# Kylerec – Day 5

## SH & SH^+ [Momchil Konstantinov’s talk]

Let us begin with a rather informal and sketchy overview of the basics behind symplectic homology (this is by no means the most general version, and we refer the reader to the vast and growing literature, of which we give some references below).

Consider $(V,\lambda)$ a Liouville domain with contact boundary $(M=\partial V, \alpha= \lambda\vert _{\partial V})$ and its completion $(\widehat{V},\widehat{\lambda})$, obtained from $(V,\lambda)$ by attaching cylindrical ends. Given a nondegenerate Hamiltonian $H:S^1\times \widehat{V}\rightarrow \mathbb{R}$,  we have an associated action functional $\mathcal{A}^H: C^\infty(\mathbb{R}/\mathbb{Z}, \widehat{V})\rightarrow \mathbb{R}$, defined by

$\mathcal{A}^H(x)=\int_{S^1}x^*\widehat{\lambda}-\int_{S^1}H_t(x(t))dt$

Its differential is given by $d_x\mathcal{A}^H(\xi)=\int_{S^1}d\lambda(\xi(t),\dot{x}(t)-X_{H_t}(x(t)))dt$, and it follows that its critical points correspond to closed Hamiltonian orbits. Given a $d\lambda$-compatible almost complex structure $J$ which is cylindrical on the ends, this induces a metric on the loop space, for which the gradient of $\mathcal{A}^H$ can be written as $\nabla_x\mathcal{A}^H=-J(\dot{x}-X_H(x))$, so that the gradient flow equation becomes the Floer equation. We define the symplectic homology chain complex (with mod 2 coefficients) as

$CF_*(H)=\bigoplus_{x \in \mbox{crit}(\mathcal{A}^H)}\mathbb{Z}_2.x$

By simplicity, assume that $x \in \mbox{crit}(\mathcal{A}^H)$ is contractible (so that we don’t have to worry about homology classes and whatnot), and also assume that $c_1(V)=0$ (this condition can be relaxed to $c_1\vert_{\pi_2(V)}=0$, and is needed for the grading). Then we can define the Conley-Zehnder index of $x$ by choosing spanning disks for $x$ and trivializing $TV$ along this disk, and we choose the grading $|x|=\mu_{CZ}(x)-n$, which is independent on the trivialization by the assumption on $c_1(V)$. The differential is now $d_H: CF_k(H)\rightarrow CF_{k-1}(H)$, given by

$d_H(x)=\sum_{\substack{y\in \mbox{crit}(\mathcal{A}_H)\\|y|=|x|-1}}\#_{\mathbb{Z}_2}\mathcal{M}(y,x)y$

where $\mathcal{M}(y,x)$ is the moduli space of Floer trajectories joining $x$ to $y$ divided by the natural $\mathbb{R}$-translation action. This moduli space is a zero dimensional manifold when $|y|=|x|-1$ (for generic $J$). Recall that Gromov compactness requires uniform $C^0$-bounds (which in our situation are not for free, since $\widehat{V}$ is non-compact) and uniform energy bounds (which we have for $u \in \mathcal{M}(y,x)$, since $E(u)=\mathcal{A}^H(x)-\mathcal{A}^H(y)$).

Def.  The spectrum of $(M,\alpha)$ is

$spec(M,\alpha)=\{ T \in \mathbb{R}: \mbox{ there exists a }\alpha-\mbox{Reeb orbit of period }T\}$

Def. The space of admissible Hamiltonians $Ad(V,\lambda)$ is the set of Hamiltonians $H: S^1 \times \widehat{V}\rightarrow \mathbb{R}$ satisfying

$H_t(r,y)=Ae^r+B$ on $r>R>>0$,  for some $R$,  where $A>0, A \notin spec(M,\alpha)$.

Denote by $h(s)=As+B$, so that $H_t(r,y)=h(e^r)$ on the ends.

