Real projective space (${\mathbb{RP}}^n$)

Real projective space is the antipodal quotient ${\mathbb{RP}}^n=S^n/(x\sim -x)$.

This is the CW complex obtained as the increasing union of finite Real projective spaces under their standard inclusions. It has one cell in each nonnegative dimension. Its cellular chain complex has boundary multiplication by two in positive even dimensions and zero in odd dimensions. Consequently its integral cohomology is $\mathbb{Z}$ in degree zero, $\mathbb{Z}/2$ in positive even degrees and zero in odd degrees. Its mod-two cohomology is ${\mathbb{Z}}_2$ in every nonnegative degree.

Collapse the indicated standard skeleton to a point. Besides its basepoint, the quotient has one cell in dimensions $k+1$ through $m$. Its integral cellular boundary is two in even dimensions and zero in odd dimensions. Steenrod operations on its relative cohomology are inherited from real projective space.

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.

For the degree-one mod-two generator, ${\mathrm{Sq}}^{1}x=x^2$ and higher squares of $x$ vanish by instability. The Cartan formula gives the displayed binomial identity. Naturality applies it to relative classes of a stunted real projective space.

The $n+1$ manifold charts $U_i=\{[X]:X_i\ne 0\}$ have coordinates $X_k/X_i$, with $k\ne i$. Their inverse inserts $1$ as the $i$th homogeneous coordinate. On an overlap, changing the denominator from $X_i$ to $X_j$ divides each remaining coordinate by the nonzero coordinate $X_j/X_i$. These rational smooth transition maps give Real projective space its standard smooth manifold structure.

For $m\ge 0$, the cohomology ring isandThe generator $a$ reduces to the square of the degree-one generator in the mod-two cohomology ring of real projective space; $b$ is the integral orientation class in odd top dimension.

The cellular cochain complex of finite Real projective space consists of alternating maps zero and two. Its groups can be tensored using the Künneth theorem; the integral Künneth torsion polynomial efficiently counts all tensor and Tor functor summands. A product of factors of dimensions two, three and four has free-rank polynomial $1+t^3$ and torsion polynomial $3t^2+3t^3+5t^4+6t^5+5t^6+4t^7+2t^8+t^9$.

Real projective space has one cell in every dimension from zero through $n$. With integral coefficients its cellular differential $C_k\to C_{k-1}$ is multiplication by $1+(-1{)}^k$, hence is zero for odd $k$ and multiplication by two for even $k$. With coefficients in ${\mathbb{F}}_2$, every cellular differential vanishes.

The mod-two cohomology ring of real projective space is

The complement of an open three-ball in ${\mathbb{RP}}^3$ is the mapping cylinder of the antipodal covering $S^2\to {\mathbb{RP}}^2$. It deformation retracts to ${\mathbb{RP}}^2$ and has boundary $S^2$.

The integral homology groups are

Doubling ${\mathbb{RP}}^3$ minus an open ball along its spherical boundary gives ${\mathbb{RP}}^3\#{\mathbb{RP}}^3$. Its fundamental group isand its universal cover is $S^2\times \mathbb{R}$.