by \(u(s, t) \xrightarrow{c} u_c(s, t) \mathrel{\vcenter{:}}= u(s, t+c)\), noting that translating the strip from above still yields a solution.

When is the intersection count \(\# \widehat{\mathcal{M}}(x, y)\) well-defined? In Morse homology, we have two conditions:

1. \((f, g)\) is Morse-Smale, to ensure that the moduli spaces are smooth manifolds (using Sard’s theorem)

2. \(\operatorname{ind}(x) - \operatorname{ind}(y) = 1\), ensuring \(\mathcal{M}(x, y)\) is 1-dimensional

3. Compactness of \(\widehat{\mathcal{M}}(x, y)\) when 1 and 2 hold.

These were enough to guarantee that \(\widehat{ \mathcal{M}} (x, y)\) was a smooth compact 0-dimensional manifold, which allowed for point counts. In Lagrangian Floer homology, we have the following replacements:

For 2 (indices): Recall that the index in Morse homology was the dimension of the negative eigenspace of the Hessian, but we’re in infinite dimensions here. So we won’t have a well-defined index, but we’ll have something that can replace the difference of indices: the Maslov index \(\mu(x, y)\), the expected dimension of \(\mathcal{M}(x, y)\). To actually have this be the dimension will require some conditions, so it’s not always true. This will be the index of some elliptic operator defined using the Cauchy-Riemann equations.

For 1 (transversality): We’ll need some version of transversality, which will imply that for a generic \(J\) that \(\mathcal{M}(x, y)\) is smooth.

For 3 (compactness): We’ll use Gromov compactness and some extra topological assumptions, which will imply that \(\widehat{ \mathcal{M}}(x, y), \mathcal{M}(x, y)\) are both compact.

Taken together, these will make the point-count well-defined.

In order for this to be a chain complex, we’ll need \({{\partial}}^2 = 0\). We’ll look at when \(\mu(x, y) = 2\), and we’ll compactify \(\widehat{ \mathcal{M}}(x, y)\) in order to show this holds. Gromov’s compactness will give us \begin{align*} {{\partial}}\overline{ \mathcal{M}(x, y) } = \bigcup_{\mu(x,z) = \mu(z, y) = 1} \mathcal{M}(x, z) \times\mathcal{M}(z, y) ,\end{align*} much like the broken trajectories from Morse homology. Here we’ll need to add in broken \(J{\hbox{-}}\)holomorphic discs:

Using the same argument as in Morse homology, we can obtain \({{\partial}}^2 = 0\).

Goal: use this as an invariant of closed 3-manifolds in the form of Lagrangian Floer homology, defined by Osvath-Szabo. Note that Floer’s theorem requires topological assumptions which make the homology well-defined, but we don’t have these available in the HF setup. In particular, the assumptions on \(\pi_2\) won’t hold.

Goal: we want to use Lagrangian Floer homology to defined invariants of closed 3-manifolds, where here closed means that \({{\partial}}M^3 = \emptyset\). One example of Lagrangian Floer homology is Heegard Floer homology. We’ll want some symplectic manifold with two Lagrangian submanifolds. Oszvath-Szabo used a 2-dimensional description of closed 3-manifolds called Heegard diagrams. We’ll need Heegard splittings to define these, and handlebodies to define the splittings.

There are some moves to relate different Heegard splittings.

Last time we saw that \(M_3 = H_1 {\coprod}_{\varphi} H_2\) as two handlebodies glued along their boundary by a diffeomorphism \(\varphi: {{\partial}}H_1 \to {{\partial}}H_2\). This is referred to as a Heegard splitting for \(M\). We can specify a genus \(g\) handlebody as \(( \Sigma, \left\{{ \gamma_1, \cdots \gamma_g }\right\}\) where \(\Sigma\setminus\left\{{ \gamma_1, \cdots, \gamma_g }\right\}\) is connected and each \(\gamma_i\) bounds a disc in \(H\).

Moreover, we can go backwards: given such data, we can build a handlebody \(H\) by

1. Thickening \(\Sigma\) to obtain \(\Sigma \times[0, 1]\) This yields \({{\partial}}( \Sigma \times[0, 1] ) = (\Sigma\times\left\{{0}\right\} ) {\coprod}(\Sigma\times\left\{{1}\right\} )\).

2. Attach thickened discs to \(\gamma_i \times\left\{{0}\right\}\). This makes the boundary \((\Sigma \times\left\{{1}\right\} ) {\coprod}S_2\)

3. Fill in the \(S^2\) boundary with a \(B^3\).

Note that given \((\Sigma, \alpha, \beta\) we can construct \(M\) in the following way:

• \((\Sigma, \alpha\) builds \(H_ \alpha\) with \({{\partial}}H_{\alpha} = \Sigma\).
• \((\Sigma, \beta\) builds \(H_ \beta\) with \({{\partial}}H_{\beta} = \Sigma\).

Note that critical points can be used to compute the Euler characteristic, using the fact the \(\chi(C) = \chi(H_*(C))\), i.e. it can be computed on dimensions of chains or ranks of homology, along with the fact that Morse homology is isomorphic to singular homology. So e.g. for a 3-manifold \(M^3\), we can show \begin{align*} \chi(M^3) &= \sum_{i=0}^3 {\operatorname{rank}}H_i \\ &= \sum_{i=0}^3 {\operatorname{rank}}CM_i \\ &= 1 - \# \operatorname{crit}_1(f) + \# \operatorname{crit}_2(f) - 1 \\ &= 0 ,\end{align*} since the number of index 2 and index 3 critical points will be the same.

Let \(M^3\) be a closed 3-manifold, then there is a Heegard splitting \begin{align*} (\Sigma_g, \alpha = \left\{{ \alpha_1, \cdots, \alpha_g }\right\}, \beta = \left\{{ \beta_1, \cdots, \beta_g }\right\} =( \Sigma_g, H_ \alpha, H_ \beta) && {{\partial}}(H_ \alpha) = {{\partial}}( H_ \beta) = \Sigma ,\end{align*} where \(M^3 = H_{ \alpha} \coprod_{ \Sigma} H_ \beta\) and \(g\) is the genus of \(HD\). We refer to \(\Sigma\) as a Heegard surface, and this set of data as a Heegard diagram.

We’ll define \(\operatorname{Sym}^g( \Sigma)\) by letting \(S_g \curvearrowright\Sigma^{\times g}\) where if \(\varphi\in S_g\) we set \(\varphi(x_1, \cdots, x_g) = x_{ \varphi(1)}, \cdots, x_{ \varphi(g) }\). Then set \(\Sigma^{\times g} \mathrel{\vcenter{:}}=\Sigma^{\times g} / S_g\). Why does this yield a smooth manifold? Is this action free? The diagonal \(D \subseteq \Sigma^{\times g}\) consists of the points with at least 2 equal coordinates, and it’s easy to see that \(S_g\curvearrowright D\) can not be free. However, this still yields a smooth submanifold!

