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Over Real projective space ${\mathbb{RP}}^n$, the real tautological line bundle isOn the standard chart $U_i=\{[x]:x_i\ne 0\}$, let $s_i([x])=x/x_i$. A local trivialization sends $([x],t)$ to $([x],t{s}_i([x]))$, with inverse $([x],v)\mapsto ([x],v_i)$. These maps prove that this is a real line bundle. Its unit sphere bundle is the antipodal map quotient $S^n\to {\mathbb{RP}}^n$. Its mod-two Euler class of a real line bundle is the degree-one generator of the mod-two cohomology ring of real projective space.

A simply connected space is path connected and has trivial fundamental group: every closed loop contracts to a point. The three-sphere is simply connected, whereas Real projective space ${\mathbb{RP}}^3$ has fundamental group ${\mathbb{Z}}_2$.

The Adjoint double cover from SU(2) to SO(3) has kernel $\{I,-I\}$. Under SU(2) as the three-sphere, multiplication by $-I$ is the antipodal map, so the quotient is Real projective space ${\mathbb{RP}}^3$. Equivalently, an axis-angle ball of radius $\pi$ has opposite boundary points identified. The fundamental group is ${\mathbb{Z}}_2$.

For the real tautological bundle $\gamma$ over Real projective space, the tangent space at a line is $\mathrm{Hom}(\gamma ,{\gamma }^{\perp })$. Adding $\mathrm{Hom}(\gamma ,\gamma )\cong {\epsilon }^1$ gives $\mathrm{Hom}(\gamma ,{\epsilon }^{m+1})\cong (m+1)\gamma$, using a metric to identify a real line with its dual. Hence its total Stiefel–Whitney class is $(1+a{)}^{m+1}$, with $a=w_1(\gamma )$.

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 a real rank-$r$ vector bundle $E\to B$, the Thom isomorphism theorem with ${\mathbb{F}}_2$ coefficients identifies the relative cohomology sequence of its disk bundle and sphere bundle withThe multiplier is the top Stiefel–Whitney class, also called the mod-two Euler class. No orientation of $E$ is required. For the real tautological line bundle, the sphere bundle is the antipodal cover of Real projective space; this sequence proves that all powers of the degree-one generator through the dimension of the base are nonzero.