Non-orientable quotients of the product of two 2-spheres by Z/4Z
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Off the diagonal, the structure of $S^2 \times S^2/\langle \sigma \rangle$ is that of a mapping cylinder. Namely, the mapping cylinder of the double cover of the lens space $L(8,1)$ by the lens space $L(4, 1)$. | Off the diagonal, the structure of $S^2 \times S^2/\langle \sigma \rangle$ is that of a mapping cylinder. Namely, the mapping cylinder of the double cover of the lens space $L(8,1)$ by the lens space $L(4, 1)$. | ||
− | So $N \cup \rm{MCyl} (L(4, 1) \to L(8, 1))$ is | + | So $N \cup \rm{MCyl} (L(4, 1) \to L(8, 1))$ is a model for the quotient $M = S^2\times S^2/ \langle \sigma \rangle$. |
− | Modifying that mapping cylinder by taking the | + | Modifying that mapping cylinder by taking the double covering $L(4, 1) \to L(8, 3)$, it can be shown that $N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and $N \cup \rm{MCyl} (L(4, 1) \to L(8, 3))$ are homotopy equivalent. |
− | In | + | In {{cite|Hambleton&Hillmann2017}} it is shown that there are at most four topological manifolds in this homotopy type, half of which are stably smoothable. |
− | + | ''Question'': Are $N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and $N \cup \rm{MCyl}(L(4, 1) \to L(8, 3))$ diffeomorphic? | |
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+ | This question was posed by Jonathan Hillmann at the [[:Category:MATRIX 2019 Interactions|MATRIX meeting on Interactions between high and low dimensional topology.]] | ||
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+ | [[Category:Problems]] | ||
[[Category:Questions]] | [[Category:Questions]] | ||
[[Category:Research questions]] | [[Category:Research questions]] |
Latest revision as of 08:48, 31 August 2020
[edit] 1 Problem
Let be a generator of and consider the free action of on defined by
Let be the quotient of obtained from this free action.
To understand the structure of this quotient, first, notice that restricted to the diagonal copy of is the antipodal map.
So the diagonal projects down to the projective plane inside the quotient. Denote a normal disk bundle neighbourhood of this projective plane by .
Off the diagonal, the structure of is that of a mapping cylinder. Namely, the mapping cylinder of the double cover of the lens space by the lens space .
SoTex syntax erroris a model for the quotient . Modifying that mapping cylinder by taking the double covering , it can be shown that
Tex syntax errorand
Tex syntax errorare homotopy equivalent.
In [Hambleton&Hillmann2017] it is shown that there are at most four topological manifolds in this homotopy type, half of which are stably smoothable.
Question: AreTex syntax errorand
Tex syntax errordiffeomorphic?
This question was posed by Jonathan Hillmann at the MATRIX meeting on Interactions between high and low dimensional topology.
[edit] 2 References
- [Hambleton&Hillmann2017] I. Hambleton and J. Hillmann, Quotients of . Available at the arXiv:17172.04572.