Non-orientable quotients of the product of two 2-spheres by Z/4Z
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1 Problem
Let be the generator of the free action of on determined by
There is a (unique) geometric quotient obtained by this free action.
To understand the structure of this quotient, first, notice that is just the antipodal map on the diagonal. So the diagonal projects down to the projective plane inside the quotient. Denote a 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 .So
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Tex syntax errorand
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In \url{https://arxiv.org/pdf/1712.04572.pdf}, 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.
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