\(\operatorname{Sym}^g( \Sigma)\) is a complex manifold of complex dimension \(g\) (or real dimension \(2g\)). We want to find half-dimensional submanifolds to do Lagrangian-Floer homology. Using the Heegard splitting, write \({\mathbb{T}}_ \alpha \mathrel{\vcenter{:}}=\prod_{i=1}^g \alpha_i \subset \Sigma^{\times g}\), which is a \(g{\hbox{-}}\)dimensional torus such that \({\mathbb{T}}_ \alpha \cap D = \emptyset\) since the \(\alpha_i\) are pairwise disjoint. Composing the inclusion above with \(\pi\), we can note that the action of \(S^g\) is free away from the diagonal \(D\), so this composition is an embedding \({\mathbb{T}}_ \alpha \hookrightarrow\operatorname{Sym}^g( \Sigma)\). Similarly, \({\mathbb{T}}_ \beta\mathrel{\vcenter{:}}=\prod_{i=1}^g \beta_i \hookrightarrow\operatorname{Sym}^g( \Sigma)\).

Note that we’re only working with complex structures now, and haven’t upgraded it to a symplectic structure yet. But we don’t really need this to count holomorphic discs. Lagrangians \(L\) were defined as submanifolds where \({ \left.{{\omega}} \right|_{{L}} } = 0\), how do we do this without a symplectic form?

We’ll write \(\Delta \mathrel{\vcenter{:}}=\pi(D) \subseteq \operatorname{Sym}^g( \Sigma)\). Note that if \(\alpha\pitchfork\beta\), then \({\mathbb{T}}_ \alpha \pitchfork{\mathbb{T}}_ \beta\). Any intersection point \(x \in {\mathbb{T}}_{\alpha} \cap{\mathbb{T}}_{\beta}\) is of the form \(x = \left\{{ x_1, \cdots, x_g}\right\} \subseteq \Sigma\) such that each \(\alpha_i, \beta_j\) contain exactly one of the coordinates of \(x\).

For Lagrangian Floer homology, we’ll have a triple \((\operatorname{Sym}^g(\Sigma), {\mathbb{T}}_{ \alpha}, {\mathbb{T}}_{\beta} )\). We’ll define \begin{align*} CF( \Sigma, \alpha, \beta) \mathrel{\vcenter{:}}=\bigoplus_{x\in {\mathbb{T}}_{\alpha} \cap{\mathbb{T}}_{\beta} } {\mathbb{Z}}/2{\mathbb{Z}}\left\langle{ x }\right\rangle \\ \\ {{\partial}}(x) \mathrel{\vcenter{:}}=\sum_{y \in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{\beta}, \mu = 1} \# \widehat{\mathcal{M}} y .\end{align*}

We’ll first figure out how to count continuous discs up to homotopy classes, since holomorphic discs are much more restrictive. We’ll see that \(\pi_2\) plays a role, and define the topology of \(\operatorname{Sym}^g\).

Today: topology of symmetric product spaces \(\operatorname{Sym}^g\). We had an assignment \begin{align*} ( \Sigma_g, \alpha, \beta) &\mapsto ( \operatorname{Sym}^g( \Sigma), {\mathbb{T}}_ \alpha, {\mathbb{T}}_ \beta) ,\end{align*} where if \(\alpha, \beta\) are all transverse then so far \({\mathbb{T}}_ \alpha, {\mathbb{T}}_ \beta\), since e.g. \({\mathbb{T}}_ \alpha = \prod_{i=1}^g \alpha_i\). We wanted to define a chain complex \begin{align*} CF( \sigma, \alpha, \beta) \mathrel{\vcenter{:}}=\bigoplus _{x\in {\mathbb{T}}_ \alpha \cap{\mathbb{T}}_{ \beta } } {\mathbb{Z}}/2{\mathbb{Z}}\left\langle{ x }\right\rangle \\ {{\partial}}x \mathrel{\vcenter{:}}=\sum_{ \substack{ y \in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta } \\ \mu(x, y) = 1} } \# \mathcal{M}(x, y) y ,\end{align*} where \(\mu\) is the Maslov index and we want to count holomorphic discs. We’ll first talk about continuous (topological) discs.

For a proof of the first isomorphism, see Lemma 2.6 in [OSZ04a?]. Idea of proof for the second isomorphism: we’ll define a map \begin{align*} \iota: H_1 ( \Sigma) &\to H_1( \operatorname{Sym}^g( \Sigma) ) \\ x &\mapsto \left\{{ x, z, \cdots, z }\right\} ,\end{align*} for some fixed \(z \in \Sigma\), along with its inverse. Note that we’re identifying an embedding \(\iota( \Sigma ) = \Sigma \times\left\{{ z }\right\}^{\times g-1} \subseteq \operatorname{Sym}^g( \Sigma)\). Now define \(j \mathrel{\vcenter{:}}=\iota_*\) the induced map on homology. \begin{align*} j: H_1( \operatorname{Sym}^g (\Sigma) ) \to H_1 ( \Sigma) \\ .\end{align*} Picking a loop \(\gamma: S^1 \to \operatorname{Sym}^g( \Sigma )\), note that \(\Delta\subset \operatorname{Sym}^g( \Sigma)\) has codimension 2, and so we can perturb \(\gamma\) to be disjoint from \(\Delta\). We can arrange so that \(\gamma\) is the union of \(g\) paths \(\gamma_1, \cdots, \gamma_g\) such that each \(\gamma_i\) connects \(x_i \in \gamma(0)\) to \(x_{ \sigma(i) } \in \gamma(0)\) where \(\gamma_0 = \left\{{ x_1, \cdots, x_g }\right\}\) and \(\sigma\in S_g\) is a permutation.

For example, for \(g=3\):

Then \(\left\{{ \gamma_1(t), \gamma_2(t), \gamma_3(t) }\right\}\) is a loop from \(\gamma(0) \to \gamma(0) \in \operatorname{Sym}^3( \Sigma)\).

This means that \(\bigcup_{i=1}^g \gamma_i\) is a 1-cycle in \(\Sigma\), and thus \([ \cup g_i ] \in H_1( \Sigma)\). So we’ll define this as \(j([ \gamma ]) = [ \cup\gamma_i ]\).

