Constructing smooth manifolds of loop spaces

Abstract

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists.We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations. Being inThat is, for a loop γ : S 1 → R, then γ ∈ L x R if there exists an open cover U of S 1 and loops γ U ∈ L x R for U ∈ U such that γ agrees with γ U on U.2. The set L x R is a subspace of Map(S 1 , R).3. The vector space L x R can be given a topology with respect to which it is a locally convex topological vector space. The locally convex topological vector spaceThis is a completeness condition. We have phrased it in the language of [KM97] but it is the same as a concept known as locally complete which is

The space of Heegaard splittings

“…The analogous statement holds for the space ΩX of smooth based free loops [31,Section 4.3]). The map LX → X given by evaluation at the basepoint is a locally trivial fibration [31,Corollary 4.8], with fiber ΩX. Since LX → X has an obvious section, the exact sequence of this fibration shows that π q (LX) ∼ = π q+1 (X) ⊕ π q (X) for all q ≥ 1.…”

On somes characteristic classes of flat bundles in complex geometry

“…Proof. -This is proved in great generality in [21,Theorem 4.6], for untwisted free loops spaces. This paper also deals with based loop spaces in Section 4.3…”

Topological complexity of symplectic manifolds

“…both equipped with the respective Whitney topology. As a consequence of the results of [22], see also [30], both M and P M possess the structure of infinite-dimensional Fréchet manifolds, locally modelled on the Fréchet spaces For k ∈ N we let k (M) denote the space of smooth real-valued k-forms on M. For each ω ∈ k (M) we letω ∈ k (M × M) denote the form given bȳ…”

“…resp. There are homotopy equivalences M L M and P M P M, see[30, Section 4.2] for details. We may thus consider M and P M as "differentiable replacements" of spaces of continuous paths and loops.…”