As shown in evaluation fibration of mapping spaces and tubular neighbourhoods of mapping spaces, if we carve out a submanifold of a mapping space by specifying “coincidences”, we often get a tubular neighbourhood. On this page, we shall give an example of a submanifold with no tubular neighbourhood. The example is simple to describe. To make it concrete, we shall fix as our source space the circle, . For our target space, we shall take a finite dimensional smooth manifold, . The full smooth mapping space, is known as the smooth loop space. For simplicity, let us take based loops within this, which we write as . Within that, we consider the space of based smooth maps which are infinitely flat at the point . Let us write this as . As we are using based loops, we can identify the tangent space of at the basepoint with and so we have a sequence, which is exact by Borel's theorem:
It is easy to show that this does not admit a tubular neighbourhood. If it did, there would be a splitting of the induced map on tangent spaces:
but as the second map is surjective, this cannot split as a splitting map would induce a continuous injection from to a normed vector space and that is impossible.