Let \(M \mathrel{\vcenter{:}}=\left\{{ (\mathbf{x}, y) {~\mathrel{\Big|}~}\mathbf{x} \in \operatorname{Sym}^g( \Sigma), y\in \mathbf{x} }\right\}\), then we’ll define a \(g:1\) branched cover away from \(\pi ^{-1} \Delta\) that yields a fiber bundle:

Link to Diagram

This can be restricted to \(M \setminus\pi ^{-1} (\Delta) \xrightarrow{g:1} \operatorname{Sym}^g( \Sigma) \setminus\Delta\). Here \(j([ \gamma ]) = [ \pi_2 \circ \gamma]\) and \(j \circ \iota_* = \one\).

The generator comes from hyperelliptic involution:

Then consider the quotient \(\Sigma / \tau\). To identify this quotient, since the top half is identified with the bottom half, we can first forget about the bottom half, and then forget about half of the arcs along the axis of rotation:

Note that this results in a copy of \(S^2\). We can define a map \begin{align*} \Sigma &\to \Sigma^{\times g} \\ x &\mapsto (x, \tau(x), z, \cdots, z) .\end{align*} This extends to a map to \(\operatorname{Sym}^g( \Sigma)\), since \(\tau(x) \mapsto (\tau(x), x, z, \cdots, z)\) and these will be equal in \(\operatorname{Sym}^g\). So we can factor this through the quotient from above:

We can find obstructions to holomorphic discs by just looking at the topology. For \(x, y\in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta}\), choose two paths connecting them: \begin{align*} a: I &\to {\mathbb{T}}_{ \alpha}\\ b: I &\to {\mathbb{T}}_{ \beta} .\end{align*}

We can consider the homology class \([a-b]\) to investigate \(\pi_1\). This is well-defined as a loop \begin{align*} \varepsilon(x, y) \mathrel{\vcenter{:}}=[a-b] \in { H_1 ( \operatorname{Sym}^g ( \Sigma ) ) \over \left\langle{ H_1( {\mathbb{T}}_{ \alpha } ) \oplus H_1 ( {\mathbb{T}}_{ \beta} ) }\right\rangle } \cong H_1(M) .\end{align*} This turns out to be independent of the choice of \(a, b\), and thus \begin{align*} \varepsilon(x, y) \neq 0 \implies \pi_2(x, y) = \emptyset ,\end{align*} and there are no continuous discs.

For \(x,y \in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta}\), recall that we had the following situation:

Then \(\pi_2(x, y)\) was defined to be the homotopy classes of discs connecting \(x\) to \(y\). The obstruction to the existence of such discs was denoted \(\varepsilon(x, y) \in H_1(M)\) for \(M\in {\mathsf{Mfd}}^3\). We’re checking if there exist two paths connecting \(x\) to \(y\), \begin{align*} a: I &\to {\mathbb{T}}_{ \alpha} \\ b: I &\to {\mathbb{T}}_{ \beta} \end{align*} such that \(a-b\) is nullhomotopic. In this case, \(\pi_2(x, y) \neq \emptyset\).

We had a theorem that \(\pi_1(\operatorname{Sym}^g \Sigma) \cong H_1( \operatorname{Sym}^g \Sigma)\), so we can replace nullhomotopic with nullhomologous above. We can also use the fact that \(H_1( \operatorname{Sym}^g \Sigma) \cong H_1 \Sigma\). Note that \([a-b]\) isn’t well-defined, since we can append any loop to \(a\) for example, but the following is well-defined: \begin{align*} \varepsilon(x, y) \mathrel{\vcenter{:}}=[a-b] \in { H_1 \operatorname{Sym}^g \Sigma \over H_1 {\mathbb{T}}_{ \alpha} \oplus H_1 {\mathbb{T}}_{ \beta} } \cong {H_1 \Sigma \over \left\langle{ [ \alpha_1], \cdots, [ \beta_1 ], \cdots }\right\rangle} \cong H_1 M .\end{align*}

How can we compute \(\varepsilon\) using the Heegard diagrams? Recall that a path in \(\operatorname{Sym}^g \Sigma\) was a union of \(g\) paths in \(\Sigma\). So choose arcs \(a_1 \cup\cdots \cup a_g\) on \(\Sigma\) such that \(a_i \subseteq \alpha_i\) is sub-arc and \({{\partial}}( a_1 \cup\cdots \cup a_g ) = y_1 + \cdots + y_g - x_1 - \cdots - x_g\), and similarly choose \(b_1 \cup\cdots \cup b_g\). Note that if \(\varepsilon(x, y) \neq 0\) then \(\pi_2(x, y) = \emptyset\).

We’ll define \(x\sim y \iff \varepsilon(x, y) = 0\), and this turns out to be an equivalence relation which partitions the set of paths.

• \(\varepsilon(x, y) = 0 \implies \varepsilon(y, x) = 0\), which follows from \(\varepsilon(x, y) = [a-b] = [b-a] = \varepsilon(y, x)\)

• \(\varepsilon(x, x) = 0\) by picking \(a,b\) constant.

For \(\varphi\in \pi_2(x, y)\), the shadow is the 2-chain \(D( \varphi )\) on \(\Sigma\) defined in the following way: remove the \(\alpha, \beta\) arcs to obtain \begin{align*} \Sigma \setminus(\alpha\cup\sigma) = \displaystyle\coprod_{i=1}^m D_i ,\end{align*}

where \({}^{o}\) denotes that the set is open. Then \(D( \varphi) = \sum_{i=1}^m a_i D_i\).

The following comes from “Introduction to Heegard Floer Homology” (Osvath-Szabo), which we’ve been following relatively closely so far.

This will characterize the coefficients \(a_i\) for which discs exist. Next time we’ll talk about holomorphic discs.

Recall that we discussed the domains of discs: for \(\varphi\in \varphi_2(x, y)\) we defined the 2-chain \(D( \varphi) = \sum_{i=1}^n a_i D_i\) where we’ve written \begin{align*} \Phi \setminus\alpha\cup\beta = {\coprod}_{i=1}^m \overset{\circ}{D_i} \end{align*} and \(a_i\) is the number of points in \(\operatorname{im}( \varphi) \cap L_{z_i}\) for \(z_i \in D_i\).

So if you want to check to see if some 2-chain could be the domain of a Whitney disc, this local condition can be checked, i.e. this is an obstruction to existence. It turns out that this is an if and only if condition.

See proof in Osvath-Szabo paper.

Ultimately these are what we want to define the differential in the chain complex.

