4. Quantum and Floer type cohomologies

For a generic Hamiltonian function <sup>H</sup>, the space <sup>M</sup>ð~a, <sup>~</sup>b<sup>Þ</sup> is a finite dimensional manifold of

where the function <sup>μ</sup> : <sup>~</sup> <sup>P</sup>ðHÞ ! <sup>Z</sup> is a version of the Maslov index defined by the path of

If <sup>μ</sup>ð~aÞ−μð~bÞ ¼ 1, then the space <sup>M</sup>ð~a, <sup>~</sup>b<sup>Þ</sup> is a one-dimensional manifold with <sup>R</sup> action by time

<sup>η</sup>ð~a, <sup>~</sup>b<sup>Þ</sup> :<sup>¼</sup> # <sup>M</sup>ð~a, <sup>~</sup>b<sup>Þ</sup>

where the connection orbits are to be counted with signs determined by a system of coherent orientation <sup>s</sup> of the moduli space <sup>M</sup>ð~a, <sup>~</sup>bÞ. These numbers give us a Floer type cochain com-

<sup>ð</sup>M, <sup>H</sup>Þ ! FC<sup>k</sup>þ<sup>1</sup>

<sup>μ</sup>ð~aÞ¼μ<sup>ð</sup>

Lemma 3.2.2. Let ðM,ϕÞ be a semipositive almost contact metric manifold with a closed functional 2-

<sup>∘</sup>δ<sup>k</sup> <sup>¼</sup> 0, for all k.

~bÞþ<sup>1</sup>

<sup>ξ</sup>Þð~aÞ ¼ ∑

Definition - Theorem 3.2.3. (1) For a generic pair ðH,ϕÞ on M, the cochain complex ðFC�

R !


unitary matrices generated by the linealized Hamiltonian flow along aðtÞ on D .

shift and the quotient <sup>M</sup>ð~a, <sup>~</sup>bÞ=<sup>R</sup> is a finite set. In fact, <sup>μ</sup>ð~aÞ∈π1ðUðnÞÞ≃Z.

δ<sup>k</sup> : FCk

ðδk

<sup>ð</sup>M, <sup>H</sup>Þ, <sup>μ</sup>ð~aÞ ¼ <sup>k</sup> <sup>þ</sup> 1 and <sup>μ</sup>ð~bÞ ¼ <sup>k</sup>.

dimMð~a, <sup>~</sup>bÞ ¼ <sup>μ</sup>ð~aÞ−μð~bÞ, (78)

, (79)

ðM, HÞ, (82)

, δÞ defines

<sup>η</sup>ð~a, <sup>~</sup>bÞξð~b<sup>Þ</sup> (83)

<sup>ξ</sup> : <sup>~</sup> <sup>P</sup>ðHÞ ! <sup>R</sup> (80)

#{~x<sup>∈</sup> <sup>~</sup> <sup>P</sup>ðHÞjξð~xÞ≠0, aHð~xÞ≤c} <sup>&</sup>lt; <sup>∞</sup> (81)

dimension

104 Manifolds - Current Research Areas

If Ht≡H is a C<sup>2</sup>

plex.

Let FC�

for all c∈R.

where ξ∈FCk

cohomology groups

uðDÞ ¼ a with index indHðaÞ.

If <sup>μ</sup>ð~aÞ−μð~bÞ ¼ 1, <sup>~</sup>a, <sup>~</sup>b<sup>∈</sup> <sup>~</sup> <sup>P</sup>ðHÞ, then we denote

ðM, HÞ be the set of functions

that satisfy the finiteness condition

Now we define a coboundary operator

forms. The coboundary operators satisfy δ<sup>k</sup>þ<sup>1</sup>

In this section, we assume that our manifold M is a compact either almost cosymplectic or contact or C-manifold. In Section 3.1, we study quantum type cohomology of M and in Section 3.2 Floer type cohomology of M. Consequently, we have:

Theorem 4.1. Let ðM, g,ϕ, η, ξ, φÞ be a compact semipositive almost contact metric manifold with a closed fundamental 2-form φ. Then, for every regular pair ðH,ϕÞ, there is an isomorphism between Floer type cohomology and quantum type cohomology

$$\Phi: \mathcal{F}H^\*(M, \phi, H, \varphi) \xrightarrow{\sim} \mathcal{Q}H^\*(M, \Lambda\_{\phi}). \tag{85}$$

