Abstract
We axiomatize the properties enjoyed by the system of moduli spaces of solutions of Floer’s equation.
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Notes
- 1.
- 2.
We note that \({\mathcal M}({\alpha },{\alpha }) = \emptyset \) in particular.
- 3.
In the case of the linear K-system obtained from periodic Hamiltonian systems, we take \(o_R=\Theta _R^{-}\) for each critical submanifold R, where \(\Theta _R^{-}\) is defined as the determinant of the index bundle of a certain family of elliptic operators. See [FOOO4, Definition 8.8.2] for the precise definition. Then [FOOO4, Proposition 8.8.6] yields the isomorphism (16.2). Note the moduli space written as \(\mathcal M(\alpha _-,\alpha _+)\) in this book was written as \(\mathcal M(R_{\alpha _+},R_{\alpha _-})\) in [FOOO4]. See Remark 16.2 (1). On the other hand, in [FOOO4, Proposition 8.8.7] we take \(o_{R}=\det TR \otimes \Theta _R^{-}\). Note that we use singular chains in [FOOO4], while we use differential forms in the current book. So the choices of o R are slightly different.
- 4.
See [FOOO4, Section 8.8].
- 5.
Note that \(o_{R_{\alpha }}\) may not coincide with the principal O(1) bundle giving an orientation of R α. For example, \(o_{R_{\alpha }}\) may be nontrivial even in the case when R α is orientable. On the other hand, if \({\mathcal M}(\alpha _-,\alpha _+)\) is orientable then \(o_{{\mathcal M}(\alpha _-,\alpha _+)}\) is trivial.
- 6.
The right hand side is a finite union by Condition (IX).
- 7.
We do not assume any compatibility of the orientation isomorphism at the corner, because what we will use is Stokes’ formula where the boundary but not the corner appears.
- 8.
Namely, we collect the same data as in Conditions 16.1 as far as critical submanifolds are concerned.
- 9.
We use \(\mathcal F\) to denote a linear K-system. Here F stands for Floer.
- 10.
In Item (5) we put deg e = 2. The Novikov ring \(\Lambda _{\mathrm {nov}}^R\) here is the same as the one introduced in [FOOO3], where the indeterminate e has degree 2.
- 11.
Here the case \(\mathcal C_1=\mathcal C_2\) is also included.
- 12.
The right hand side is a finite sum by Condition (IX).
- 13.
In Condition 16.17 (V) \(\mathcal N(\alpha _1,\alpha _2)\) is required to be the empty set if E(α 1) = E(α 2) + c.
- 14.
We denote by \({\mathcal N}_{12}\) the interpolation space of the morphism \(\mathfrak N_{21}\). See Remark 18.33.
- 15.
- 16.
Hereafter we use the superscript i to label the elements in the inductive system.
- 17.
In other words, it is a homotopy of homotopies.
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). Linear K-Systems: Floer Cohomology I – Statement. In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_16
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