We’ll set up a correspondence: \begin{align*} \left\{{\substack{ (\text{Riemann surfaces } F, {\color{red} {{\partial}}_{ \alpha}}F, {\color{blue}{{\partial}}_{\beta}}F) \xrightarrow{\pi_{\Sigma}} ( \Sigma, {\color{red} \alpha}, {\color{blue} \beta }) \\ {\big\Downarrow} \hspace{4em} {\scriptsize \text{$g{\hbox{-}}$fold branched cover $\pi_D$} } \\ (D, e_1, e_2) \\ {{\partial}}F = ({{\partial}}_{ \alpha} F) {\coprod}_{ {{\partial}}} ({{\partial}}_{ \beta} F) \\ \pi_D( {{\partial}}_{ \alpha} ) = e_1 \hspace {2em} \pi_D( {{\partial}}_{ \beta} ) = e_2 }}\right\} &\rightleftharpoons \left\{{\substack{ \text{holomorphic } u: (D^2, e_1, e_2) \to (\operatorname{Sym}^g(\Sigma), {\mathbb{T}}_{ \alpha}, {\mathbb{T}}_{ \beta}) \text{} }}\right\} \end{align*}

To do this, we define \(u(z) = \pi_{\Sigma}( \pi_D ^{-1}(z) ) \in \operatorname{Sym}^g(\Sigma)\). Check that if \(\pi_D, \pi_\Sigma\) are holomorphic, then \(u\) is holomorphic.

Link to Diagram

Then if \(u\) is holomorphic, it can be shown that \(\pi_D, \pi_{\Sigma}\) are also holomorphic. Given \(\varphi\in \pi_2(x, y)\), define \(\mathcal{M}( \varphi)\) to be the moduli space of holomorphic discs connecting \(x\) to \(y\) in the same homotopy class as \(\varphi\) (i.e. such discs represent \(\phi\)). After perturbing the complex structure \(\operatorname{Sym}^g(j)\) to make it generic, \(\mathcal{M}( \varphi)\) will be smooth. We’ll have a notion of dimension, the Maslov index \(\mu( \varphi)\), which is the expected dimension of \(\mathcal{M}( \varphi)\). There will be an \({\mathbb{R}}{\hbox{-}}\)action on \(\mathcal{M}( \varphi)\), where we remember the biholomorphism between the disc and the vertical strip:

We’ll define \(\widehat{ \mathcal{M}}( \varphi) \mathrel{\vcenter{:}}=\mathcal{M}( \varphi) / {\mathbb{R}}\). The chain complex will be defined as \begin{align*} \operatorname{HF}( \Sigma, \alpha, \beta) &\mathrel{\vcenter{:}}=\bigoplus_{x \in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta} } {\mathbb{Z}}/2 \left\langle{ x }\right\rangle \\ {{\partial}}x &\mathrel{\vcenter{:}}=\sum_{y\in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta} } \sum _{ \varphi\in \pi_2(x, y) ?} \# \widehat{ \mathcal{M}}(\varphi) .\end{align*}

We’ll need

• Check that \({{\partial}}\) is well-defined and \({{\partial}}^2=0\),

• Check independence of choices, e.g. the Heegard the diagram, the complex structure, the perturbations of \(\operatorname{Sym}^g(j)\), etc.

Osvath-Szabo picked a basepoint \(z\in \Sigma\setminus\qty{ \alpha\cup\beta}\) and work with pointed Heegard diagrams \((\Sigma, \alpha, \beta, z)\). Perturb the differential to obtain \begin{align*} \tilde {{\partial}}x \mathrel{\vcenter{:}}=\sum_{y \in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta} } \sum_{ \substack{ \varphi \in \pi_2(x, y), \\ \mu( \varphi) = 0, \\ n_z( \varphi) = 0 }} \# \widehat{\mathcal{M}}(\varphi) y .\end{align*} where \(n_z\) denotes the coefficient of \(\phi\) at the basepoint \(z\), i.e. the number of intersection points \(\# (\operatorname{im}\varphi \cap L_z)\).

Defining \(\widehat{\operatorname{HF}}\) as the same chain complex with the new differential now gets interesting! We’ll define \(\widehat{\operatorname{HF}}\) as the homology of this new complex.

Last time: to strengthen the homology theory, take a pointed Heegard diagram \((\Sigma, \alpha, \beta, z \in \Sigma\setminus\alpha\cup\beta\) and define a new chain complex \begin{align*} \widehat{\operatorname{HF}}( \Sigma, \alpha, \beta, z) &= \bigoplus_{ x\in {\mathbb{T}}_ \alpha \cap{\mathbb{T}}_ \beta} {\mathbb{Z}}/2 \left\langle{ x }\right\rangle \\ {{\partial}}x &= \sum_{y \in {\mathbb{T}}_ \alpha \cap{\mathbb{T}}_ \beta} \sum_{ \substack{ \varphi\in \pi_2(x, y), \\ \mu( \varphi) = 1, \\ n_z(\varphi) = 0 }} \# \widehat{ \mathcal{M}}(\varphi) y .\end{align*} Note that \(n_z( \varphi) = 0\) means that the coefficient attached to the region containing \(z\) is zero. Recall that we had diagram moves, how do they translate to the pointed setting?

• Allow pointed isotopies, which are isotopies disjoint from \(z\).
• Allow pointed handleslides, where now the bounded pair-of-pants is disjoint from \(z\):

• Allow isotopies of the base point.

Recall that we had a natural concatenation operation on Whitney discs: \begin{align*} \ast: \pi_2(x, y) \times\pi_2(y, z) \to \pi_2(x, z) ,\end{align*} using the identification of these discs with paths in the path space and using concatenation of paths there. Note that the domains of concatenations are given by \(D( \varphi_1 \ast \varphi_2) = D( \varphi_1) + D( \varphi_2)\), since this amounts to adding algebraic intersection numbers.

There is an inverse \begin{align*} \pi_2(x, y) &\to \pi_2(y, x)\\ \varphi &\mapsto \varphi^{-1}(s, t) \mathrel{\vcenter{:}}=\phi(s, -t) ,\end{align*} which reverses the parameterization on \((s, t) \in I \times{\mathbb{R}}\) and runs the path backward. Here \(D( \varphi^{-1}) = -D( \varphi)\).

There is also a sphere addition \begin{align*} \pi_2( \operatorname{Sym}^g( \Sigma), x) \times\pi_2(x, y) &\to \pi_2(x, y) \\ (\Omega, \varphi) &\mapsto \Omega \ast \varphi .\end{align*}

Note that for \(g\geq 2\), the \(\pi_2\) on the left-hand side is isomorphic to \({\mathbb{Z}}\), which came from quotienting by the hyperelliptic involution several lectures ago. Writing the positive generator as \(S\), we have \(\Omega = kS\) for some \(k\in {\mathbb{Z}}\).

