The de Rham complex on infinite dimensional manifolds.

*(English)*Zbl 0636.58004The de Rham complex of a smooth manifold is called split if the space of n-forms \(\Omega^ n\) can be decomposed as \(\Omega^ n=B^ n\oplus H^ n\oplus \tilde B^{n+1}\) where \(B^ n=d(\Omega^{n+1})\) is the space of coboundaries in \(\Omega^ n\), \(H^ n\subset \ker (d)\) is the cohomology in dimension n and \(\tilde B^{n+1}\) is a subspace mapped isomorphically to \(B^{n+1}\) by d. If M is compact and oriented the de Rham complex is split by the Hodge de Rham decomposition theorem. The paper deals with the extension of this splitting result for compact manifolds to certain infinite dimensional manifolds. The infinite dimensional manifolds are \(C^{\infty}\)-manifolds in the sense of J. Milnor [“On infinite dimensional Lie groups”; preprint]. They are modelled on topological vector spaces which are Hausdorff and locally convex. The space of n-forms is given the smooth compact open topology such that the coboundary operators of the de Rham complex are continuous operators.

The author proves theorems on the splitting of this de Rham complex. The following are examples of the type of results. (1) The de Rham complex of M is split if M is a finite union of smoothly contractible sets with all intersections smoothly contractible; (2) The de Rham complex is split for all open submanifolds of certain topological vector spaces. The general results are applied to the smooth manifolds \(C^{\infty}(M,N)\) (smooth maps from M to N) and D(M) (diffeomorphisms of M), where M is a compact manifold and N a separable finite dimensional manifold. The last section deals with the Van Est spectral sequence, which gives a relation between different cohomology theories of Lie groups.

The author proves theorems on the splitting of this de Rham complex. The following are examples of the type of results. (1) The de Rham complex of M is split if M is a finite union of smoothly contractible sets with all intersections smoothly contractible; (2) The de Rham complex is split for all open submanifolds of certain topological vector spaces. The general results are applied to the smooth manifolds \(C^{\infty}(M,N)\) (smooth maps from M to N) and D(M) (diffeomorphisms of M), where M is a compact manifold and N a separable finite dimensional manifold. The last section deals with the Van Est spectral sequence, which gives a relation between different cohomology theories of Lie groups.

Reviewer: K.H.Mayer