= Stunted real projective space
{title2=$\mathbb{RP}^m/\mathbb{RP}^k$}
For $0\leq k<m$, collapse the standard <skeleton of a CW complex> $\mathbb{RP}^k$ inside <Real projective space> $\mathbb{RP}^m$. The quotient has one <base point> and one <topological cell> in each <dimension> $k+1,\ldots,m$.
Between two surviving consecutive cells, the integral <cellular boundary> is multiplication by two in even source <dimension> and zero in odd source <dimension>. The reduced <cellular boundary> \b[out of the bottom surviving cell] is zero, since the lower <cellular chain group> has been collapsed. This exception must be applied before computing <homology>.
For example, $\mathbb{RP}^4/\mathbb{RP}^1$ has cells in dimensions two, three and four, with reduced chain sequence
$$
0\longrightarrow\mathbb Z\xrightarrow{2}\mathbb Z\xrightarrow{0}\mathbb Z\longrightarrow0.
$$
Thus its nonzero reduced integral <homology groups> are $\widetilde H_2=\mathbb Z$ and $\widetilde H_3=\mathbb Z/2$. <Steenrod squares> on the relative mod-two <cohomology> are inherited from <Real projective space>. If $k=m$, the quotient is a single <point>.
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