The Maslov index is the “expected” dimension of \begin{align*} \mathcal{M}( \varphi) = \left\{{ u: I \to {\mathbb{R}}\to \operatorname{Sym}^g( \Sigma) {~\mathrel{\Big|}~} [u] = \varphi \,du\circ i = J \circ \,du }\right\} \end{align*} where \(i\) is the standard complex structure on the strip and \(J\) will be a perturbation of the complex structure over the Heegard surface. This will yield an operator \begin{align*} \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J: B &\to \mathcal{L} \\ u & \mapsto du \circ i - J \circ du \end{align*} for some appropriate infinite dimensional spaces. The elements of \(\mathcal{M}( \varphi )\) will be in the kernel of this operator. We want 0 to be a regular value (surjective derivative) for \(\mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J\), since in finite dimensions the inverse image would be a smooth manifold. In the infinite dimensional setting, we’ll have by the inverse function theorem that \(\mathcal{M} (\phi) = \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu^{-1}_J(0)\) will be a smooth manifold. We’ll want the following derivative to be surjective: \begin{align*} D_u \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J: T_u B \to T_{\mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J u} \mathcal{L} \end{align*} for all \(u \in \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu^{-1}_J(0)\), which is referred to as transversality of the operator, and can be made to hold by perturbing the complex structure. Since the dimension of a manifold is the dimension of the tangent spaces, we’ll have \(\mathcal{M}( \varphi)\) smooth of dimension equal to \(\dim \ker D_u \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J\) for any \(u \in \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J^{-1}(0)\). This will be an order 2 elliptic operator (or more generally a Fredholm operator), for which we have a notion of index: \begin{align*} \operatorname{ind}( D \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J) = \dim( \ker D\mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J) - \dim (\operatorname{coker}D \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J) .\end{align*} If surjectivity holds, the cokernel will be zero, so it will suffice to compute the dimension of the kernel to get the dimension of the moduli space. The index of this operator will be the Maslov index.

Take a look at Gromov compactness again!

Recall that for \(x,y \in {\mathbb{T}}_ \alpha \cap{\mathbb{T}}_ \beta\), there is a map \begin{align*} \mu: \pi_2(x, y) &\to {\mathbb{Z}}\\ &\mu &= \operatorname{ind}(D \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu_J) .\end{align*}

This index is the expected dimension of \(M(\varphi)\). The following theorem can be found in the paper “A cylindrical reformulation of Heegard Floer homology”:

Recall that we were working with a diagram for \(S^1 \times S^2\):

Here we have \({{\partial}}x = 2y = 0\) since we’re working mod 2, and \({{\partial}}y = 0\), so we have \begin{align*} \widehat{\operatorname{HF}}(H_1) = {\ker {{\partial}}\over \operatorname{im}{{\partial}}} = { \left\langle{ x, y }\right\rangle\over 1} = ({\mathbb{Z}}/2)^{\oplus 2} .\end{align*}

However, with a different diagram, we get a different result:

Here \(\widehat{\operatorname{HF}}(H_2) = 0\). To prevent this, we’ll have some class of admissible diagrams.

Note that for (2), the boundary could involve 1-chains, so this condition avoids corners on \({{\partial}}P\). The local picture is the following:

For any Whitney disc \(\varphi\in \pi_2(x, x)\) with \(n_z( \varphi) = 0\), \(D( \varphi)\) is a periodic domain. For any periodic domain \(P\), we can associate a homology class \(H(P) \in H_2(M)\). Writing \begin{align*} {{\partial}}P = \sum_{i=1}^g a_i \alpha_i + \sum_{i=1}^g b_i \beta_i \xrightarrow{H} H(P) \mathrel{\vcenter{:}}=[ P + \sum_{i=1}^g a_i A_i + \sum_{i=1}^g b_i B_i] .\end{align*} using that each \(\alpha_i\) is the boundary of some disc \(A_i\) in one handlebody, and \(\beta_i = {{\partial}}B_i\) similarly. Noting that \(P\) is a boundary, this amounts to adding a number of discs to get a closed nontrivial cycle.

Let \(P = \sum_{i=1}^m n_i D_i\) be a 2-chain that satisfies condition 2, so \({{\partial}}P = \sum_{i=1}^m a_i \alpha_i + \sum_{i=1}^m b_i \beta_i\). Then we can obtain a periodic domain: \begin{align*} P_0 \mathrel{\vcenter{:}}= P - n_z(P) \qty{ \sum_{i=1}^m D_i } \mathrel{\vcenter{:}}= P - n_z(P) [ \Sigma ] .\end{align*}

This is because \(H_2(M) = 0\) means there are no periodic domains.

Recall the example from last time: we are trying to show that changing a diagram by isotopy doesn’t change the homology.

Here we have \(g=1\) and so \(\operatorname{Sym}^1(T^2) = T^2\), and \(\alpha \cap\beta = \left\{{ a,b,c,d,e }\right\}\). So \(\widehat{\operatorname{HF}}( \Sigma, \alpha, \beta, z) = {\mathbb{Z}}/2 \left\langle{ a,b,c,d,e }\right\rangle\).

First mark the component that contains the base point \(z\) and give it a coefficient of zero:

We can make this part bigger, and find that there are only two bigons involving \(a\).

This is because starting at a point and following the orientation should yield red first and then blue, matching up with the orientation on the disc.

So \({{\partial}}a = {\color{yellow} b} + {\color{purple} d}\), since we require 90 degree corners. Similarly,

• \({{\partial}}e = b + d\)
• \({{\partial}}b = c\)
• \({{\partial}}d = c\)
• \({{\partial}}c = 0\)

We can simplify this information with a graph with arrows pointing toward boundaries:

Then any linear combination with the same image will have zero boundary, so we have \begin{align*} \ker {{\partial}}&= \left\langle{ a + e, b + d, c }\right\rangle \\ \operatorname{im}{{\partial}}&= \left\langle{ b+d, c }\right\rangle ,\end{align*} and thus \(\widehat{\operatorname{HF}}(\Sigma, \alpha, \beta, z) = {\mathbb{Z}}/2\).

Recall that given a rectangle, there is a 2-to-1 branched cover:

Such branched coverings bijectively correspond to biholomorphic involutions \begin{align*} a &\rightleftharpoons e \\ b &\rightleftharpoons f .\end{align*}

This is because there is a unique involution exchanging them by the Schwarz lemma, since any pole of the involution must lie along the line connecting points it exchanges, and exchanging each pair of corners in the rectangle forces to pole to be precisely the point in the center of the rectangle. So these correspond got biholomorphic involutions of \({\mathbb{D}}\) using complex analysis.

