TITLE: Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?
QUESTION [55 upvotes]: 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 $\pi_2$?

REPLY [41 votes]: Apparently the asphericity is due to Zastrow (see Cannon-Conner-Zastrow). 
Also apparently the result that the higher homology groups vanish is due to Zastrow, but his habilitation thesis never seems to have appeared.