hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
Let $M$ be a finite dimensional smooth manifold. Its smooth loop space, $L M \coloneqq C^{\infty}(S^{1},M)$, is an example of a mapping space and is one of the simplest and most commonly encountered mapping spaces. It has many interesting submanifolds: some are defined by coincidences, such as the based loop space, and others are defined by considering fixed point sets of the natural circle action. An extremely useful tool when studying a submanifold is a tubular neighbourhood. Not all submanifolds have tubular neighbourhoods, so it is good to establish some results that establish that they exist. The situation for coindicence submanifolds is considered in tubular neighbourhood of a mapping space. Here we shall consider submanifolds defined by the circle action.
Although in the above we concentrated on the loop space, these results generalise to more general mapping spaces.
Let $E \to B$ be a fibre bundle with typical fibre $S$. Suppose that $E$ is compact, and that $S$ has a measure such that the transition functions are measure-preserving. Consider the submanifold of $C^{\infty}(E,M)$ of maps $E \to M$ which are constant on the fibres of $E \to B$. Then this submanifold has a tubular neighbourhood which is invariant under fibrewise measure-preserving diffeomorphisms of $E \to B$.
As a corollary of this, $G \subseteq S^{1}$ is a compact subgroup then the fixed point set $L M^{G}$ is a submanifold of $L M$ with an equivariant tubular neighbourhood.
This theorem is itself a corollary of a more specific situation where the base consists of a single point. In this case, we consider maps $S \to M$ and look at the submanifold of constant maps. We construct a tubular neighbourhood for this that is invariant as described, and the invariance allows us to extend the result to the more general situation.
The construction of the tubular neighbourhood depends on the existence of a local averaging function. This should be compared with the local addition which was needed to construct the manifold structure on the mapping space in the first place. A local addition is a way of saying: if two points are close, we can move one of them to the other. A local averaging function generalises this to a family of points indexed by some group and says that if such a family is close then we can move all of them towards each other in an equivariant fashion.
The reason that this is needed is because we want to define what it means for a map to be close to one in the fixed point set. The simplest case is where the source is a two point set, say $\{1,-1\}$. We want to say what it means for a map $\{1,-1\} \to M$ to be “close” to a constant map, but we want to say this in such a way that if we swap the labels, nothing changes.
The crucial ingredient in this is what we call a local averaging function. For a local averaging function, we need to be able to take a smooth map $\alpha \colon S \to M$ which is “small” in some sense, and define a single point in $M$ which is the “average” of $\alpha$. This should invariant under the action of measure-preserving diffeomorphisms of $S$ on $\alpha$.
Let us illustrate this with $S = \{1,-1\}$. So we have $\alpha \colon \{1,-1\} \to M$, which means two points $a_{1}, a_{-1} \in M$. We need to give a general rule as to how to move them together to a point $a_{0} \in M$ with the property that if we swap $a_{1}$ and $a_{-1}$ then we produce the same point. Once we have $a_{0}$ we can use a local addition (based at $a_{0}$) to move both $a_{1}$ and $a_{-1}$ towards it. This will be invariant under the group action because it only depends on the points $a_{1}$ and $a_{-1}$ and not on the labels. Thus the key is to find $a_{0}$, the “average” of $a_{1}$ and $a_{-1}$. If $M = \mathbb{R}^{n}$ then there is an obvious such point: $\frac{1}{2}a_{1} + \frac{1}{2}a_{-1}$. However, this will not work on $M$ because the sum does not make sense. A local addition could solve this, except that a local addition requires a choice of point and that is precisely what we are trying to avoid.
Our solution is to put $M$ inside $\mathbb{R}^{n}$. In $\mathbb{R}^{n}$, we can form $a_{0} \coloneqq \frac{1}{2}a_{1} + \frac{1}{2}a_{-1}$. However, the point $a_{0}$ may not lie on $M$, so we need a way to return it to $M$. We can do that if we know that $a_{0}$ is in a tubular neighbourhood of $M$. Then having found $a_{0}$, we can use a local addition at $a_{0}$ to move $a_{1}$ and $a_{-1}$ together.