Next week: more about the Maslov index and \(\mathrm{Spin}^{\mathbb{C} }\) structures, then invariance under diagram moves.

Let \(M\in {\mathsf{Mfd}}^3({\mathbb{R}})\) be a closed oriented 3-manifold and \(\mathcal{H} = (\Sigma, \alpha, \beta, z)\) a Heegaard diagram for \(M\). Letting \(b_i\) be the Betti numbers, note that \(b_1 = 0 \iff M \in \operatorname{QHS}^3\) is a rational homology 3-sphere, i.e. \(H_i(M; {\mathbb{Q}}) \cong H_i(S^3; {\mathbb{Q}})\) for all \(i\). This also implies that \(H_2(M; {\mathbb{Z}}) = 0\). Under this condition, we can define a relative \({\mathbb{Z}}{\hbox{-}}\)grading (i.e. we have a difference of grading between any two elements) on \(\widehat{CF}\) in the following way: for \(x, y\) two generators, we set \begin{align*} {\mathsf{gr}\,}(x) - {\mathsf{gr}\,}(y) \mathrel{\vcenter{:}}=\mu( \varphi) -2n_z( \varphi) && \text{for some } \varphi\in \pi_2(x,y) .\end{align*}

Recall that \(\mu({-})\) denotes the Maslov index, \(n_z( {-})\) is the local multiplicity of a Whitney disc at \(z\), and \(x, y\) denote tuples of points.

This involves a choice of disc, so why is it well-defined? We’ll also see why we need \(M\in \operatorname{QHS}^3\).

Note that the relative grading is only defined if \(\pi_2(x, y) \neq \emptyset \iff \varepsilon(x, y) = 0 \in H_1(M; {\mathbb{Z}})\). This generated an equivalence relation of elements in \({\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta}\) by \(x\sim y \iff \varepsilon(x, y) = 0\), so we have a decomposition \begin{align*} \widehat{CF}( \mathcal{H} ) = \bigoplus _{?} \widehat{CF}( \mathcal{H}, ?) .\end{align*} which is preserved by \({{\partial}}\), so \(\widehat{HF}(\mathcal{H})\) will split similarly as \begin{align*} \widehat{HF}( \mathcal{H} ) = \bigoplus _{?} \widehat{CF}( \mathcal{H}, ?) .\end{align*}

It turns out that the right thing to replace the “?” with will be \(\mathrm{Spin}^{\mathbb{C} }\) structures.

We’ll discuss Turaev’s (?) reformulation of \(\mathrm{Spin}^{\mathbb{C} }\) structure for \({\mathsf{Mfd}}^3\). Note that \(\chi(M) = 0\), so there exists nowhere vanishing vector fields on \(M\) by Poincaré-Hopf.

Recall that given a Heegard diagram \(( \Sigma, \alpha, \beta, z )\) gives an equivalence relation \begin{align*} x \sim y \iff \varepsilon(x, y) = 0 \in H_1(M) \overset{\mathrm{PD}}{=} H^2(M) .\end{align*} This yields a decomposition of \(\widehat{\operatorname{HF}}\) into a direct sum over equivalence classes of subcomplexes defined by \(\mathrm{Spin}^{\mathbb{C} }\) structures. Note that the differential will preserve each direct summand. We defined \(\mathrm{Spin}^{\mathbb{C} }(M)\) as the set of nowhere vanishing vector fields on \(M\) modulo being homotopic outside finitely many 3-balls in \(M\). We had a map \begin{align*} {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{\beta} \xrightarrow{s_z} \mathrm{Spin}^{\mathbb{C} }(M) ,\end{align*} recalling that the left-hand side are the generators of \(\widehat{\operatorname{HF}}\). We took a self-indexing Morse function on \(M\), took the inverse image of \(3/2\) to get the Heegard surface, and each intersection point \(x_i\) gave a flow line from an index 2 critical point to an index 1 critical point passing through \(x_i\):

We proceeded by cancelling adjacent flow lines (at the level of vector fields), and then modifying \(\gamma_z\) (the flow line passing through the basepoint \(z\) connecting the index 0 to the index 3) to get a nowhere vanishing vector field. We then took a trivialization \(\tau: TM \to M \times{\mathbb{R}}^3\) defined a map \begin{align*} \mathrm{Spin}^{\mathbb{C} }(M) &\xrightarrow{\gamma^ \tau} H^2(M) \\ s = [v] &\mapsto f_v^*( \alpha) .\end{align*} where \(\alpha\) is the volume form of \(S^2\) and \begin{align*} f_v: M &\to S^2 \\ x &\mapsto \widehat{v_x} \mathrel{\vcenter{:}}={ v_x \over {\left\lVert {v_x} \right\rVert} } .\end{align*} Note that \(\delta^\tau\) a priori depends on \(\tau\), but \begin{align*} \delta(s_1, s_2) = \delta^{ \tau}(s_1) - \delta^{ \tau}(s_2) \in H^2(M) ,\end{align*} and the difference is independent of \(\tau\).

We thus have \begin{align*} \widehat{\operatorname{HF}}( \Sigma, \alpha, \beta, z) = \bigoplus _{{\mathfrak{s}}\in \mathrm{Spin}^{\mathbb{C} }(M)} \widehat{\operatorname{HF}}( \Sigma, \alpha, \beta, z, {\mathfrak{s}}) .\end{align*}

We have several properties of \(\mathrm{Spin}^{\mathbb{C} }\) structures. There is a map \begin{align*} J: \mathrm{Spin}^{\mathbb{C} }(M) &\to \mathrm{Spin}^{\mathbb{C} }(M) \\ s = [v] &\mapsto {\overline{{s}}} \mathrel{\vcenter{:}}=[-v] .\end{align*} There is also a first Chern class \begin{align*} c_1: \mathrm{Spin}^{\mathbb{C} }(M) &\to H^2(M) \\ s &\mapsto s - {\overline{{s}}} ,\end{align*} i.e. \(c_1(s) = \delta(s, {\overline{{s}}})\).

