Non-orientable quotients of the product of two 2-spheres by Z/4Z
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== Problem == | == Problem == | ||
<wikitex>; | <wikitex>; | ||
− | Let $\sigma$ be | + | Let $\sigma$ be a generator of $\mathbb{Z}/ 4\mathbb{Z}$ and consider the free action of $\Z/4$ on $S^2 \times S^2$ defined by |
$$\sigma(x, y) = (y, -x), \text{ where } (x,y)\in S^2 \times S^2.$$ | $$\sigma(x, y) = (y, -x), \text{ where } (x,y)\in S^2 \times S^2.$$ | ||
− | + | Let $M := S^2\times S^2/ \langle \sigma \rangle$ be the quotient of $S^2 \times S^2$ obtained from this free action. | |
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− | To understand the structure of this quotient, first, notice that $\sigma^2$ is | + | To understand the structure of this quotient, first, notice that $\sigma^2$ restricted to the diagonal copy of $S^2 \subset S^2 \times S^2$ is the antipodal map. |
− | So the diagonal projects down to the projective plane $\mathbb{R}P^2$ inside the quotient. Denote a disk bundle neighbourhood of this projective plane by $N$. | + | |
+ | So the diagonal projects down to the projective plane $\mathbb{R}P^2$ inside the quotient. Denote a normal disk bundle neighbourhood of this projective plane by $N$. | ||
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 $L(4, 1)$. | ||
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So $N \cup \rm{MCyl} (L(4, 1) \to L(8, 1))$ is homotopy equivalent to the quotient $M = S^2\times S^2/ \langle \sigma \rangle$. | So $N \cup \rm{MCyl} (L(4, 1) \to L(8, 1))$ is homotopy equivalent to the quotient $M = S^2\times S^2/ \langle \sigma \rangle$. | ||
− | Modifying that mapping cylinder by taking the map to $L(8, 3)$, | + | Modifying that mapping cylinder by taking the map to $L(8, 3)$, it can be shown that $N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and $N \cup \rm{MCyl} (L(4, 1) \to L(8, 3))$ have 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. | 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. |
Revision as of 06:02, 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 homotopy equivalent to the quotient . Modifying that mapping cylinder by taking the map to , it can be shown that
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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
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