If one chooses an admissible $H$ and a $J$ which is cylindrical on the ends, one gets $C^0$-bounds, as follows from the maximun principle: indeed, consider $\Omega \subseteq \mathbb{R}\times S^1$ an open subset, and $u: \Omega \rightarrow \widehat{V}$ a holomorphic map, which has a portion lying on the cylindrical ends. This portion can be parametrized by $u(s,t)=(a(s,t),v(s,t))\in \mathbb{R} \times M$, and a computation gives

$\Delta a + \partial_s(h^\prime(e^a))=\Delta a + h^{\prime\prime}(e^a)e^a\partial_sa=||\partial_s v ||^2\geq 0$

The maximum principle then implies that a sequence of Floer cylinders with fixed asymptotics cannot escape to infinity, since we would get a maximum of $a$, which implies $\Delta a\geq 0$, and this cannot happen if one assumes that the maximum is non-degenerate (a clever trick then gets rid of this assumption). So we get the $C^0$-bounds, which leads to compactness by Gromov, which implies $d_H$ well-defined and $d_H^2=0$ (as follows by studying the boundary of 1-dimensional moduli spaces of Floer trajectories). From this, one gets the Floer homology group

$HF_k(H):=H_k(CF_*(H),d_H)$

The first thing one asks is: is it independent of $H$? And the answer is…well… nope. BUT…

Consider two different $H_+, H_- \in Ad(V,\lambda)$, and choose a smooth path of Hamiltonians $H^s:\widehat{V}\rightarrow \mathbb{R}$ for $s \in \mathbb{R}$, such that $H^s=H_-$ for $s<<0$, $H^s=H_+$, for $s>>0$, and $H^s(r,y)=h_s(e^r)=A_se^r+B_s$ for $A_s,B_s \in \mathbb{R}$, on the cylindrical ends. This gives the parametrized Floer equation $\partial_su + J(\partial_tu - X_{H^s}(u))=0$ and a corresponding moduli space $\mathcal{M}_{\{H^s\}}(x_-,x_+)$ joining the orbits $x_-$ and $x_+$, which is zero dimensional when $|x_-|=|x_+|$ (now we don’t have a translation action). This ideally would allow us to define a map

$\Phi: CF_*(H_+)\rightarrow CF_*(H_-)$

given by

$\Phi(x_+)=\sum_{\substack{x_- \in \mbox{crit}(\mathcal{A}_H)\\ |x_-|=|x_+|}} \#_{\mathbb{Z}_2} \mathcal{M}_{\{H_s\}}(x_-,x_+)x_-$

satisfying $d_{H_-}\circ \Phi = \Phi \circ d_{H_+}$, as follows by studying how trajectories in 1-dimensional moduli spaces can break. But this, again, requires Gromov compactness. A similar computation gives

$\Delta a + \partial_s(h_s^\prime (e^a))=\Delta a + h_s^{\prime\prime}(e^a)e^a\partial_sa + (\partial_s h_s^\prime)(e^a)=||\partial_s v||^2$

So, to have $\Delta a + h_s^{\prime\prime}(e^a)e^a\partial_sa\geq 0$ it suffices with

$\partial_sh_s^\prime=\partial_s A_s<0$

In other words, the slope of $H_-$ is necessarily steeper than that of $H_+$. This means that we only get compactness in “one direction”, and we do not get a homotopically inverse map.

If we define a partial order $\prec$ on $Ad(V,\lambda)$ by $H_1\prec H_2$ if $H_1 outside of a compact set, the previous discussion gives us a map $HF_*(H_1)\rightarrow HF_*(H_2)$. Moreover, we get commutative diagrams for any $H_1 \prec H_2 \prec H_3$, giving a direct system, so that we may define the symplectic homology of $(V,\lambda)$ as

$SH_k(V,\lambda)=\varinjlim_{H \in Ad(V,\lambda)} HF_k(H)$

Observe that, as with any direct limit, one can compute it by taking cofinal sequences. Now we identify the generators of this homology. Let us recall the following fact from Floer theory:

Fact. If $H$ is sufficiently $C^2$-small then all the 1-periodic orbits of $X_H$ are critical points of $H$, and every Floer trajectory between them is a Morse flow-line.

This means that if $H$ is sufficiently $C^2$-small and positive on $V$, then the generators on this region of $SH_k$ will correspond to critical points (graded by $|x|=\mu_{CZ}(x)-n=n-ind_x(H)-n=-ind_x(H)$), and observe that $\mathcal{A}^H(x)=-H(x)<0$. On the cylindrical ends, we have $X_H=h^\prime(e^r)e^{-r} R_\alpha$, where $R_\alpha$ is the Reeb vector field of $\alpha$ on $r=0$, so that closed Hamiltonian orbits lie in the contact slices $\{r=r_0\}$ and are reparametrizations of closed Reeb orbits of period $T:=h^\prime(e^{r_0})$, and these have action

