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In general, introduce the graded differential complexes: RW2m = Λ (h1 , . . , hm−1 , χm ) ⊗ I(SO(q))2m RW2m+1 = Λ (h1 , . . , hm ) ⊗ I(SO(q))2m+1 February 17, 2009 16:46 WSPC - Proceedings Trim Size: 9in x 6in viii-icdg 23 where dW (hj ⊗1) = 1⊗pj and dW (χm ⊗1) = em ⊗1. For I = (i1 < · · · < i ) and J = (j1 ≤ · · · ≤ jk ) set hI ⊗ p J = h 1 1 ∧ · · · ∧ h i ⊗ p j 1 ∧ · · · ∧ p j k (11) Note that deg(hI ⊗ pJ ) = 4(|I| + |J|) − , and that dW (hI ⊗ pJ ) = 0 exactly when 4i1 + 4|J| > q. In the following, the expression hI ⊗ pJ will always assume that the indexing sets I and J are ordered as above.

Barth, C. Peters, A. Van de Ven, Compact complex surface Springer-Verlag, Berlin 1984. 2. A. Candel, Adv. Math. 176 (2003), no. 2, 187-247. 3. A. Candel, L. Conlon, Foliations. II. Graduate Studies in Mathematics, 60. AMS, Providence, RI, 2003. 4. B. Deroin, V. Kleptsyn, GAFA vol. 17 (2007) 4 1043-1105. 5. B. Deroin, C. Vernicos, In preparation. 6. J. E. Fornaess, N. Sibony, arXiv:0606744. 7. H. Furstenberg, Amer. J. Math. 83 (1961), 573-601. 8. H. Furstenberg, Ann. of Math. (2) 77 (1963) 477-515.

Given any smooth map ϕ : M → SO(q), we obtain a new framing s = s · ϕ : M → P by setting s (x) = s(x) · ϕ(x). Thus, ϕ can be thought of as a gauge transformation of the normal bundle Q → M . The cohomology of the Lie algebra so(q) is isomorphic to an exterior algebra, generated by the cohomology classes of left-invariant closed forms τj ∈ Λ4j−1 (so(q)) for j < q/2, and the Euler form χm ∈ Λ2m−1 (so(q)) when q = 2m. The map ϕ pulls these back to closed forms ϕ∗ (τj ) ∈ Ω4j−1 (M ). 2 (Lazarov33,34 ).