ScalingStacks

Example 2.10 (Positive fibration) . [04HU]

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Example 2.10 (Positive fibration).

This model is the other possible fibration over a neighborhood of a point in Δd\Delta_{d} –in [7] this is called (1,2)(1,2) fibration. Let Y=S1×BY=S^{1}\times B with BB and Δ⊂B\Delta\subset B as in Example 2.8. Let Y′=Y∖({p}×Δ)Y^{\prime}=Y\setminus(\{p\}\times\Delta), where p∈S1p\in S^{1}. Let L≅ℤ2L\cong\mathbb{Z}^{2} and define T⁡(L)=L⊗ℤℝ/LT(L)=L\otimes_{\mathbb{Z}}\mathbb{R}/\penalty L. Now consider a principal T⁡(L)T(L)-bundle π′:X′→Y′\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}. Under some mild assumptions on π′\pi^{\prime} (cf. [7] Prop. 2.9), there is a unique manifold XX with X′⊂XX^{\prime}\subset X extending the topology of X′X^{\prime} and a proper extension π:X→Y\pi:X\rightarrow Y of π′\pi^{\prime}. The composition of π\pi with the projection Y→BY\rightarrow B defines a topological T3T^{3}-fibration, f:X→Bf:X\rightarrow B. The fibre of ff over b∈B∖Δb\in B\setminus\Delta is T3T^{3}. The fibre over b∈Δib\in\Delta_{i} is homeomorphic to S1×I1S^{1}\times I_{1}, whereas the fibre over the vertex b0∈Δb_{0}\in\Delta is homeomorphic to S1×T2/({p​o​i​n​t}×T2)S^{1}\times T^{2}/\penalty(\{point\}\times T^{2}). It is proved in [7] that the monodromy group of this model is generated, in some basis, by the inverse transpose of the matrices (3). The reader should not worry, at this point, for the lack of details in this description as we will give explicit Lagrangian models for this example later on.

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