On Groups of Volume‐Preserving Diffeomorphisms and Foliations with Transverse Volume Form

Let Ω be a smooth nowhere‐zero n ‐form on a non‐compact n ‐dimensional manifold Y . We study the homology of the group DiffΩ Y , consisting of all diffeomorphisms of Y preserving Ω, provided with the discrete topology. In particular, if Ω is the standard volume form dx 1 ∧ … ∧ dx n on R n , and if n ¦ 2,3, we show that the natural maps B D i f f ΩR n ↪ B E m b ΩR n⟶ ≃ ⁡ B Γ Ω ninduce isomorphisms of integral homology. Here EmbΩ R n is the discrete monoid of smooth embeddings of R n preserving Ω, and BT Ω n is the Haefliger classifying space for codimension‐ n smooth foliations with a smooth non‐vanishing transverse closed n ‐form. (Analogous statements for the group of all diffeomorphisms of R n were proved by Segal in [ 16 ] with no restriction on dimension.) It follows that if n ¦ 2,3 the natural map B Diff Ω R n → B SO ( n ), where SO ( n ) is given its usual topology, induces an isomorphism on integral homology H i for 0 ⩽ i < n , and a surjection on H n with kernel isomorphic to R . If Y is a non‐compact manifold which is diffeomorphic to the interior of a compact n ‐dimensional manifold with boundary, and if every end of Y has infinite ‐volume, we show thatH 1 ( B D i f f Ω 0 Y ; Z ) ≅ H n ‐ 1 ( Y ; R ) ,providing that n ¦ 2,3. Here Diff ο0 ⊂ Diff ο denotes the subgroup consisting of elements smoothly isotopic to the identity.