$\mathcal{A}^H(x)=T-h(e^{r_0})>0$

Since we assume that the slope of $H$ does not lie in the spectrum, there are no closed orbits for $r>R>>0$, and between $0$ and $R$ we see potential closed Hamiltonian orbits of bounded action. Since the differential decreases action, we have a subcomplex $CF_*^{-}(H)$ of $CF_*(H)$ generated by orbits of negative action (critical points), and an exact sequence of chain complexes

$0\rightarrow CF_*^{-}(H)\rightarrow CF_*(H)\rightarrow CF_*^+(H)\rightarrow 0$

where $CF_*^+(H)=\frac{CF_*(H)}{CF_*^{-}(H)}$. If we define

$SH_*^+(H)=\varinjlim_{H \in Ad(V,\lambda)}H_*(CF_*^+(H),d_H)$

and we take direct limit in the resulting long exact sequence (which preserves exactness), we get an induced exact triangle

Here we have used the Floer theory fact, and the maximum principle, to say that $CF_*^-(H)$ computes $H^{-k}(V)$ for every $H$ ($C^2$-small on $V$). Observe that we get cohomology of $V$ rather than homology, since we get a minus in the grading ($-ind_x(H)$ goes to $-ind_x(H)-1=-(ind_x(H)+1)$ under the differential). Yes, it’s confusing.

We can now state a few theorems.

Thm. [Bourgeois-Oancea] If all Reeb orbits of $(M,\alpha)$ satisfy

$\mu_{CZ}(x)+n-3>0$

that is, if $(M,\alpha)$ is dynamically convex, and $V,W$ are two Liouville fillings of $M$ with $c_1(V)=c_1(W)=0$, then $SH_*^+(V)\simeq SH_*^+(W)$.

In other words, $SH_*^+$ is an invariant of $M$, rather than the fillings (with $c_1=0$). The idea is to show that no critical points can be connected to a non-constant orbit by a Floer trajectory, and that no cylinder connecting two of the latter ventures into the filling $V$ (there is a stretching the neck argument here).

Thm. [ML Yau] If $(M,\xi)$ is subcritically Stein fillable (for a filling with $c_1=0$), then $M$ admits a dynamically convex contact form.

Thm. [Cieliebak] If $V$ is subcritically Stein (with $c_1=0$), then it has vanishing symplectic homology.

Cieliebak proves that $V$ is isomorphic to a split Stein manifold $W \times \mathbb{C}$, for $W$ Stein, and using a version of the Künneth formula for $SH_*$, the result follows from the fact that $SH_*(\mathbb{C})=0$, which one can compute by hand.

Cor. If $V,W$ are subcritical Stein fillings of $(M,\xi)$ with $c_1(V)=c_1(W)=0$, then $H^*(V)\simeq H^*(W)$.

This follows from the exact triangle, and all theorems stated above, since $H^{-*}(V)\simeq SH^+_*(V)$ for a subcritical Stein manifold with $c_1(V)=0$.

References

A few references on symplectic homology (by all means very much non-exhaustive):

A begginer’s overview: https://www.mathematik.hu-berlin.de/~wendl/pub/SH.pdf

A nice survey: https://arxiv.org/abs/math/0403377

A Morse-Bott version (relevant for Cédric’s talk below): https://arxiv.org/abs/0704.1039

A related theory (Rabinowitz Floer homology): https://arxiv.org/abs/0903.0768

## Contact manifolds with flexible fillings [Scott Zhang’s talk]

The main reference for this post is this paper: https://arxiv.org/pdf/1610.04837.pdf.

Let us recall the following result, which appeared in Momchil’s talk:

Thm. [M.L Yau] If $W_1, W_2$ are two subcritical fillings of a contact manifold $(M^{2n-1},\xi)$, (with $c_1(W_1)=c_1(W_2)=0$) then $H^*(W_1)\simeq H^*(W_2)$.

The goal for this talk was to discuss the following generalization to the $\emph{flexible}$ case:

Thm 1. [O. Lazarev] If $W_1,W_2$ are two flexible fillings of $(M,\xi)$, then $H^*(W_1)\simeq H^*(W_2)$.

Remark: The same conclusion is true if we consider fillings with vanishing symplectic homology.

