Emb(m, N)

Aim of this note is to gain cohomological information about the manifold Emb(M, N) from the topology of the target manifold N . For special conditions, a monomorphism H(N)→ H(Emb(M,N)) is constructed. Let Embg(M,N) be the manifold of all embeddings of a manifold M into a manifold N that differ from a fixed reference embedding g only within a compact subset of M . Note that if M is compact then Embg(M,N) = Emb(M,N), the space of all embeddings. Now and in the following M should be nonempty and N at least one-dimensional to give sense to the notion of 1-forms. Embg(M,N) is infinitedimensional iff dim(M) ≥ 1. Note. The following theorem 2 is a straightforward consequence within the framework of fiber integration (cf. [2], [1]), nevertheless it may be instructive to see another proof for the particular case. For our construction, a crucial tool is the identification of a tangent vector V resp. the value of a vector field V on Embg(M,N) at a fixed map f with a vector field along f , i.e. a section of fTN : ˆ|f : Vect(Embg(M,N)) → Γ(f TN), V̂ |f (p) := LV evp or equivalently, V̂ |f : p 7→ ∂t(f(p)), where p ∈ M , ft a curve representing V (f). This means, we just fix a point p ∈ M and note the direction in which it is moved infinitesimally by the family of maps ft. V̂ |f has compact support because ⋃ t∈[−1;1]{ft 6= g} is compact. Conversely, a vector field V on N maps to a vector field Ṽ on Embg(M,N) by ̃: Vect(N) → Vect(Embg(M,N)) Ṽ : f 7→ ∂t|t=0(Fl V t ◦ f), where Fl t is the flow of V .