# Non-orientable quotients of the product of two 2-spheres by Z/4Z

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''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? | ''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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## Revision as of 06:00, 8 January 2019

## 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 exactly four topological manifolds in this homotopy type, two of which are smoothable and two which have non-trivial Kirby-Siebenmann invariant.

*Question*: Are

Tex syntax errorand

Tex syntax errordiffeomorphic?

This question was posed by Jonathan Hillmann at the MATRIX meeting on Interactions between high and low dimensional topology.

## 2 References

- [Hambleton&Hillmann2017] I. Hambleton and J. Hillmann,
*Quotients of*. Available at the arXiv:17172.04572.