The idea is to replace the dynamical convexity condition in Bourgeois-Oancea’s result by an asymptotic version. In the following, given $\alpha_1,\alpha_2$ contact forms for the same contact structure, we will denote $\alpha_1\geq \alpha_2$ if $\alpha_1=f \alpha_2$ for some smooth function $f\geq 1$, and by $\mathcal{P}^{ the set of $\alpha$-Reeb orbits $\gamma$ with action $\int_\gamma \alpha . The degree of a Reeb orbit $\gamma$ is $|\gamma|=\mu_{CZ}(\gamma)+n-3$.

Def.  $(M^{2n-1},\xi)$ is asymptotically dynamically convex (ADC) if there exists a sequence of contact forms $\alpha_1\geq \alpha_2\geq \dots$ for $\xi$ and a sequence $0 with $\lim_{i}D_i=\infty$ such that every element in $\mathcal{P}^{ has positive degree.

We have the following:

Thm 2. [O. Lazarev] If $(M,\xi)$ is ADC, then $SH^+$ is independent of the Stein filling with $c_1=0$.

Recall that  flexible Weinstein manifolds have vanishing symplectic homology. This follows by the Bourgeois-Ekholm=Eliashberg surgery formula (https://arxiv.org/pdf/0911.0026.pdf), but there are alternative arguments not using the SFT machinery, based on an h-principle for exact codimension zero embeddings, and the Künneth formula for symplectic homology, which even works for twisted coefficients (see e.g. Murphy-Siegel https://arxiv.org/abs/1510.01867). From the exact triangle for $SH_+$, we know that $SH_*^+(W)\simeq H^{-*}(W)$ for flexible $W$, so to get thm. 1 it suffices to show that flexible fillings induce ADC contact structures on their boundaries.

Thm 3. [O. Lazarev] If  $(M^\prime,\xi^\prime)$ is obtained from $(M,\xi)$ by flexible surgery and $(M,\xi)$ is ADC, then so is $(M^\prime,\xi^\prime)$.

Remark. The subcritical case where the ADC condition is replaced by DC (dynamical convexity) is already due to Yau.

Since the standard sphere is ADC, thm. 1 follows.

Here are a few ingredients in the argument. Let us recall first the following:

Prop. [Bourgeois-Ekholm-Eliashberg] After surgery along a Legendrian sphere $\Lambda^{n-1} \;(n\geq 3)$, we have a 1-1 correspondence between the newly created Reeb orbits with action bounded by $D>0$, and words of Reeb chords on $\Lambda$ with action bounded by $D$ (up to cyclic permutation). Moreover, we have $|\gamma_{c_1\dots c_n}|=\left(\sum_i |c_i|\right)+n-3$, where $\gamma_{c_1\dots c_n}$ denotes the Reeb orbit corresponding to the word $c_1\dots c_n$.

The idea is to slightly perturb the data so that given a collection of ordered chords, there is a closed Reeb orbit which enters the handle and is close to the original chords in the complement of the handle (the fact that all closed orbits that enter the handle have to leave it boils down to the fact that the geodesics on the flat disk leave the disk).

Key lemma. If $\Lambda$ is loose, there exists a Legendrian isotopy such that (action bounded) Reeb chords have positive degree.

The point is that stabilizing a loose Legendrian, which in general does not change the formal homotopy type, actually does not change the genuine isotopy type, by Murphy’s h-principle, and one can explicitly see that the degree of the resulting Reeb chords is greater or equal than 1 after the stabilization. The fact that we get decreasing contact forms comes form this stabilization process.

## Computations on Brieskorn manifolds [Cédric De Groote’s talk]

The goal for this talk, much more computational in spirit, was to discuss how invariants like contact and symplectic homology can be used to distinguish contact structures on Brieskorn manifolds, specially when the underlying manifolds are diffeomorphic, and in certain cases even when the contact structures are homotopic as almost contact structures.  A useful tool is a Morse-Bott version of symplectic homology, which applies in many cases where a lot of symmetry in present in the setup.

Brieskorn manifolds and Ustilovsky exotic contact spheres

The Brieskorn manifold associated to $a=(a_0,\dots,a_n)$, where $a_i\geq 2$ is an integer, is defined by $\Sigma(a)^{2n-1}=\{z_0^{a_0}+\dots + z_n^{a_n}=0\}\cap S^{2n+1}\subseteq \mathbb{C}^{n+1}$. In other words, it is the link of the (isolated) singularity associated to the complex polynomial $f(z)=z_0^{a_0}+\dots + z_n^{a_n}$. It is the binding of an open book on $S^{2n+1}$, with pages which are diffeomorphic to $\{f(z)=\epsilon\}\cap \mathbb{D}^{2n+2}$, for small $\epsilon>0$ (the Milnor fiber of $f$, see Milnor’s classic book: “Singular points of complex hypersurfaces”).