Proof. Let h : M ! R be a Morse function such that the negative gradient flow of h with respect to the metric φ � ,ϕð�Þ þ η⊗η is Morse-Smale and consider the time-independent Hamiltonian

$$H\_t := -\varepsilon h, t \in \mathbb{R}.\tag{86}$$

If ε is sufficiently small, then the 1-periodic solutions of

$$
\dot{a}(t) = X\_t(a(t)) \tag{87}
$$

are precisely the critical point of h. The index is

$$\mu(a,\mu\_a) = n\text{-ind}\_h(a) = \text{ind}\_{-h}(a) \text{-} n \tag{88}$$

where ua : D ! M is the constant map uaðzÞ ¼ a.

The downward gradient flow lines u : R ! M of h are solutions of the ordinary differential equation

$$
\dot{\mu}(\mathbf{s}) = f(\mu)\mathbf{X}\_t(\mu). \tag{89}
$$

These solutions determine a coboundary operator

$$\delta: \mathbb{C}^\*(M, h, \Lambda\_\phi) \longrightarrow \mathbb{C}^\*(M, h, \Lambda\_\phi). \tag{90}$$

This coboundary operator is defined on the same cochain complex as the Floer coboundary δ, and the cochain complex has the same grading for both complex C� ðM, h, ΛφÞ, which can be identified with the graded Λφ module of all functions

$$\xi: \text{Crit}(h)H\_2(M) \to R \tag{91}$$

that satisfy the finiteness condition

$$\#\{(a,A)|\xi(a,A)\rtimes 0,\phi(A)\cong\}<\infty\tag{92}$$

for all c∈R. The Λφ-module structure is given by

$$(\nu \* \xi)(a, A) = \sum\_{B} \nu(B)\xi(a, A + B), \tag{93}$$

the grading is degða, AÞ ¼ indhðaÞ−2c1ðAÞ, and the coboundary operator δ is defined by

$$\xi(\delta\xi)(a,A) = \sum\_{b} n\_{h}(a,b)\xi(b,A), (a,A)\in \text{Crit}(h)H\_{2}(M),\tag{94}$$

where nhða, bÞ is the number of connecting orbits from a to b of shift equivalence classes of solutions of

$$\frac{1}{2}\left(\dot{u}(\mathbf{s}) + \nabla u(\mathbf{s}) = 0, \lim\_{s \to -\infty} u(\mathbf{s}) = a, \lim\_{s \to +\infty} u(\mathbf{s}) = b\right) \tag{95}$$

counted with appropriate signs.

Here we assume that the gradient flow of h is Morse-Smale and so the number of connecting orbits is finite when indhðaÞ−indhðbÞ ¼ 1. Then the coboundary operator δ is a Λφ-module homomorphism of degree one and satisfies δ∘δ ¼ 0. Its cohomology is canonically isomorphic to the quantum type cohomology of M with coefficients in Λφ.

For each element <sup>~</sup>a<sup>∈</sup> <sup>~</sup> <sup>P</sup>ðH<sup>Þ</sup> we denote <sup>M</sup>ð~a, <sup>H</sup>,ϕ<sup>Þ</sup> by the space of perturbed <sup>ϕ</sup>-cohomomorphic maps <sup>u</sup> : <sup>C</sup> ! <sup>M</sup> such that <sup>u</sup>ðre<sup>2</sup>πit<sup>Þ</sup> converges to a periodic solution <sup>a</sup>ðt<sup>Þ</sup> of the Hamiltonian system Ht as r ! ∞. The space Mð~a, H,ϕÞ has dimension n−μð~aÞ. Now fix a Morse function h : M ! R such that the downward gradient flow u : R ! M satisfying (95) is Morse-Smale. For a critical point <sup>b</sup>∈Critðh<sup>Þ</sup> the unstable manifold <sup>W</sup><sup>u</sup>ðb, <sup>h</sup><sup>Þ</sup> of <sup>b</sup> has dimension indhðb<sup>Þ</sup> and codimension 2n−indhðbÞ in the distribution D.

