Abstract
As we mentioned in the Introduction, we study systems of K-spaces (K-systems) so that the boundary of each of its members is described by a fiber product of other members. We will obtain an algebraic structure on certain cochain complexes which realize the homology groups of certain spaces. They are the spaces over which we take fiber products between members of the system of K-spaces.
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Notes
- 1.
- 2.
Such a gluing process is more involved for Kuranishi structures, for example.
- 3.
The condition |ω x|≥ 0 (Definition 7.4 (4)) is used to show \(\Pi (\mathrm {Supp}(\vert \omega _x\vert )) = \mathrm {Supp}(\vert \overline \omega _x\vert )\), where \((\overline W,\overline \omega _x,\overline {\mathfrak s^{\epsilon }})\) is a projection of \((W,\omega _x,{\mathfrak s^{\epsilon }})\).
- 4.
In fact for any x ∈ U 1 the restriction defines a map between germs:
, which induces the sheaf morphism (7.20). - 5.
The tubular distance function is, roughly speaking, the distance from the image \(\varphi _{\mathfrak p\mathfrak q}(U_{\mathfrak p\mathfrak q})\). We do not give the precise definition here since we do not use this notion. See [Ma].
- 6.
The condition |ω x|≥ 0 (Definition 7.4 (4)) is used to show \(\Pi (\mathrm {Supp}(\vert \omega _x\vert )) = \mathrm {Supp}(\vert \overline \omega _x\vert )\), where \((\overline W,\overline \omega _x,\overline {\mathfrak s^{\epsilon }})\) is a projection of \((W,\omega _x,{\mathfrak s^{\epsilon }})\).
- 7.
The existence of such δ 0 > 0 follows from Definition 3.15 (6).
- 8.
The integration here is taken on the virtual fundamental chain.
, which induces the sheaf morphism (References
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). CF-Perturbations and Integration Along the Fiber (Pushout). In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_7
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