Brieskorn manifolds come with a contact form $\alpha_a=\frac{i}{8}\sum_{j=0}^na_j(z_jd\overline{z_j}-\overline{z_j}dz_j)$, which is induced by the “weighted” exact symplectic form $\omega_a=\frac{i}{4}\sum_{j=0}^n a_j dz_j\wedge d\overline{z_j}$ on $\mathbb{C}^{n+1}$, with associated Liouville vector field $V(z)=z/2$, which is transverse to $\Sigma(a)$. The corresponding Reeb vector field is $R_a=(\frac{4i}{a_0}z_0,\dots,\frac{4i}{a_n}z_n)$, which has flow $\phi_a^t(z)=(e^{\frac{4it}{a_0}}z_0,\dots,e^{\frac{4it}{a_n}}z_n)$. We also have a filling for $\Sigma(a)$, given by $W_a=\{f(z)=\epsilon \varphi(|z|)\}$, where $\varphi: [0,+\infty)\rightarrow \mathbb{R}$ satisfies $\varphi\equiv 1$ close to $0$, and vanishes close to $1$ (so that $W_a$ is a non-singular interpolation between the Milnor fiber and the singular hypersurface $\{f=0\}$). It comes endowed with the restriction of $\omega_a$, and is therefore an exact filling (it is actually Stein). By thm. 5.1 in Milnor’s book, it is parallelizable, and hence $c_1(W_a)=0$.

Some interesting facts:

1. $\pi_1(\Sigma(a))=\dots=\pi_{n-1}(\Sigma(a))=0$, i.e $\Sigma(a)$ is $(n-1)$-connected (lemma 6.4 in Milnor, which works for any Milnor fiber).
2. If $n\neq 2$, $\Sigma(a)$ is homeomorphic to a sphere if and only if it is a homology sphere (For $n \geq 3$ it follows by 1. above -which implies simply connectedness-, and the generalized Poincaré hypothesis, and is trivial for $n=1$). By 1., Poincaré duality and Hurewicz’ theorem, this is equivalent to the reduced homology $\widetilde{H}_{n-1}(\Sigma(n))=0$.
3. There exist conditions on $a$ which are equivalent to $\Sigma(a)$ being homeomorphic to the sphere $S^{2n-1}$. Namely, If there exist $a_i,a_j$ which are relatively prime to all other exponents, OR there exist $a_i$ which is relatively prime to all others and a set $\{a_{j_1},\dots,a_{j_r}\} (r\geq 3 \mbox{ odd })$ such that every $a_{j_k}$ is relatively prime to every exponent not in the set, and $gcd(a_{j_k},a_{j_l})=2$ for $k\neq l$.
4. $\Sigma(2,2,2,3,6k-1)$ for $k=1,\dots,28$ gives all smooth structures in $S^7$ (it is homeomorphic to the sphere by the previous criterion).
5. Any simply connected spin 5-manifold is a connect sum of Brieskorn 5-manifolds.

Thm.[Brieskorn] If $p \equiv \pm 1 (mod \;8)$ then $\Sigma(p,2,\dots,2)$, where the number of 2’s is $2m+1$, is diffeomorphic to $S^{4m+1}$.

Denote by $\xi_p$ the contact structure on $\Sigma(p,2,\dots,2)$ that we obtain by the weighted symplectic form, as above. Observe that by the above criterion these manifolds are all homeomorphic to spheres.

Thm.[Ustilovsky] If $p_1 \neq p_2$, then $\xi_{p_1}$ is not contactomorphic to $\xi_{p_2}$.

The proof uses contact homology. One can take an explicit perturbation making the contact form non-degenerate, and compute the degrees of the resulting non-degenerate Reeb orbits, which are all even. This implies that the differential vanishes, so that contact homology is isomorphic to the underlying chain complex. For different values of $p$, the degrees of the generators differ, and hence contact homology does also (and this is an invariant of the contact structure).

