# Homology groups of compact subset of $\mathbb{R}^2$

I am working over the paper: Target Enumeration via Euler Characteristic Integrals and in order to follow a proof I need to prove:

If $A$ is compact nonempty subset of $\mathbb{R}^2$, then the singular homology groups of $A$, $H_k(A)$ vanish for $k\geq 2$.

The result seems seems reasonable for me (there are non obvious counterexamples in dimension grater than three but not in dimension two). But I am having difficulties finding a proof.

What I have tried or thought so far:

• Using the long exact sequence in homology and it didn't work.
• Maybe trying luck with de Rham cohomology and using some kind of isomorphism followed by the Universal Coefficients theorem in cohomology....
• Using some dimension theory but I have no clue how...

Any ideas? Thanks in advance and any help would be appreciated.

• Please do not ask the same question here and on math.stackexchange.coom at the same time. Please delete this one and if we have to we will move the other one here. – Mariano Suárez-Álvarez Feb 26 '17 at 18:43
• Link: math.stackexchange.com/questions/2162579/… . Unlike Mariano I'd rather suggest you erase the MathSE question and leave it here. – YCor Feb 26 '17 at 18:48
• Well, I took the time to add a couple of comments, chase a reference and explain why two of the ideas have no chance of working there. There is a reason why we do not like double posting, and it is precisely that: wasted and duplicated effort. – Mariano Suárez-Álvarez Feb 26 '17 at 18:49
• As I said in the other post, there are obvious counterexamples in dimension $\ge 2$: the 2-sphere. To be a counterexample, it is enough to have $H_2\neq 0$. Mariano mentioned more fancy counterexamples, with subsets of the 3-space with nonvanishing higher homology groups. – YCor Feb 26 '17 at 19:05
• A related fact: $\pi_i$ of planar sets (with basepoints) vanishes for $i\ge 2$: Cannon, Conner, Zastrow, One-dimensional sets and planar sets are aspherical. Topology Appl. 120 (2002), no. 1-2, 23-45. (MR link: ams.org/mathscinet-getitem?mr=1895481; ScienceDirect link: sciencedirect.com/science/article/pii/S0166864101000050. – YCor Feb 27 '17 at 0:59