Non-orientable quotients of the product of two 2-spheres by Z/4Z
1 Problem
Let be a generator of
and consider the free action of
on
defined by
![\displaystyle \sigma(x, y) = (y, -x), \text{ where } (x,y)\in S^2 \times S^2.](/images/math/4/1/0/410b7a611ca68037f161f579b45ce172.png)
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
.
Tex syntax erroris homotopy equivalent to the quotient
![M = S^2\times S^2/ \langle \sigma \rangle](/images/math/c/b/5/cb5a7c02ff675d7663d34edb7b0436ba.png)
![L(8, 3)](/images/math/9/2/0/9204a51a19df9eda18cddf97fc229c88.png)
Tex syntax errorand
Tex syntax errorhave the same homotopy type.
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.
The question is ifTex syntax errorand
Tex syntax errorare homeomorphic or even diffeomorphic.