$W_q$ and framed foliations

These algebras are useful for foliation $(M, \sF)$ with $Q$ trivializable, when one can transgress to a flat Riemannian connection and get

$$\mu_E\colon W_q\to\Omega^{\bullet}(M),$$

$$\mu_E(u_i):=T\lambda_E(\nabla^{\flat}, \nabla^{\sharp, 0})(c_i),$$

$$\mu_E(c_i):=\lambda_E(\nabla^{\flat})(c_i).$$

Notation: for $\underbrace{i_1<\hdots<i_r}_I$, $\underbrace{j_1\leqslant \hdots\leqslant j_s}_J$ we denote

$$u_Ic_J=u_{i_1}\hdots u_{i_r}c_{j_1}\hdots c_{j_r}.$$

Consequences of (a) for $\HH^*(W_q)$.

1. 
   
   $$\deg(u_Ic_J)=(2i_1-1)+\hdots+(2i_r-1)+(2j_1+\hdots+2j_s)\leqslant$$
   
   
   
   
   $$\leqslant 2(1+\hdots +q)-q+2\vert J\vert\leqslant q(q+1)-q+2q=q^2+2q.$$
   
   
   Hence
   
   $$\HH^m(W_q)=0,\mbox{ for }m>q^2+2q.$$
   
   

2. On the other hand
   
   $$\deg(u_Ic_J)\geqslant 2\vert J\vert>2q,$$
   
   
   hence
   
   $$\HH^m(W_q)=0,\mbox{ for }1\leqslant m< 2q.$$
   
   
   With a little more work we can elliminate $m=2q$ which can occur only if $\vert I\vert$ even.
3. The product structure is trivial.
4. In $\HH^{2q+1}(W_q)$ the classes $u_1 c_1^{\alpha_1}\hdots c_k^{\alpha_k}$ with $\sum_{i=1}^k \alpha_i=q$ are linearly independent

Similar conclusions hold for $\HH^*(WO_q)$:

1. 
   
   $$\HH^m(WO_q)=0,\mbox{ for }m>q^2+2q.$$
   
   

2. For $m\leqslant 2q$ one gets the Pontryagin classes
   
   $$\{1, p_1,\hdots, p_{\left[\frac{q}{2}\right]}\}.$$
   
   

3. The product structure is trivial in 'high degree'.
4. In $\HH^{2q+1}(WO_q)$ the classes $u_1 c_1^{\alpha_1}\hdots c_k^{\alpha_k}$ with $\sum_{i=1}^k \alpha_i=q$ are linearly independent.