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

1 Problem

Let $\sigma$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\sigma$ be a generator of $\mathbb{Z}/ 4\mathbb{Z}$$\mathbb{Z}/ 4\mathbb{Z}$ and consider the free action of $\Z/4$$\Z/4$ on $S^2 \times S^2$$S^2 \times S^2$ defined by

$\displaystyle \sigma(x, y) = (y, -x), \text{ where } (x,y)\in S^2 \times S^2.$ $$\displaystyle \sigma(x, y) = (y, -x), \text{ where } (x,y)\in S^2 \times S^2.$$

Let $M := S^2\times S^2/ \langle \sigma \rangle$$M := S^2\times S^2/ \langle \sigma \rangle$ be the quotient of $S^2 \times S^2$$S^2 \times S^2$ obtained from this free action.

To understand the structure of this quotient, first, notice that $\sigma^2$$\sigma^2$ restricted to the diagonal copy of $S^2 \subset S^2 \times S^2$$S^2 \subset S^2 \times S^2$ is the antipodal map.

So the diagonal projects down to the projective plane $\mathbb{R}P^2$$\mathbb{R}P^2$ inside the quotient. Denote a normal disk bundle neighbourhood of this projective plane by $N$$N$.

Off the diagonal, the structure of $S^2 \times S^2/\langle \sigma \rangle$$S^2 \times S^2/\langle \sigma \rangle$ is that of a mapping cylinder. Namely, the mapping cylinder of the double cover of the lens space $L(8,1)$$L(8,1)$ by the lens space $L(4, 1)$$L(4, 1)$.

So

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$N \cup \rm{MCyl} (L(4, 1) \to L(8, 1))$ is a model for the quotient $M = S^2\times S^2/ \langle \sigma \rangle$$M = S^2\times S^2/ \langle \sigma \rangle$. Modifying that mapping cylinder by taking the double covering $L(4, 1) \to L(8, 3)$$L(4, 1) \to L(8, 3)$, it can be shown that

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$N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and

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$N \cup \rm{MCyl} (L(4, 1) \to L(8, 3))$ are 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: Are

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$N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and

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$N \cup \rm{MCyl}(L(4, 1) \to L(8, 3))$ diffeomorphic?

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 $S^2 \times S^2$$S^2 \times S^2$. Available at the arXiv:17172.04572.