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

Question

Recall that Bott's obstruction for integrability [Bott70] 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 $r> 1$.)

(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.

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References

[Bott70] Bott, Raoul. On a topological obstruction to integrability. Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968), pp. 127–131 Proc. Sympos. Pure Math., XIV-XVI American Mathematical Society, Providence, RI, 1970

[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; Haefliger. A, On characteristic classes of $\Gamma$-foliations. Bull. Amer. Math. Soc.78(1972), 1039–1044.

Answer 1

This is explained very nicely in [Law77]. Here is a sketch.

We will use the following crucial lemma.

Lemma (Haefliger). Let $M$ be a smooth manifold and let $\mathcal{H}$ be a $\Gamma_q$-structure on $M$. There are a smooth fiber bundle $p:E\to M$ with fiber $\mathbb{R}^q$, a $\Gamma_q$-foliation $\mathcal{F}$ on $E$, and a continuous section $s:M\to E$ such that $s^*\mathcal{F}=\mathcal{H}$.

(Some people seem to call $E$ the foliated microbundle.)

Accepting the leemma for now, we can deduce assertion ($\ast$) from Bott's theorem, as follows. Let $c\in H^n(B\Gamma_q;\mathbb{R})$ be any element which is a polynomial of Pontryagin classes, where $n>2q$. We wish to show that $c=0$. Let $\sigma\in H_n(B\Gamma_q;\mathbb{R})$ be an arbitrary cocycle. We must show that $\langle c,\sigma \rangle=0$. Since every finite CW complex has the homotopy type of a smooth manifold, we can find a smooth manifold $M$, an element $\sigma'\in H_n(M;\mathbb{R})$, and a continuous map $f:M\to B\Gamma_q$ such that $f_\ast(\sigma')=\sigma$. Using the lemma (and replacing $M$ if necessary), we may assume further that $f$ classifies a $\Gamma_q$-foliation on $M$. But then Bott's theorem shows that $f^*c=0$, so that $\langle c, \sigma\rangle =\langle f^*c,\sigma'\rangle=0.$ The proof is now complete. $\square$

And here's a sketch proof of the lemma, for the sake of completeness.

(Proof of the lemma)

Represent $\mathcal{H}$ by a cocycle $(\{U_i\}_{i\in I}, \{\gamma_{ij}\}_{i,j\in I}),$ where $\{U_i\}_{i\in I}$ is a locally finite cover of $M.$ For each point $x\in M$, let $I(x)\subset I$ denote the set of indices $i\in I$ such that $x\in U_i.$ By the definition of $\Gamma_q$-structures, there are a neighborhood $V_x\subset \bigcap_{i\in I(x)}U_i,$ an open set $O_i^x\subset \mathbb{R}^q$ which contains $\gamma_i(x)$ and is diffeomorphic to $\mathbb{R}^q$ for each $i\in I(x)$, and diffeomorphisms $\phi_ij^x:O_i^x\to O_j^x$ such that $\gamma_ij(x')=[\phi_{ij}^x,\gamma_i(x')]$ for all $x'\in V_x$. ($[-,-]$ denotes a germ.) By shrinking the $O_i^x$'s and $V_x$ if necessary, we may assume that $\phi_{ii}^x=\operatorname{id}$ and $\phi_{jk}^x\phi_{ij}^x=\phi_{ik}^x$ for all $i,j,k\in I(x)$.

Now construct a smooth manifold $E$ from the disjoint union $\coprod_{i\in I, x\in U_i}V_x \times O_i ^x$ by identifying $(x,y)\in V_{x_0}\times O_{i_0}^{x_0}$ with $(x,\phi _{ii'}^x(y))$ if $x\in V_{x_0}\cap V_{x_1}.$ This $E$ and the evident projections, sections, and foliations of $E$, have the desired properties. $\square$

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Reference

[Law77] Lawson, H. Blaine, Jr. The quantitative theory of foliations. Expository lectures from the CBMS Regional Conference held at Washington University, St. Louis, Mo., January 6–10, 1975 Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 27 American Mathematical Society, Providence, RI, 1977. v+65 pp.