Def.  An almost contact structure on $Y^{2n+1}$ is  a pair $(\alpha,\beta)$ of a 1-form $\alpha$ and a 2-form $\beta$ such that $\beta\vert_{\ker \alpha}$ is non-degenerate. This is equivalent to having a reduction of the structure group of $TY$  to $U(n)\times 1$.

Def. A contact sphere $(S^{2n+1},\xi)$ is called exotic if it is not contactomorphic to  $(S^{2n+1},\xi_{std})$, the standard contact structure on $S^{2n+1}$. It is homotopically trivial if it is homotopic to $(S^{2n+1},\xi_{std})$ as almost contact structures.

An almost contact structure on $S^{4m+1}$ is equivalent to a lift of the classifying map $S^{4m+1}\rightarrow BSO(4m+1)$ to a map $S^{4m+1}\rightarrow B(U(2m)\times 1)$, under the natural map $B(U(2m)\times 1) \rightarrow BSO(4m+1)$ induced by inclusion. This map has fibers $S0(4m+1)/ (U(2m)\times 1)$, and therefore almost contact structures are classified by the group $G:= \pi_{4m+1}( S0(4m+1)/ (U(2m)\times 1))$.

Thm.[Massey] $G$ is cyclic of order $d=(2m)!$ if $m$ even, and $d=(2m)!/2$ if $m$ odd.

Thm.[Morita] The contact structure $\xi_p$ on $\Sigma(p,2,\dots,2)$ represents $\frac{p-1}{2} (mod \; d)$ in $G$ when viewed as an almost contact structure.

It follows that if $p\equiv 1 (mod \; 2(2m)!)$ and $p\equiv \pm 1 (mod \; 8)$ then $\xi_p$ is homotopically trivial.  Since there are infinitely many $p$‘s satisfying these conditions, we obtain:

Thm.[Ustilovsky] There exist infinitely many exotic but homotopically trivial contact structures on $S^{4m+1}$.

Morse-Bott techniques

The Morse-Bott condition is morally the next best thing to having non-degeneracy (in fact, one can argue that it is the best thing when one wishes to do computations), and it can be thought of as a manifestation of symmetry.

Recall that a function $f:M \rightarrow \mathbb{R}$ is Morse-Bott if its critical set $\mbox{crit}(f)=\bigsqcup_i C_i$ is a disjoint union of connected submanifolds $C_i$, such that, if we denote by $\nu(C_i)$ the normal bundle of $C_i$ inside $M$, then $Hess_p(f)\vert_{\nu(C_i)}$ is non-degenerate.

Loosely speaking, the degeneracies are “well-controlled”, and come in “families”. In general, in the Morse-Bott situation, one hopes for a perturbation scheme which recovers the non-degenerate/Morse case, by a small perturbation of the data, in such a way that one gets a 1-1 correspondence between the symmetric (i.e Morse-Bott) data, and the generic (i.e Morse) one, and so that compuations can be carried out in the Morse-Bott setting in the first place. For instance, if one wishes to compute Morse homology from a Morse-Bott function $f$, one can choose a Morse function $h$ on $\mbox{crit}(f)$, and consider $f_\epsilon:=f+\epsilon \rho h$, for $\epsilon>0$ small, and $\rho$ is a bump function with support near $\mbox{crit}(f)$. The critical points of $f_\epsilon$ are exactly those of $h$, and there is a well-defined notion of convergence of flow-lines of $f_\epsilon$ to “cascades” (when the perturbation parameter $\epsilon$ is taken to go to zero). The latter consist of a flow-line of $f$ hitting a critical manifold, followed by a flow-line segment of $h$ along this manifold, followed by another flow-line of $f$ hitting another critical manifold, and so on, finishing in a critical point of $f$ (see the figure below). One can define the index of a cascade in such a way that the index is preserved under this convergence, and there is a 1-1 correspondence between index $I$ cascades and index $I$ Morse flow-lines of $f_\epsilon$. Hence, one can define a Morse-Bott differential which counts cascades, and the resulting Morse-Bott (co)homology coincides with the usual Morse (co)homology.

In the setting of symplectic homology, if $W$ is a Liouville filling of a contact manifold $(M,\xi)$ and $H$ is an admissible autonomous Hamiltonian, then we have closed Hamiltonian orbits in the contact slices $\{r\}\times M$ corresponding to closed Reeb orbits, which come in $S^1$-families obtained by reparametrizations (since $H$ is time-independent). This is then a Morse-Bott situation.