There is one more subtlety. As we move $a_{1}$ and $a_{-1}$ together, we want to ensure that their “average” point stays at $a_{0}$. That is, we need to ensure that the local addition and the local average are independent. To see that this is a reasonable thing, note that the local addition is defined using the tangent space of $M$ whilst the tubular neighbourhood uses the normal bundle of the embedding. Thus their effects should be separable. To make this concrete, we use the orthogonal structure on the ambient Euclidean space.
Thus we start with an embedding of $M$ in some Euclidean space, say $\iota \colon M \to \mathbb{R}^{n}$. Then $d_{p} \iota \colon T_{p} M \to T_{\iota(p)} \mathbb{R}^{n}$ embeds $T_{p} M$ in $T_{\iota(p)} \mathbb{R}^{n}$. Using the orthogonal structure on $\mathbb{R}^{n}$, we can form the orthogonal complement of $T_{p} M$ in $T_{\iota(p)} \mathbb{R}^{n}$. Let us write this as $N_{p}$. Then $N_{p}$ is the fibre of the normal bundle of the embedding at $p$.
Let us identify $M$ with its image. Let us also identify $T_{x} \mathbb{R}^{n}$ with $\mathbb{R}^{n}$ except that we translate it so that the origin lies at $x$. The identification of $M$ with its image also allows us to identify $T_{p} M$ and $N_{p}$ with their images in $T_{p} \mathbb{R}^{n}$, and thus with their images in $\mathbb{R}^{n}$. They are then affine spaces anchored at $p$ and are orthogonal (once the translation is taken into account).
Our first task is to construct a tubular neighbourhood. From the above, we have a map $\nu \colon N \to \mathbb{R}^{n}$ which restricts on the zero section to the inclusion of $M$. At the zero section, the derivative of this map is the identification of $T_{p} M \oplus N_{p}$ with $T_{p} \mathbb{R}^{n}$. As this is an isomorphism, the inverse function theorem comes in to play and says that there is a neighbourhood $W$ of the zero section in $N$ and a neighbourhood $X$ of $M$ in $\mathbb{R}^{n}$ such that $\nu$ restricts to a diffeomorphism $W \to X$, which we shall also denote by $\nu$. Reversing $\nu$, we get a map $X \to W \subseteq N$ which we can combine with the projection to $M$. Let us write $\mu \colon X \to M$ for this composition.
This map $\mu$ is the local averaging function that we need. Well, technically the local averaging function is a bit more complicated but to define it in full one needs more data. However, that data will depend on the source of our maps so $\mu$ is the data that is solely dependent on $M$ that goes in to the mix that defines the local averaging function and so deserves that name.
The local averaging function is only half of what we need to construct a suitable tubular neighbourhood. The other half is provided by a local addition. It needs to be compatible with the local averaging function and so we assume that the construction of the previous section has been carried out.
Turning to the tangent spaces, for $p \in M$, the space $T_{p} M$ sits as an affine subspace in $\mathbb{R}^{n}$ with complement $N_{p}$. There is an orthogonal projection map $\mathbb{R}^{n} \to T_{p} M$, projecting along $N_{p}$. Restricting this to $M$, we obtain a map $\lambda_{p} \colon M \to T_{p} M$.
Note that $\lambda_{p}(p) = p$, which is the zero of $T_{p} M$, and that $d \lambda_{p} \colon T_{p} M \to T_{p}(T_{p} M) = T_{p} M$ is the identity. Allowing $p$ to vary, we obtain a smooth map $\lambda \colon M \times M \to T M$. If we think of $M \times M$ as a bundle over $M$ (via the projection on to the first factor) then $\lambda \colon M \times M \to T M$ is a bundle map.