The submanifold <sup>M</sup>ðb, <sup>~</sup>a<sup>Þ</sup> of all <sup>u</sup>∈Mð~a, <sup>H</sup>,ϕ<sup>Þ</sup> with <sup>u</sup>ð0Þ∈W<sup>u</sup>ðb<sup>Þ</sup> has dimension

$$\dim \mathcal{M}(b,\tilde{a}) = \text{ind}\_h(b) - \mu(\tilde{a}) \neg n. \tag{96}$$

If indhðbÞ ¼ μð~aÞ þ n, then Mðb, ~aÞ is 0zero-dimensional and hence the numbers nðb, ~aÞ of its elements can be used to construct the chain map defined by

$$\Phi: \mathcal{FC}^\*(M, H) \to \mathcal{C}^\*(M, h, \Lambda\_\phi) \tag{97}$$

$$(\Phi \xi)(b, A) \land \sum\_{\text{ind}\_h(b) = \mu(\tilde{a}) + n} n(b, \tilde{a}) \xi(A \# \tilde{a}) \tag{98}$$

which is a Λφ-module homomorphism and raises the degree by n. The chain map Φ induces a homomorphism on cohomology

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds http://dx.doi.org/10.5772/65663 107

$$\Phi: FH^\*(M, \Lambda\_\phi) \to H^\*(M, h, \Lambda\_\phi) = \frac{\text{Ker}\delta}{\text{Im}\delta} \simeq QH^\*(M, \Lambda\_\phi). \tag{99}$$

Similarly, we can construct a chain map,

$$\Psi: \mathbb{C}^\*(M, h, \Lambda\_\phi) \to \mathbb{C}^\*(M, H) \tag{100}$$

$$(\Psi \xi)(\tilde{a}) := \sum\_{\mu(\tilde{a}) + n = \text{ind}\_{\hbar}(b) - 2c\_{1}(A)} n((-A) \# \tilde{a}, b) \xi(b, A). \tag{101}$$

Then Φ∘Ψ and Ψ∘Φ are chain homotopic to the identity. Thus we have an isomorphism Φ.

We have studied the Gromov-Witten invariants on symplectic manifolds ðM, ω, JÞ using the theory of J-holomorphic curves, and the Gromov-Witten type invariants on almost contact metric manifolds ðN, g,ϕ, η, ξ, φÞ with a closed fundamental 2-form φ using the theory of ϕcoholomorphic curves. We also have some relations between them. We can apply the theories to many cases.

#### Examples 4.2.

ξ : CritðhÞH2ðMÞ ! R (91)

nhða, bÞξðb, AÞ,ða, AÞ∈CritðhÞH2ðMÞ, (94)

dimMðb, ~aÞ ¼ indhðbÞ−μð~aÞ−n: (96)

ðM, h, ΛφÞ (97)

nðb, ~aÞξðA#~aÞ (98)

νðBÞξða, A þ BÞ, (93)

uðsÞ ¼ b, (95)

#{ða, AÞjξða, AÞ≠0, φðAÞ≥c} < ∞ (92)

that satisfy the finiteness condition

106 Manifolds - Current Research Areas

counted with appropriate signs.

solutions of

for all c∈R. The Λφ-module structure is given by

<sup>ð</sup>δξÞða, <sup>A</sup>Þ ¼ ∑

b

<sup>f</sup> <sup>u</sup>\_ðsÞ þ <sup>∇</sup>uðsÞ ¼ <sup>0</sup>, lim<sup>s</sup>!−<sup>∞</sup>

to the quantum type cohomology of M with coefficients in Λφ.

elements can be used to construct the chain map defined by

codimension 2n−indhðbÞ in the distribution D.

homomorphism on cohomology

<sup>ð</sup><sup>ν</sup> � <sup>ξ</sup>Þða, <sup>A</sup>Þ ¼ ∑

B

where nhða, bÞ is the number of connecting orbits from a to b of shift equivalence classes of

Here we assume that the gradient flow of h is Morse-Smale and so the number of connecting orbits is finite when indhðaÞ−indhðbÞ ¼ 1. Then the coboundary operator δ is a Λφ-module homomorphism of degree one and satisfies δ∘δ ¼ 0. Its cohomology is canonically isomorphic

For each element <sup>~</sup>a<sup>∈</sup> <sup>~</sup> <sup>P</sup>ðH<sup>Þ</sup> we denote <sup>M</sup>ð~a, <sup>H</sup>,ϕ<sup>Þ</sup> by the space of perturbed <sup>ϕ</sup>-cohomomorphic maps <sup>u</sup> : <sup>C</sup> ! <sup>M</sup> such that <sup>u</sup>ðre<sup>2</sup>πit<sup>Þ</sup> converges to a periodic solution <sup>a</sup>ðt<sup>Þ</sup> of the Hamiltonian system Ht as r ! ∞. The space Mð~a, H,ϕÞ has dimension n−μð~aÞ. Now fix a Morse function h : M ! R such that the downward gradient flow u : R ! M satisfying (95) is Morse-Smale. For a critical point <sup>b</sup>∈Critðh<sup>Þ</sup> the unstable manifold <sup>W</sup><sup>u</sup>ðb, <sup>h</sup><sup>Þ</sup> of <sup>b</sup> has dimension indhðb<sup>Þ</sup> and

