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Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gam...

I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations. … A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044. …
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Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gam...

.$ This $E$ and the evident projections, sections, and foliations of $E$, have the desired properties. $\square$ Reference [Law77] Lawson, H. Blaine, Jr. The quantitative theory of foliations. …
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