Now we don’t actually need this to be defined on the whole of $M$, just on a small piece of it near $p$, but what “small” means here depends on $X$. Using the metric on $\mathbb{R}^{n}$ coming from the orthogonal structure, we choose a continuous function $\epsilon \colon M \to (0,\infty)$ with the property that the closed ball of radius $\epsilon(p)$ at $p$ is contained in $X$. Let us write $X_{p}$ for the corresponding open ball. Then we restrict $\lambda_{p}$ to $X_{p} \cap M$ and thus restrict $\lambda$ to the set
which is an open neighbourhood of the diagonal.
From the above, we see that (again by the inverse function theorem), there is a neighbourhood $V$ of the diagonal and a neighbourhood $U$ of the zero section such that $\lambda$ restricts to a diffeomorphism $\lambda \colon V \to U$. As we have already restricted $\lambda$ to $X_{M}$, $V$ is a subset of it.
Let $\eta \colon U \to M$ be the composition:
where the projection is on to the second factor. As $\lambda$ is a bundle map, $\lambda^{-1} = \pi \times \eta$ and thus $\eta$ is a local addition.
Let $U_{p} = U \cap T_{p} M$. Then $\eta(U_{p})$ is a subset of $\{p\} \times M$. Let $\eta_{p} \colon U_{p} \to V_{p} \subseteq M$ be the obvious restriction. Since $\lambda^{-1} = \pi \times \eta$ and the domain of $\lambda$ is a subset of $X_{M}$, the set $\eta(U_{p})$ is a subset of $X_{M} \cap \{p\} \times M = \{p\} \times X_{p}$. Thus $\eta_{p}(U_{p}) \subseteq X_{p}$. What this means is that the set $\eta_{p}(U_{p})$, which is a subset of $M$ and therefore may be all twisted and turned, nevertheless has the property that the closure of its convex hull is contained inside $X$, the tubular neighbourhood of $M$.
Thus we have the maps
The map $\eta_{p}$ works by projecting $U_{p} \subseteq T_{p} M$ on to $M$ along $N_{p}$. The map $\mu$ projects $X$ on to $M$ also by projecting along $N_{p}$.
What we have so far can be portrayed in the following diagram.
At this point it is time to we reintroduce our group, although at this stage it isn’t important that our source be a group. We need a compact smooth space; let us write this as $S$. We also need a measure on $S$ with the property that $S$ has measure $1$. The idea is to consider smooth maps $S \to M$ with the property that the closed convex hull of the image of $S$ is contained in $X$. Then given such a smooth map $\alpha \colon S \to M$ we can consider it as a map in to $\mathbb{R}^{n}$ and take its integral. As the closed convex hull of the image of $S$ is in $X$, the value of this integral will also be in $X$ and thus can be projected to $M$ via $\mu$. Let us write $\Cvx(S,M)$ for the set of smooth maps $S \to M$ with the property that the closed convex hull of $S$ lies in $X$ and let us write $\tau \colon \Cvx(S,M) \to M$ for this local averaging map.
Now that we know what is the “average” of a “small” map $\alpha \colon S \to M$, we can consider the question as to whether or not we can use the local addition to contract $\alpha$ to its average. To be able to do this, we need to further restrict our attention to those maps $\alpha \colon S \to M$ with the property that $(\tau \alpha, \alpha(s)) \in V$ for all $s \in S$. Knowing this, we can compose $\alpha$ with $\eta_{\tau \alpha}^{-1}$ to produce a map $S \to U$.
The construction is set up so that the resulting map $S \to U$ lies in a single fibre and has the property that it averages to $0$ (viewed in $T_{p} M$). This will define a diffeomorphism between the set of smooth maps $S \to U$ with those properties and an open subset of $C^{\infty}(S,M)$. Let us define:
The reverse map is straightforward to describe: given a map $\alpha \in C^{\infty}_{0}(S,U)$, we compose it with $\eta_{\pi \alpha}$ to produce a map $S \to M$.
