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.



Fundamental groups and homology groups of closed subsets of the plane
**Fundamental Group:** The fundamental group of a planar set naturally injects into the first Cech homotopy group, which is an inverse limit of free groups. In particular, the algebraic restrictions gained from this fact are: the fundamental group must be locally free, fully residually free (and thus torsion free), and residually finite. See:



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> 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, doi:[10.2140/agt.2005.5.1655](https://doi.org/10.2140/agt.2005.5.1655), arXiv:[math/0512343](http://arxiv.org/abs/math/0512343).
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**Cech Homology:** It follows that the first homology pro-group is an inverse system of of finitely generated free groups. The first Cech homology group $\check{H}\_1(X)$ is an inverse limit of free abelian groups.


**Singular Homology:** This is more complicated. Certain cases are understood.


*Easy case.* If $X$ is locally path-connected and semilocally simply connected, then $H\_1(X)\cong \check{H}\_1(X)$ is free abelian.


*Harder case.* Suppose your planar set $X$ is path-connected but not semilocally simply connected. To simplify things, let's at least suppose $X$ is a Peano continuum (locally path-connected compact metric space). It is a general result of Katsuya Eda that for any Peano continuum $X$ (planar or otherwise) the canonical map $\phi:H\_1(X)\to \check{H}\_1(X)$ is surjective. See:



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> Katsuya Eda, Kazuhiro Kawamura, *The surjectivity of the canonical homomorphism from singular homology to Cech homology*, Proc. Amer. Math. Soc. **128** No. 5 (1999) pp 1487-1495, doi:[10.1090/S0002-9939-99-05670-1](https://doi.org/10.1090/S0002-9939-99-05670-1)
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This result helps because it means that if know $\phi$ splits and we can identify $\ker(\phi)$, then we can "compute" $H\_1(X)$.


Now, if $X$ happens to also be one-dimensional (such as the Hawaiian earring or Sierpinski carpet) then we can do exactly that. In particular, it turns out that $\check{H}\_1(X)\cong \prod\_{n\in\mathbb{N}}\mathbb{Z}$ and $\ker(\phi)\cong \prod\_{n\in\mathbb{N}}\mathbb{Z}\Big/ \bigoplus\_{n\in\mathbb{N}}\mathbb{Z}$ where the second isomorphism is purely abstract. Moreover, since $\prod\_{n\in\mathbb{N}}\mathbb{Z}\Big/ \bigoplus\_{n\in\mathbb{N}}\mathbb{Z}$ is algebraically compact, the homomorphism $\phi$ splits. We can conclude that for any (planar) one-dimensional Peano continuum $X$, that
$$H\_1(X)\cong \prod\_{n\in\mathbb{N}}\mathbb{Z}\oplus \left(\prod\_{n\in\mathbb{N}}\mathbb{Z}\Big/ \bigoplus\_{n\in\mathbb{N}}\mathbb{Z}\right).$$


The fact that all of these spaces have exactly the same first singular homology group tells you that abelianizing $\pi\_1$ for these spaces actually kills all of the geometry remembered by $\pi\_1$.


For details, see:



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> Katsuya Eda, Kazuhiro Kawamura, *The Singular Homology of the Hawaiian Earring*, Journal of the London Mathematical Society **62** Issue 1 (2000) pp 305–310, doi:[10.1112/S0024610700001071](https://doi.org/10.1112/S0024610700001071) ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/eda-kawamura2.pdf))
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> Katsuya Eda, *Singular homology groups of one-dimensional Peano continua*, Fundamenta Mathematicae **232** Issue 2 (2016) pp 99–115, doi:[10.4064/fm232-2-1](https://doi.org/10.4064/fm232-2-1) ([pdf](http://www.logic.info.waseda.ac.jp/%7Eeda/pdf/singular.pdf))
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*Possibly unknown case.* As of 2019, I don't think $H\_1$ is known for a general 2-dimensional planar set (or Peano continuum) because path reduction is not as straightforward as in the 1-dimensional case. However, in the end, the answer is likely to be similar to the one-dimensional case.