There are few things to discuss:

1. The almost complex structure \(J\):

2. Isotopies

3. Handle slides

4. Stabilization

Remarks on these:

1. This involves a standard argument from Lagrangian Floer homology.
2. There are two cases:
   • If the isotopy doesn’t create a new intersection, we have a 1-to-1 correspondence between generators for any two choices, and changing \(J\) to \(J'\) will give a correspondence between the differentials. This just involves picking a diffeomorphism that maps \(\alpha\) circles to \(\alpha'\) circles, and so on. So this reduces to showing 1.
   • If is does create new intersection points, there are again standard arguments in Lagrangian Floer homology for this.
3. This involves the following situation, which induces a map

For an appropriate choice of \(J\) on \(\Sigma \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{\#}}}T^2\), the map \(f\) above will induce a chain homotopy equivalence \begin{align*} \tilde f: \widehat{\operatorname{HF}} (\Sigma, \alpha, \beta, z) \xrightarrow{\sim} \widehat{\operatorname{HF}} (\Sigma \mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{\#}}}T^2, \alpha', \beta', z) .\end{align*}

4. What’s the picture?

This will yield a map \begin{align*} (\Sigma, \alpha, {\color{blue} \beta}, z) \leadsto (\Sigma, \alpha, {\color{green} \gamma}, z) .\end{align*} For \(i = 1, \cdots, g-1\), we’ll have \(\gamma_i\) isotopic to \(\beta_i\), and for \(i=g\), \(\gamma_g\) is obtained by sliding \(\beta_g\) over \(\beta_{g-1}\). We’ll combine these into the same diagram with different colors to compare them, yielding a Heegard triple: \begin{align*} (\Sigma, {\color{red} \alpha}, {\color{blue} \beta}, { \color{green} \gamma}, z) .\end{align*} We can think of this as three separate diagrams: \begin{align*} (\Sigma, {\color{red} \alpha}, {\color{blue} \beta}, z) &\leadsto M\\ (\Sigma, {\color{blue} \beta}, { \color{green} \gamma}, z) &\leadsto ? \\ (\Sigma, {\color{red} \alpha}, { \color{green} \gamma}, z) &\leadsto M .\end{align*} What does the middle one represent?

Here this is a diagram for \((S^1 \times S^2)^{\mathop{ \Large\scalebox{0.8}{\raisebox{0.4ex}{\#}}}2}\).

Note that we draw \(\gamma_i\) such that it intersects \(\beta_i\) in two transverse intersection points to make sure the diagram is admissible.

Give a Heegard triple \(( \Sigma, \alpha, \beta, \gamma, z\), pick three intersection points \begin{align*} x &\in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta} \\ y &\in {\mathbb{T}}_{ \beta} \cap{\mathbb{T}}_{ \gamma} \\ w &\in {\mathbb{T}}_{ \gamma} \cap{\mathbb{T}}_{ \alpha} .\end{align*} We can use Whitney triangles to connect \(x,y,w\):

We then define \(\pi_2(x,y,z)\) to be the homotopy class of Whitney triangles connecting \(x,y,w\). We can similarly define \({\mathcal{M}}( \psi )\) to be the moduli space of \(J{\hbox{-}}\)holomorphic representatives of \(\psi\in \pi_2(x,y,w)\), along with a chain map \begin{align*} f_{\alpha \beta \gamma}: \widehat{\operatorname{HF}}(\Sigma, \alpha, \beta, z) \otimes \widehat{\operatorname{HF}}(\Sigma, \beta, \gamma z) &\to \widehat{\operatorname{HF}}(\Sigma, \alpha, \gamma, z) \\ x\otimes y &\mapsto \sum_{ w \in {\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{ \beta}} \sum_{\substack{ \psi\in \pi_2(x,y,w) \\ \mu( \psi) = 0 \\ n_z( \psi) = 0 }} \# {\mathcal{M}}( \psi) \cdot w .\end{align*}

Next time: we’ll show how to get a chain homotopy equivalence from the first tensor term above to the codomain. We’ll also see surgery exact triangles.

Recall that we have several variants: namely \(\widehat{\operatorname{HF}}, \operatorname{HF}^-, \operatorname{HF}^+, \operatorname{HF}^{\infty }\). Let \(M\in {\mathsf{Mfd}}^3\) and take a Heegard diagram \((\Sigma, \alpha, \beta, z)\). Note that \(\operatorname{HF}^-( \Sigma, \alpha, \beta, z)\) is the free \({\mathbb{Z}}/2[u]{\hbox{-}}\)module generated by \({\mathbb{T}}_{ \alpha} \cap{\mathbb{T}}_{\beta}\) with differential given by ?.

For nice Heegaard diagrams, the Maslov index 1 holomorphic discs with \(n_z( \varphi) = 0\) are embedded bigons and rectangles.

Some properties:

1. \(\mathrm{Spin}^{\mathbb{C} }\) structures: we have a decomposition \begin{align*} \operatorname{HF}^-(M) = \bigoplus _{s\in \mathrm{Spin}^{\mathbb{C} }(M) } \operatorname{HF}^-(M, s) \end{align*} which induces \begin{align*} \operatorname{HF}^\star(M) = \bigoplus_{s\in \mathrm{Spin}^{\mathbb{C} }(M)} \operatorname{HF}^\star(M, s) \end{align*} where \(\star = +, -, \infty\).

2. Maslov grading: For a \(\operatorname{QHS}^3\), \(\operatorname{HF}^-(M)\) is relatively \({\mathbb{Z}}{\hbox{-}}\)graded. The degree of \(u\) is -2, and this grading can be lifted to an absolute \({\mathbb{Q}}{\hbox{-}}\)grading.

3. There is a SES \begin{align*} 0 \to \operatorname{HF}^-(M, s) \xrightarrow{\cdot u} \operatorname{HF}^-(M, s) \to \widehat{\operatorname{HF}}(M, s) \to 0 .\end{align*}

This yields an exact triangle

Link to Diagram

4. There is a short exact sequence

\begin{align*} 0 \to \operatorname{HF}^-(M) \to \operatorname{HF}^{\infty }(M) \to \operatorname{HF}^+(M) \to 0 .\end{align*}

yielding an exact triangle

Link to Diagram

5. \({\mathbb{Z}}/2[u]\) is a PID, so by the structure theorem, any module over it will decompose and we have \begin{align*} \operatorname{HF}^-(M, s) = \bigoplus_{i} {\mathbb{Z}}/2[u] \oplus \bigoplus_j {{\mathbb{Z}}/2[u] \over \left\langle{u^{n_j}}\right\rangle } .\end{align*}

Supposing that \(M \in \operatorname{QHS}^3\), then by Osvath-Szabo, for any \(s\in \mathrm{Spin}^{\mathbb{C} }(M)\) there is exactly one free summand. Let \(d\) be the Maslov grading of the free generator, and \(c_j\) be the grading of the torsion part. We write the \(u{\hbox{-}}\)torsion part as \(\operatorname{HF}_{ \text{red} }(M, s)\).

Note that this is only a monoid without the equivalence relation, but this equivalence creates inverses.

