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

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

The outward-normal-first volume form on the unit sphere pulls back under the antipodal map by $(-1{)}^{n+1}$. For $n\ge 1$, the sphere is connected, and its orientation descends to the twofold quotient exactly when this deck transformation preserves it. Thus Real projective space is orientable precisely in odd positive dimensions; the zero-dimensional case is one point and is also orientable.

The quotient of the three-sphere by its antipodal map is a closed oriented three-manifold. Equivalently it is the lens space obtained by integer surgery of coefficient two on the unknot. Its fundamental group is cyclic of order two, from the sphere double covering. This makes it a basic example for Jones polynomial surgery invariants.

// Target: knot-theory.bigb

For $n\ge 2$, a nonsurjective self-map of Real projective space ${\mathbb{RP}}^n$ has zero mod-two degree of a map between closed manifolds. In the mod-two cohomology ring of real projective space, the nonzero degree-one class $a$ has $a^n\ne 0$. If pullback fixed $a$, it would fix $a^n$, contradicting zero degree. Thus the map is trivial on the fundamental group and lifts to $S^n$. A point missed by the original map has both its inverse images missed by the lift. The lift therefore takes values in a punctured sphere, which is contractible, and is null-homotopic. Composing its contraction with the covering projection proves the conclusion.

Choose a complex structure $J$ on ${\mathbb{R}}^{2m}$, with $J^2=-I$. Its induced map on $\mathbb{R}{P}^{2m-1}$ has no fixed point: $Jx=\lambda x$ for a real nonzero vector would give ${\lambda }^2=-1$. The maps $\cos tI+\sin tJ$, for $0\le t\le \pi /2$, give a homotopy from the identity to $J$, because they are invertible throughout. Thus this fixed-point-free map is homotopic to the identity.

Cover the circle by two arcs whose intersection has two components and take the product with the projective plane. The Mayer–Vietoris sequence has maps $(a,b)\mapsto (a+b,-a-b)$ on the two copies of the homology of the plane. Its kernels and cokernels, together with $H_1({\mathbb{RP}}^2;\mathbb{Z})=\mathbb{Z}/2$, give the displayed groups; all higher groups vanish. The product is a closed nonorientable three-manifold, but is homotopy equivalent to a noncompact oriented four-manifold.

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 unit sphere bundle of the tautological line is the double cover $S^n\to {\mathbb{RP}}^n$. In its Gysin sequence, multiplication by the mod-two Euler class $a$ is an isomorphism in successive interior degrees, since the sphere has no interior cohomology. Connectedness in degree zero makes multiplication into degree one injective. At the top, vanishing above dimension $n$ makes the transfer from $H^n(S^n;{\mathbb{F}}_2)$ surjective; together with that injection it makes the top group one-dimensional. Thus every $a^j$, $0\le j\le n$, is the nonzero generator and the next power vanishes.

For any continuous map between the displayed Real projective spaces, the induced map on cohomology sends the degree-one generator to $\alpha x+\beta y$. Its $2n$th power is $\left(\genfrac{}{}{0ex}{}{2n}{n}\right){\alpha }^n{\beta }^n{x}^n{y}^n$, since all other terms exceed the truncation exponent of one factor. The coefficient is zero modulo two because $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)=2\left(\genfrac{}{}{0ex}{}{2n-1}{n-1}\right)$. Naturality of the evaluation pairing then makes the top induced map on homology zero. This obstruction does not apply to $n=0$, when both spaces are points.

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 $S^2\times \mathbb{R}$.