Timeline for Can the product of an exotic torus and a circle be the standard torus?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2023 at 16:09 | answer | added | Michael Albanese | timeline score: 16 | |
| Dec 8, 2022 at 17:06 | comment | added | Igor Belegradek | If $M$, $N$ are closed manifolds such that $M\times\mathbb R$ can also be written as $N\times \mathbb R$, and if we pick $t$ so that $M\times 0$, $N\times t$ are disjoint, then the manifold bounded by these two "slices" is an h-cobordism. Just argue that the inclusion of boundary components is injective and surjective on homotopy groups. Note that the product deformation retracts to every slice. | |
| Dec 8, 2022 at 16:32 | comment | added | Michael Albanese | @IgorBelegradek: I can see how a diffeomorphism from $T\times\mathbb{R}$ to $T^{n-1}\times\mathbb{R}$ gives rise to a cobordism between $T$ and $T^{n-1}$, but I don't see why it is an h-cobordism. Is it always the case that a cobordism constructed this way is an h-cobordism, or are you using the asphericity of $T$ and $T^{n-1}$ somehow? | |
| Nov 22, 2022 at 16:11 | comment | added | Igor Belegradek | The existence of an exotic 4-torus (or more generally any exotic aspherical closed 4-manifold) is a well-known open problem. I seem to recall that most exotic 4-manifolds must become standard after taking product with a circle. Please feel free to write the answer yourself - I am not sufficiently motivated. | |
| Nov 22, 2022 at 16:02 | comment | added | Michael Albanese | @IgorBelegradek: Thanks! Even though it doesn't cover the four-dimensional case, would you be willing to expand your comments into an answer? I wouldn't be surprised if the four-dimensional case is open. | |
| Nov 22, 2022 at 15:39 | comment | added | Igor Belegradek | The above assumes that $T$ is smoothable (to apply the smooth h-cobordsim theorem to any smooth structure on $T$). For homotopy tori of dimension $\ge 4$ smoothability is proved in [Fake tori, Hsiang-Shaneson], corollary 4.3, maths.ed.ac.uk/~v1ranick/papers/hsiashan.pdf. | |
| Nov 22, 2022 at 15:12 | comment | added | Igor Belegradek | If $T$ is a homotopy torus, then any diffeomorphism $f:T\times S^1\to T^n$ lifts to a diffeomorphism of covering spaces corresponding to $\pi_1(T)\times 1$ and its $f_*$-image. It is a diffeomorphism from $T\times\mathbb R$ to $T^{n-1}\times\mathbb R$. This defines an h-cobordism between $T$ and $T^{n-1}$, and since tori have trivial Whitehead group, by the s-cobordism theorem we get a diffeomorphism of $T$ and $T^{n-1}$ if $n\ge 6$. The case $\dim(T)=3$ follows by the Poincare conjecture (due to Perelman). The only remaining case is when $T$ is $4$-dimensional. | |
| Nov 22, 2022 at 14:36 | history | asked | Michael Albanese | CC BY-SA 4.0 |