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.

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. The first homology progroup 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. Fundamental group of subsets of the plane. Topology and its Applications Volume 84, Issues 13, 24 April 1998, Pages 283306 This is more or less a duplicate of the math.se thread: http://math.stackexchange.com/questions/36279/thefundamentalgroupofeverysubsetofmathbbr2istorsionfree 

