nLab
A Not-So-Nice Submanifold

Idea

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, $S^1$. For our target space, we shall take a finite dimensional smooth manifold, $M$. The full smooth mapping space, $C^{\mathrm{\infty }}(S^1,M)$ is known as the smooth loop space. For simplicity, let us take based loops within this, which we write as $\Omega M$. Within that, we consider the space of based smooth maps $S^1\to M$ which are infinitely flat at the point $1\in S^1$. Let us write this as ${\Omega }_{♭}M$. As we are using based loops, we can identify the tangent space of $M$ at the basepoint with ${ℝ}^n$ and so we have a sequence, which is exact by Borel's theorem:

(1)$${\Omega }_{♭}M\to \Omega M\to \prod _{j=1}^{\mathrm{\infty }}{ℝ}^n$$\Omega_♭ M \to \Omega M \to \prod_{j = 1}^\infty \mathbb{R}^n

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:

(2)$$T_{\alpha }{\Omega }_{♭}M\to {\Omega }_{\alpha }M\to \prod _{i=1}^{\mathrm{\infty }}{ℝ}^n$$T_\alpha \Omega_♭ M \to \Omega_\alpha M \to \prod_{i = 1}^\infty \mathbb{R}^n

but as the second map is surjective, this cannot split as a splitting map would induce a continuous injection from ${\prod }_{i=1}^{\mathrm{\infty }}{ℝ}^n$ to a normed vector space and that is impossible.