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It seems that the following is true : if $V$ and $W$ are compact differentiable manifold of the same dimension, and $\mathbb{R} \times V$ is diffeomorphic to $\mathbb{R} \times W$, then $V$ and $W$ are diffeomorphic.

Where Can I find a proof of this to cite ?

Can I replace diffeomorphic with ambient isotopic, assuming $V$ and $W$ are sub variety of some projective space ?

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  • $\begingroup$ Off the top of my head I suspect that result isn't true. Presumably if you're in the dimensions where s-cobordism applies it would fail when the Whitehead group of the fundamental group is non-trivial. $\endgroup$
    – Ryan Budney
    Commented Sep 17 at 5:07
  • $\begingroup$ Can you elaborate, the only counter example I see from whitehead is with V and W non compact (and in fact non simply connected at infinity). $\endgroup$
    – Christophe Raffalli
    Commented Sep 17 at 5:11
  • $\begingroup$ And topological is may be a bit to weak. My manifold are DIFF. I edit the post. $\endgroup$
    – Christophe Raffalli
    Commented Sep 17 at 5:17
  • $\begingroup$ I'll think about it again in the morning. What I said might be nonsense, I'm tired. $\endgroup$
    – Ryan Budney
    Commented Sep 17 at 5:33
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    $\begingroup$ Let me just comment in passing that this isn't true without the “compact” hypothesis: if $V=\mathbb{R}^4$ and $W$ is an exotic $\mathbb{R}^4$, then $V\times\mathbb{R}$ and $W\times\mathbb{R}$ are both the standard $\mathbb{R}^5$. $\endgroup$
    – Gro-Tsen
    Commented Sep 17 at 7:22

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Let me convert my comment to an answer: Your expectation is false at least if the dimension of your manifold is $\ge 5$. Indeed, in every odd dimension $m\ge 5$ Milnor (Corollary 12.9 and Example 12.14 in the reference below) constructs examples of pairs of smooth compact manifolds $(L_1, L_2)$ and smooth $h$-cobordisms between these manifolds such that $L_1, L_2$ are not homeomorphic. Now, consider two pairs $(L_1, L_2), (L_2, L_1)$ and $h$-cobordisms $W_1, W_2$ between these pairs which are the same apart from the orientation. In particular, they have opposite torsions. Now, glue $W_1, W_2$ along the copies of $L_2$ via the identity map. The result is an $h$-cobordism from $L_1$ to $L_1$ with zero torsion. Hence, by the $s$-cobordism theorem, $W$ is diffeomorphic to the product $L_1\times [0,1]$. Then, glue together infinitely many such $h$-cobordisms. The resulting manifold $N$ will be diffeomorphic to $L_1\times \mathbb R$. At the same time, $N$ is also a union of trivial $s$-cobordisms between copies of $L_2$, hence, is diffeomorphic to $L_2\times \mathbb R$. Hence, $L_1\times \mathbb R$ is diffeomorphic to $L_2\times \mathbb R$, but $L_1$ is not homeomorphic to $L_2$.

Milnor, John W., Whitehead torsion, Bull. Am. Math. Soc. 72, 358-426 (1966). ZBL0147.23104.

A famous open problem along the lines of your question is: Suppose that $M_1, M_2$ are compact aspherical topological manifolds such that $M_1\times \mathbb R$ is homeomorphic to $M_2\times \mathbb R$. Is it true that $M_1$ is homeomorphic to $M_2$?

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  • $\begingroup$ thanks for this answer. and for lower dimension? $\endgroup$
    – Christophe Raffalli
    Commented Sep 18 at 4:59
  • $\begingroup$ In your example, there is a surjection from $L_1$ to $L_2$, because all points in $L_2$ are on exactly one of the fiber of $L_1 \times {t}$ of $W$. And this surjection seems differentiable. Right ? $\endgroup$
    – Christophe Raffalli
    Commented Sep 18 at 5:10
  • $\begingroup$ @ChristopheRaffalli: Of course, it is also a homotopy-equivalence. $\endgroup$
    – Moishe Kohan
    Commented Sep 18 at 13:58
  • $\begingroup$ You answer and Michael comment, let me think that I shoud add that in fact I know that $M$ and $N$ are h-cobordant. I even know that the cobordism $W \subset M \times [0,1]$ with the border $\partial W \simeq (M \times {0}) \cup N$. $\endgroup$
    – Christophe Raffalli
    Commented Sep 20 at 1:56
  • $\begingroup$ @ChristopheRaffalli: There is no need to add this, as h-cobordism follows from your assumptions. $\endgroup$
    – Moishe Kohan
    Commented Sep 20 at 3:23

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