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Let $U$, $V$ and $W$ be algebraic varieties of finite dimensions (in the case I am really interested, $U = \mathbb R$ and $V$ and $W$ are defined by a system of homogeneous polynomials in $\mathbb R ^{n+1}$, but the question is more general).

Assume that $U \times V$ and $U \times W$ are homeomorphic. Is it true then that $V$ and $W$ are homeomorphic?

Remark: $V$ and $W$ have the same Betti numbers as the Newton polynomials of the Cartesian product is the product of the polynomials.

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A contractible complex affine variety of dimension at least three is homeomorphic to the affine space of the same dimension. So take $U$ and $V$ to be affine spaces of suitable dimension and $W$ to be the Ramanujan surface (if you want real varieties take Weil restrictions).

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  • $\begingroup$ Is this a counter example ? You inversed U and W ? U is the variety that appears on both side in my question. Moreover, in my specific case, U is fixed to be the real line although the general question is interesting. $\endgroup$
    – Christophe Raffalli
    Commented Oct 12, 2021 at 20:19
  • $\begingroup$ $U$ appears on both sides. I was replying to the general question; I don't know a counterexample with $U=\mathbb{R}$. $\endgroup$
    – user409739
    Commented Oct 13, 2021 at 5:59

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