[Bourgeois-Oancea] In the Morse-Bott situation described above, if we assume that the orbits come in $S^1$-families (and there are no further directions of degeneracy), then there is a Morse-Bott version of symplectic homology of $W$, $SH_{MB}(W)$.

More generally, one can ask the following Morse-Bott conditions: $\mathcal{N}_T:=\{m|\varphi^T(m)=m\}$ is closed submanifold (where $\varphi^T$ is the time $T$ Reeb flow), such that $rank(d\alpha\vert_{\mathcal{N}_T})$ is locally constant and $T\mathcal{N}_T=\ker(d\varphi^T-id)$. Informally, one can think of this as an infinite-dimensional version of the Morse-Bott conditions, applied to the action functional defined on the loop space, whose critical points are closed Hamiltonian orbits. Assuming that $c_1(W)=0$ and the closed orbits are contractible (so we get an integer grading), fix a choice of Morse functions $f_T$ on $\mathcal{N}_T$ for every $T$. The generators will correspond to pairs $(\gamma,T)$  where $\gamma \in \mbox{crit}(f_T)$, and the differential counts “Floer cascades”, consisting of a Floer cylinder, followed by a flow-line segment of a $f_T$, followed by another Floer cylinder…(finitely many times). The grading is defined by $|(\gamma,T)|=\mu_{RS}(\mathcal{N}_T)+ind_{\gamma}(f_T)-\frac{1}{2}(\dim(\mathcal{N}_T)-1)$, where $\mu_{RS}$ is the Robin-Salamon index, and with this definition the differential has degree -1. Under these conditions, we have a Morse-Bott version of symplectic homology $SH_{MB}$.

Uebele’s computation

We focus now on the Brieskorn manifolds $\Sigma_l^n:=\Sigma(2l,2,\dots,2)$, where there are $n$ 2’s, for odd $n$, endowed with the contact structure discussed in the first part of this talk. Randell’s algorithm gives $H_{n-1}(\Sigma_l^n)=\mathbb{Z}$, and it follows from Wall’s classification of highly-connected manifolds that $\Sigma_l^n \simeq S^{n-1}\times S^n$ if $l\equiv 0 (mod 4)$, $\Sigma_l^n \simeq S^*S^n$ if $l\equiv 1 (mod 4)$, $\Sigma_l^n \simeq S^{n-1} \times S^n \# K$ if $l\equiv 2 (mod 4)$, $\Sigma_l^n \simeq S^*S^n \#K$ if $l\equiv 3 (mod 4)$. Here, $K=\Sigma(2,\dots,2,3)$ is Kervaire’s sphere. If $n=3$, $K$ is diffeomorphic to $S^5$, and hence $\Sigma_l^5$ is always $S^2 \times S^3$.

These contact manifolds manifolds are actually not distinguishable by contact homology. However, we have:

Thm. [Uebele] The manifolds $\Sigma_l^n$ are pairwise non-contactomorphic.

This uses the following lemma:

Lemma. For $\Sigma_l^n$, $SH_{MB}^+$ is independent of the filling, as long as $c_1(W)\vert_{\pi_2(W)}=0$.

This is proved by showing that these manifolds are dynamically convex, and using an analogous version of Bourgeois-Oancea result. Therefore one can regard $SH_{MB}^+$ as a contact invariant.

The idea now is to compute $SH_{MB}^+$ of the natural filling of these Brieskorn manifolds, using the Morse-Bott techniques, and showing that they are pairwise different. One can choose perfect Morse functions along the critical manifolds (or “formally pretend” that one can, by a spectral sequence argument due to Fauck), making the Morse differential trivial, and between different critical manifolds, one sees that for each consecutive degrees $N, N+1$ there exists a unique pair of generators having these degrees, the one with bigger degree $N+1$ having lower action than the one with smaller degree $N$. Since the differential has degree -1 and lowers the action, it has to vanish (this works for $n\geq 5$, and a different argument is needed for $n=3$). The upshot is that the Morse-Bott symplectic homology coincides with its chain complex, and the degrees differ for different values of $l$.

References

A nice reference for a survey of Brieskorn manifolds in contact topology can be found here: https://arxiv.org/abs/1310.0343

Ustilovsky’s exotic spheres: 1999-14-781

Uebele’s computations: https://arxiv.org/abs/1502.04547

Fauck’s thesis (related, and uses RFH): https://arxiv.org/abs/1605.07892