All Questions

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 $\pi_2(X)\ne 1$?

Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...