The map $C^{\infty}_{0}(S,U) \to C^{\infty}(S,M)$ defined by composition with $\eta$ is a diffeomorphism on to its image which is open in $C^{\infty}(S,M)$.
To see that its image is open, let us first note that $\Cvx(S,M)$ is open since $X$ is open in $\mathbb{R}^{n}$ and $S$ is compact. Then within that, the condition on a map $\alpha \colon S \to M$ that $(\tau \alpha, \alpha(s)) \in V$ for all $s \in S$ is also an open condition, again because $S$ is compact. Hence the set of smooth maps $\alpha \colon S \to M$ with the property that the closed convex hull of $\alpha(S)$ lies in $X$ (whence $\tau \alpha$ is defined) and that $(\tau \alpha, \alpha(s)) \in V$ for all $s \in S$ is an open subset of $C^{\infty}(S,M)$.
Let us write this as $Y$. We have a map $C^{\infty}_{0}(S,U) \to Y$ given by composition: $\alpha \mapsto \eta_{\pi \alpha} \circ \alpha$; and a map $Y \to C^{\infty}_{0}(S,U)$ given by $\eta_{\tau \beta}^{-1} \circ \beta$. That the second is well-defined is not completely obvious: although $\eta_{\tau \beta}^{-1} \circ \beta$ lies in $U_{\tau \beta}$ by construction, we also have to show that its average is the zero in $T_{\tau \beta} M$.
This follows from the way that $\eta_{\tau \beta}$ and $\tau$ were constructed. Since averaging is a linear map, the average of $\beta$ splits as the sum of a point in $T_{\tau \beta} M$ and a point in $N_{\tau \beta} M$. However, by definition, $\tau \beta$ lies in $N_{\tau \beta}$ and thus the contribution from $T_{\tau \beta} M$ must be the zero vector.
To show that these are inverse, it is enough to show that $\tau (\eta_{\pi \alpha} \circ \alpha) = \pi \alpha$ and $\pi(\eta_{\tau \beta}^{-1} \circ \beta) = \tau \beta$.
The second is by construction: $\eta_{\tau \beta}$ has image in $U_{\tau \beta}$. The first follows by the argument on splitting the average: the average of $\eta_{\pi \alpha} \circ \alpha$ splits as the average of its projection to $T_{\pi \alpha} M$ plus the average of its projection to $N_{\pi \alpha}$. Since the former is $0_{\pi \alpha}$, the average of $\eta_{\pi \alpha} \circ \alpha$ lies in $N_{\pi \alpha}$ and hence $\tau \eta_{\pi \alpha} \circ \alpha = \pi \alpha$.
The last step in constructing the tubular neighbourhood is to construct a diffeomorphism between $C^{\infty}_{0}(S,U)$ and $C^{\infty}_{0}(S,T M)$. This is straightforward: we restrict $U$ (if necessary) so that it is diffeomorphic as a fibre bundle to $T M$, say via $\rho \colon U \to T M$, and then define the map $C^{\infty}_{0}(S,U) \to C^{\infty}_{0}(S, T M)$ via:
The above construction produces a tubular neighbourhood of the space of constant maps $S \to M$ in the space of smooth maps $S \to M$. The construction is invariant under measure-preserving diffeomorphisms of $S$. In particular, if $S$ is a group and the measure is invariant under the group action, the tubular neighbourhood is equivariant.
This construction extends to the case where the source is a fibre bundle $E \to B$ with fibre $S$ and the submanifold under consideration are those maps $E \to M$ which are constant on the fibres of $E \to B$. We need to assume that the transition functions preserve the measure on the fibres. The required tubular neighbourhood consists of those maps $E \to M$ with the property that when restricted to a fibre (identified with $S$ via a measure-preserving diffeomorphism), the map lies in the image of $C^{\infty}_{0}(S,M)$.