all visibly onto and define the commuting square above. It remains to prove our claims about these maps.

Applying Lemma \[lem:GeqvtBS\] to the maps $BS_{\vec\alpha} \iota : BS_{\vec\alpha} Z\to G\cdot Z$, $BS_{\vec\beta} BS_{\vec\alpha} \iota :
  BS_{\vec\beta} BS_{\vec\alpha} Z\to G\cdot Z$, we see that the general fiber of $BS_{\vec\beta} BS_{\vec\alpha} \iota :
  BS_{\vec\beta} BS_{\vec\alpha} Z\to G\cdot Z$ has the same number of connected components as the general fiber of $BS_{\vec\alpha} \iota : BS_{\vec\alpha} Z\to G\cdot Z$, which (since it is generically finite-to-one) is just its degree.

In the case $Z=pt$, the following is a standard result about Bott-Samelson manifolds for partial flag manifolds.

\[lem:BSforGmodP\] Let $Z$ be a $B$-space, and $\vec \alpha$ a list of simple roots whose corresponding reflections $(r_{\alpha_i})$ give a reduced word for the Weyl group element $w_0 w_0^P$ where $w_0$ is the long element of $G$’s Weyl group and $w_0^P$ the long element of $P$’s. Then the map $BS_{\vec\alpha} \cdot Z \to G\times^P Z$ (constructed by applying $BS_{\vec\alpha}$ to the inclusion $Z {\mathop{\cong}}P\times^P Z {\operatorname*{\hookrightarrow}}G\times^P Z$ of the fiber over the basepoint) is a birational isomorphism.

These two spaces are $Z$-bundles, and the map takes fibers to fibers; as such it is equivalent to check that $BS_{\vec\alpha}\cdot 