This induces a homomorphism \begin{align*} d: \Theta_{\mathbb{Q}}^3 \to {\mathbb{Q}} .\end{align*}

Today: \(L{\hbox{-}}\)spaces and the surgery exact triangle. We’ve been loosely following [OS-1?], references for upcoming topics include [OS-2?] and Jen Hom’s survey [H?].

Recall that we were discussing \(\operatorname{HF}^-(M, s)\) for \(M\in {\mathsf{Mfd}}^3({\mathbb{R}})\) and \(s\) a \(\mathrm{Spin}^{\mathbb{C} }\) structure, and if \(M\in \operatorname{QHS}\) this decomposes as \({\mathbb{Z}}/2[u] \oplus \qty{\bigoplus_i {{\mathbb{Z}}/2[u] \over \left\langle{u^{}n_i}\right\rangle }} \mathrel{\vcenter{:}}={\mathbb{Z}}/2[u] \oplus \operatorname{HF}_{ \text{red} }(M, s)\). The Maslov grading of \(1\) in the first summand is the \(d{\hbox{-}}\)invariant, \(d(M, s)\). If one defines \(d(M) \mathrel{\vcenter{:}}=\sum_{s\in \mathrm{Spin}^{\mathbb{C} }(M)} d(M, s)\), then \(d: \Theta^3_{\mathbb{Q}}\to {\mathbb{Q}}\) is a group homomorphism. We want to talk about \(X\in {\mathsf{Mfd}}^3\) which have the “simplest” Floer theory, in the sense that the torsion summand above vanishes.

Recall that there is an exact triangle

Link to Diagram

Since multiplication by \(u\) is injective, we obtain \begin{align*} \widehat{\operatorname{HF}} = \operatorname{im}p \cong {\operatorname{HF}^-(M, s) \over \ker p} \cong {{\mathbb{Z}}/2[u] \over u {\mathbb{Z}}/2[u] } \cong {\mathbb{Z}}/2 .\end{align*}

Note that we’ve proved the forward implication but not the reverse. This is sometimes used as a definition in talks!

Sketch of the proof (\(\impliedby\)): A computation will show that \(\chi \widehat{\operatorname{HF}}(M, s) = \pm 1\) for all \(s\), since the grading can be shifted. This is proved in [OS-2?], and is the main ingredient in this proof. This implies that \({\operatorname{rank}}\widehat{\operatorname{HF}}(M, s) \geq 1\), and so adding all summands yields \({\operatorname{rank}}\widehat{\operatorname{HF}}(M) \geq \# H_1(M)\). This implies that \(\widehat{\operatorname{HF}}(M, s) \cong {\mathbb{Z}}/2\) for all \(s\), making \(M\) an \(L{\hbox{-}}\)space.

Here note that \(C_*\) is \({\mathbb{Z}}/2{\hbox{-}}\)graded, as is \((\widehat{\operatorname{HF}}(M), {{\partial}})\), so we define \(\chi(C_*) = {\operatorname{rank}}C_0 - {\operatorname{rank}}C_1\). Since we have a relative \({\mathbb{Z}}{\hbox{-}}\)grading given by \(\mu\), we get a relative \({\mathbb{Z}}/2{\hbox{-}}\)grading given by \({\mathsf{gr}\,}_{{\mathbb{Z}}/2}(x, y) = {\mathsf{gr}\,}_{\mathbb{Z}}(x, y)\), which gives us \(\chi \widehat{\operatorname{HF}}(M)\) up to sign.

So \(\widehat{\operatorname{HF}}\) can detect these two among all integral homology spheres using \(\widehat{\operatorname{HF}}\).

This allows us to specify Dehn surgeries by a rational number.

Can we get everything 3-manifold this way, as surgery on a knot? The answer is no, but yes if you allow links!

Here exactness means that e.g. \(\ker(\widehat{F}_1 = \operatorname{im}( \widehat{F}_0)\). There is a similar triangle for \(\operatorname{HF}^+\), as well as \(\operatorname{HF}^{\infty }\) and \(\operatorname{HF}^-\), although these are more complicated. However, \(\operatorname{HF}^-\) becomes easier to work with when one is looking at knot invariants instead.

Note that this gives a way to produce \(L{\hbox{-}}\)spaces.

This exercise is useful because it can be used to prove the following:

Recall: let \((M, M_0, M_1)\) be a triple of 3-manifolds corresponding to a knot \(K \subseteq M\), where \(M_0\) is 0-surgery, \(M_1\) is 1-surgery, and \(M_{\infty}\) is \(\infty{\hbox{-}}\)surgery. Here \(M\) can be chosen such that M

• \(\gamma_{\infty }\) is a meridian of \(K\),
• \(\gamma_0\) is a longitude of \(K\),
• \(\gamma_1 = -\gamma_{\infty } -\gamma_0\)

Then there exists an exact triangle:

Link to Diagram

Our goal is to define \(f: \widehat{\operatorname{HF}}(M) \to \widehat{\operatorname{HF}}(M_0)\).

Note that \(M\) admits a Heegard diagram \begin{align*} ( \Sigma_g, \vec{ \alpha} = [\alpha_1, \cdots, \alpha_g], \vec{ \beta} = [\alpha_1, \cdots, \alpha_g] ) \end{align*} such that \((\Sigma_g, \vec \alpha, [\beta_1, \cdots, \beta_{g-1}]\) is a “diagram” for \(M - \operatorname{nd}(K)\). Recall the notion of handlebodies, where each handle bounds a disc:

We can generalize this to a compression body:

• Start with \(\Sigma'_{g'} \times[0, 1]\).
• Attach a solid handle \(K\) to \(\Sigma' \times\left\{{ 1 }\right\}\)

This yields a cobordism from \(\Sigma'_{g'} \times\left\{{ 0 }\right\}\) to \(\Sigma_{g' + k}\). So we can write \({{\partial}}C = \Sigma' \times\left\{{ 0 }\right\} {\coprod}\Sigma\). Label the curves bounding the embedded discs as \(\gamma_i\):

Then we can form a diagram \((\Sigma_g, \left\{{ \gamma_1, \cdots, \gamma_k }\right\}\) where \(k\leq g\) will specify the compression body. If these are pairwise disjoint simple closed curves that are linearly independent in \(H_1( \Sigma )\), this will be a compression body from a surface with genus \(g-k\) to \(\Sigma_g\).

In this case, \((\Sigma, \vec \alpha, \left\{{ \beta_1, \cdots, \beta_{g-1} }\right\}\) will be a diagram for \(M\setminus\operatorname{nd}(K)\).