# About why Danielewski surfaces are the counterexamples for cancellation problem

The Danielewski surfaces $xy^n=1-z^2$ are the famous counterexamples for cancellation problem for affine spaces. I'm asking where to find the articles telling the whole story. Especially how to distinguish them in the sense of isomorphism (I heard it involves the fundamental group at infinity, which I'm quite curious about but not familiar with). I tried google scholar but the papers are investigating other properties of them.

Thanks!

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I wrote about this counterexample in another answer, but I suppose it'd be useful to list the relevant articles here. In addition to the sources below, a good friend of mine wrote a nice exposition of this topic, and there is a very nice overview by Kraft:

Hanspeter Kraft, Challenging problems on affine $n$-space, Astérisque (1996), no. 237, Exp. No. 802, 5, 295–317, Séminaire Bourbaki, Vol. 1994/95. MR 1423629 (97m:14042)

The original article, which to my knowledge is not available online, is

Wlodzimierz Danielewski, On a cancellation problem and automorphism group of affine algebraic varieties, preprint, Warsaw, 1989.

Fieseler describes how to compute the "homology at infinity" for the two surfaces, as well as proving why the surfaces are isomorphic after taking the product with $\mathbf{A}^1$ (see Rem. 1.5):

Karl-Heinz Fieseler, On complex affine surfaces with $\mathbf{C}^+$-action, Comment. Math. Helv. 69 (1994), no. 1, 5–27. MR 1259603 (95b:14027)

Tom Dieck describes the "fundamental group at infinity," which you are particularly interested in, and his proof of non-isomorphicity is of a more geometric flavor:

Tammo tom Dieck, Homology planes without cancellation property, Arch. Math. (Basel) 59 (1992), no. 2, 105–114. MR 1170634 (93i:14012)

Makar-Limanov discovered what he calls the "AK-invariant," which algebraically distinguishes the surfaces $n=1$ and $n>1$, and further results in the following paper distinguish surfaces with different values of $n$:

Leonid Makar-Limanov, On the group of automorphisms of a surface $x^ny = P(z)$, Israel J. Math. 121 (2001), 113–123. MR 1818396 (2001m:14086)

There is a textbook account of this last article: see Rowen's Graduate algebra: commutative view, starting at p. 201.

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