{\mathop{\cong}}{{\mathbb P}^1}$, and in particular $\dim P_\alpha\times^B C
= \dim C + 1$. This $C$-bundle is trivial if $f$ is not just $B$- but $P_\alpha$-equivariant, with the projection onto the $C$ factor given by the $B$-quotient of the action map $P_\alpha \times C \to
C$; we study this further in Lemma \[lem:GeqvtBS\] below.

Since we generally turn the crank many times in succession, using a sequence $\vec \alpha = (\alpha_1,\ldots,\alpha_k)$, we will denote products of these functors $P_{\alpha_i} \times^B$ by $BS_{\vec\alpha}
:= P_{\alpha_k} \times^B \ldots \times^B P_{\alpha_1} \times^B$. A space $BS_{\vec\alpha}\cdot pt$ is a [[****]{} Bott-Samelson manifold]{}. The natural $G$-space for a point to map to $B$-equivariantly is $G/B$, so each Bott-Samelson manifold comes with a [[****]{}Bott-Samelson map]{} to $G/B$.

Seeing a Bott-Samelson manifold as a free quotient by $B$ on the right of $P_{\alpha_k} \times^B \ldots \times^B P_{\alpha_1}$, any Bott-Samelson manifold tautologically carries a principal $B$-bundle. It is sometimes useful to see the space $BS_{\vec\alpha} Z$ as the associated $Z$-bundle over the Bott-Samelson manifold $BS_{\vec\alpha}\cdot pt$.

\[lem:GeqvtBS\] Let $G$ act on two varieties $C,Y$ (which need not be linear), and let $f: C\to Y$ be a $G$-equivariant map. Let $\alpha_1,\ldots,\alpha_j$ be a sequence of simple roots.

Then the general 