TITLE: The second homotopy group of a simple CW-complex
QUESTION [17 upvotes]: Let $X$ be a CW-complex with 

one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher. 

Is it always true that $\pi_2(X)\ne 1$?

REPLY [14 votes]: There are classic examples, coming from small cancellation theory. See
the section of the Wikipedia article on asphericity.