tubular neighbourhood of a mapping space



The result of evaluation fibration of mapping spaces extends to more general evaluation maps between mapping spaces. One way to interpret that result is that the inclusion C (S,p;M,q)C (S,M)C^\infty(S,p;M,q) \to C^\infty(S,M) has a tubular neighbourhood. Providing MM has enough diffeomorphisms, this is true of more general inclusions where they are defined by “coincidences”. That is to say, if PP is a condition on maps SMS \to M that prescribes where certain points “coincide”, then the submanifold of C (S,M)C^\infty(S,M) of smooth maps satisfying this condition will have a tubular neighbourhood in the manifold of all smooth maps.