Abstract
Before studying fiber products we consider direct products. Let X i, i = 1, 2 be separable metrizable spaces, Z i ⊆ X i compact subsets, and \(\widehat {\mathcal U}_i\) Kuranishi structures of Z i ⊆ X i.
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Notes
- 1.
Let \(p:\nu _{\Gamma _g/M \times N} \to \Gamma _g\) be the normal bundle of the graph Γg of g. We identify \(\nu _{\Gamma _g/M \times N}\) with a tubular neighborhood U( Γg) of Γg in M × N. Then N has a Kuranishi chart \((U(\Gamma _g), \nu _{\Gamma _g/M \times N}, s_{can}, \psi _{can})\), where s can is the tautological section and ψ can is the identification of the zero section with N. Then \(pr_M\vert _{U(\Gamma _g)}: U(\Gamma _g) \to M\) is a submersion.
- 2.
The fiber product in the sense of category theory is always associative if it exists. Since we do not study morphisms between K-spaces, the fiber product we defined is not the fiber product in the sense of category theory. Therefore we need to prove its associativity. However, it is obvious in our case.
References
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Fukaya, K., Oh, YG., Ohta, H., Ono, K. (2020). Fiber Product of Kuranishi Structures. In: Kuranishi Structures and Virtual Fundamental Chains. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-5562-6_4
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DOI: https://doi.org/10.1007/978-981-15-5562-6_4
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