This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about the answer.


Is it possible for a subset of $\mathbb R^2$ to have a nontrivial singular homology group $H\_2$? What about a nontrivial homotopy group $\pi\_2$?



Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?
Apparently the asphericity is due to Zastrow (see [Cannon-Conner-Zastrow](http://www.sciencedirect.com/science/article/pii/S0166864101000050#)). 


Also apparently the result that the higher homology groups vanish is [due to Zastrow](https://mat.ug.edu.pl/~zastrow/nonanorn.ps), but his habilitation thesis never seems to have appeared. 


