# Distinguishing two topological spaces

1) If $X$ is union of two unlinked circles and $Y$ is union of two linked circles, then $\pi_1(\mathbb{R}^3-X)=\mathbb{Z} * \mathbb{Z}$, and $\pi_1(\mathbb{R}^3-Y)=\mathbb{Z} \times \mathbb{Z}$ (See Algebraic Topology-Hatcher). Can we say that $X$ and $Y$ are not homeomorphic?

2)By "cut-and-paste", we can transform one space to another. Can we say that they are homeomorphic?

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$X$ and $Y$ are homeomorphic, but their complements in $\mathbb{R}^3$ aren't. Have you reviewed the definition of homeomorphism? It makes no reference to an ambient space. –  Qiaochu Yuan Sep 15 '11 at 3:51
Forget "paste" - try to think what sort of things you could cut an interval into. Or a sphere. Or anything, really. –  Alon Amit Sep 15 '11 at 7:13
If things were that easy, we wouldn't have knot theory! –  user641 Sep 15 '11 at 23:48
Disjoint unions of embedded circles (that is, a collection of knots) in a 3-dimensional surrounding space are called links. The notion of equivalence used for knots and links is ambient isotopy, not homeomorphism of the circles. Here the difference in $\pi_1$ of the complements shows that there is no homeomorphism of R^3 (including orientation reversals, and not only smooth isotopies) that carries one pair of circles onto the other. Fundamental groups are a complicated way of proving that the linked circles cannot be pulled apart, since a simpler invariant, the linking number, is zero for one pair and nonzero for the other.