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Let $X$ be a closed subset of $\mathbb{R}^2$. What restrictions are there on $\pi_1(X)$ and on the homology groups of $X$ (both singular and Cech)? This is elementary if $X$ has reasonable local properties, but the example of the Hawaiian Earring shows that things can be very complicated indeed.

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See this question: mathoverflow.net/q/189323/1345 –  Ian Agol Dec 10 at 18:32

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The fundamental group of a closed planar set naturally injects into the first Cech homotopy group, which is an inverse limit of finitely generated free groups. In particular, the algebraic restrictions gained are: the fundamental group must be locally free, fully residually free, and residually finite.

Fischer, H., Zastrow, A., The fundamental groups of subsets of closed surfaces inject into their first shape groups, Algebraic and Geometric Topology 5 (2005) 1655-1676.

The first homology pro-group then consists of finitely generated free groups and the first Cech homology group is the inverse limit of these.

When $X\subset \mathbb{R}^2$ is compact and locally path connected, the canonical map $H_1(X)\to \check{H}_1(X)$ is surjective but the kernel can be difficult to understand even for the Hawaiian earring.

Eda, K., Kawamura, K. The surjectivity of the canonical homomorphism from singular homology to Cech homology Proc. Amer. Math. Soc. 128 No. 5 (1999) 1487-1495

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Eda, K. Fundamental group of subsets of the plane. Topology and its Applications Volume 84, Issues 1-3, 24 April 1998, Pages 283-306

This is more or less a duplicate of the math.se thread: http://math.stackexchange.com/questions/36279/the-fundamental-group-of-every-subset-of-mathbbr2-is-torsion-free

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A duplicate of that question?!?!?!? Really? Other than the fact that they both concern the topology of subsets of the plane, they seem like rather different questions to me (and the questioner over there seemed really clueless). Are you suggesting that I should have asked it over there? –  Tina Oct 28 '12 at 3:20

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