CW-decomposition of quotient space

Question

Let $X$ be the space that results form $D^3$ by identifying points on the boundary $S^2$ that are mapped to one another by a $180°$-rotation about some fixed axis.

I want to calculate the cellular homology of $X$, but I have trouble finding a suitable CW-decomopistion. How does one generally approach finding a CW-decomopostion of a CW-complex, especially if they are more complex?

Answer 1

There is no general way of finding a CW-decomposition of an arbitrary space. In this specific example, note that $S^2$ with the prescribed identifications is simply $S^2$. Then in order to obtain our space $X$ from that we must glue in a 3-cell. Try to convince yourself that the attaching map of the $3$-cell is precisely the suspension of the degree $2$ map $$f:S^1\to S^1\\ z\mapsto z^2.$$ So we can give $X$ the cell structure of one $0$-cell, one $2$-cell and one $3$-cell where the $3$-cell is attached via a degree $2$ map. (The suspension of a degree $k$ map has degree $k$ - prove this, it's easy.)

This provides all the necessary ingredients to compute the (cellular) homology of this space.

Answer 2

Here's a hint: first find a CW decomposition of $S^2$ which is invariant under the $180^\circ$ rotation.