I want to compute cellular homology of the sphere $S^n$ using the CW structure where I have two cells $e^k_1,e^k_2$ for each dimension $0 \leq k \leq n$. Let me define the attaching maps to all be homeomorphisms that preserve orientation.
With this definition, when we compute the differential maps in the cellular chain complex, we get: $$\partial e^k_1 = \partial e^k_2 = e^{k-1}_1 + e^{k-1}_2.$$ The problem with this seems to be that $\partial^2 \neq 0$. Clearly, the problem must lie with defining the attaching maps to preserve orientation but I don't see where I went wrong.