If indhðbÞ ¼ μð~aÞ þ n, then Mðb, ~aÞ is 0zero-dimensional and hence the numbers nðb, ~aÞ of its

ðM, HÞ ! C�

indhðbÞ¼μð~aÞþ<sup>n</sup>

which is a Λφ-module homomorphism and raises the degree by n. The chain map Φ induces a

The submanifold <sup>M</sup>ðb, <sup>~</sup>a<sup>Þ</sup> of all <sup>u</sup>∈Mð~a, <sup>H</sup>,ϕ<sup>Þ</sup> with <sup>u</sup>ð0Þ∈W<sup>u</sup>ðb<sup>Þ</sup> has dimension

Φ : FC�

<sup>ð</sup>ΦξÞðb, <sup>A</sup>Þ<sup>Λ</sup> ∑

<sup>u</sup>ðsÞ ¼ <sup>a</sup>, lim<sup>s</sup>!þ<sup>∞</sup>

the grading is degða, AÞ ¼ indhðaÞ−2c1ðAÞ, and the coboundary operator δ is defined by


Examples 4.3. Let N be a quintic hypersurface in CP<sup>4</sup> which is called a Calabi-Yau threefold. Then N is symply connected, c1ðTNÞ ¼ 0 and its Betti numbers b<sup>0</sup> ¼ b<sup>6</sup> ¼ 1, b<sup>1</sup> ¼ b<sup>5</sup> ¼ 0, b<sup>2</sup> ¼ b<sup>4</sup> ¼ 1 and b<sup>3</sup> ¼ 204.

Let <sup>A</sup> be the standard generator in <sup>H</sup>2ðN<sup>Þ</sup> and <sup>h</sup>∈H<sup>2</sup> ðNÞ such that hðAÞ ¼ 1. The moduli space <sup>M</sup>0,3ðN, <sup>A</sup><sup>Þ</sup> has the dimension zero. The Gromov-Witten invariant <sup>Φ</sup><sup>N</sup>,<sup>A</sup> <sup>0</sup>,<sup>3</sup> ða1, a2, a3Þ is nonzero only when degðaiÞ ¼ 2, <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3. In fact, <sup>Φ</sup><sup>N</sup>,<sup>A</sup> <sup>0</sup>,<sup>3</sup> ðh, h, hÞ ¼ 5 [4, 5]. The quantum cohomology of N is QH� ðNÞ ¼ H� ðNÞ⊗Λ where Λ is the universal Novikov ring [5].

Let <sup>ð</sup>N, <sup>g</sup>1, <sup>ω</sup>1, <sup>J</sup>1<sup>Þ</sup> be the standard Kähler structure on <sup>N</sup> and <sup>S</sup><sup>1</sup> , <sup>g</sup>2,ϕ<sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>η</sup><sup>2</sup> <sup>¼</sup> <sup>d</sup>θ, <sup>ξ</sup><sup>2</sup> <sup>¼</sup> <sup>d</sup> <sup>d</sup><sup>θ</sup> , <sup>φ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup><sup>Þ</sup> the standard contact structure on <sup>S</sup><sup>1</sup> . Then the product <sup>M</sup> <sup>¼</sup> NS<sup>1</sup> has a canonical cosymplectic structure ðM, g,ϕ, η, ξ, φÞ as in Section 3. The quantum type cohomology of M is

$$QH^\*(M) = QH^\*(N) \otimes QH^\*(S^1) \tag{102}$$

Let ψ<sup>1</sup> : N ! N be a Hamiltonian symplectomorphism with nondegenerate critical points. Then #Fixðψ1Þ≥∑ 6 i¼0 biðNÞ ¼ 208.

Let ψ<sup>2</sup> : M ! M be a Hamiltonian contactomorphism with nondegenerate critical points. Then #Fixðψ2Þ≥∑ 7 i¼0 biðMÞ ¼ 416.
