Stunted real projective space (${\mathbb{RP}}^m/{\mathbb{RP}}^k$)

For $0\le k<m$, 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 $k+1,\dots ,m$.

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 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 sequenceThus its nonzero reduced integral homology groups are ${\tilde{H}}_2=\mathbb{Z}$ and ${\tilde{H}}_3=\mathbb{Z}/2$. Steenrod squares on the relative mod-two cohomology are inherited from Real projective space. If $k=m$, the quotient is a single point.

For this quotient, homotopy groups through degree six and in degree eight vanish. The degree-seven and degree-nine groups are $\mathbb{Z}/2$. A map to $K(\mathbb{Z}/2,7)$ induces an isomorphism on degree-nine integral homology because the relevant square of the bottom mod-two class is nonzero. Two relative Hurewicz steps then compute degrees eight and nine.