Recall that Bott's obstruction for integrability [Bott71] asserts that:

> Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every polynomial of Pontryagin classes of $TM/E$ vanishes in degree $>2q$. 

In [p. 144, Hae71], Haefliger says:

> ($\ast$) The theorem of Bott ... implies that the induced homomorphism 
$$H^*(BO(q);\mathbb{R})\to H^*(B\Gamma_q;\mathbb{R})$$
is zero for $\ast>2q$. (I simplified his claim a little; he makes a similar assertion for $B\Gamma_q^r,$ for all $r\geq 1$.)

(Here $\Gamma_q$ denotes the topological groupoid whose objects are the points of $\mathbb{R}^q,$ and whose morphisms $x\to y$ are the germs of a smooth diffeomorphism between open sets of $\mathbb{R}^q$ which carries $x$ to $y$. The space of groupoids is topologized as a quotient of $\coprod_{U\to V}U$, where the coproduct ranges over all $C^\infty$ diffeomorphisms of open sets of $\mathbb{R}^q$.) 

**Can anyone explain why Bott's theorem implies ($\ast$)?** I do not think this is so trivial, for $B\Gamma_q$ only classifies $\Gamma_q$-structures, something more general than codimension $q$ foliations. 

I am sure this is well-known, but I haven't been able to find a reference that explains this point. I appreciate any help or comment. Thanks in advance.

**P.S.** In the first proposition of [BH71], Bott and Haefliger even assert that characteristic classes of foliations are in *bijection* with cohomology classes of $B\Gamma_q$; again I do not see why this is the case. I would appreciate it if someone could enlighten me on this point.


----------


## References ##

[Bott71] Bott. R, On topological obstructions to integrability. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp. 27–36
Gauthier-Villars Éditeur, Paris, 1971


[Hae71] Haefliger, A, Homotopy and integrability. Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), pp. 133–163
Lecture Notes in Math., Vol. 197
Springer-Verlag, Berlin-New York, 1971

[BH71] Bott, R, On characteristic classes of 
$\Gamma$-foliations.Bull. Amer. Math. Soc.78(1972), 1039–1044.