We discuss the chemical synthesis of topological links, in particular higher order links which have the Brunnian property (namely that removal of any one component unlinks the entire system). Furthermore, we suggest how to obtain both two dimensional and three dimensional objects (surfaces and solids, respectively) which also have this Brunnian property.
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
We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E * (−). Our description is as a graded and completed version of a Tall-Wraith monoid. The E * -cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.
Abstract. In this paper we investigate bundles whose structure group is the loop group LU (n). These bundles are classified by maps to the loop space of the classifying space, LBU (n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle ξ. This is essentially a decomposition of ξ as ζ ⊗ LC, where ζ is a finite dimensional subbundle of ξ and LC is the space ofThe criterion is a reduction of the structure group to the finite rank unitary group U (n) viewed as the subgroup of LU (n) consisting of constant loops. Next we study the case where one starts with an n dimensional bundle ζ → M classified by a map f : M → BU (n) from which one constructs a loop bundle Lζ → LM classified byLf : LM → LBU (n). The tangent bundle of LM is such a bundle. We then show how to twist such a bundle by elements of the automorphism group of the pull back of ζ over LM via the map LM → M that evaluates a loop at a basepoint. Given a connection on ζ, we view the associated parallel transport operator as an element of this gauge group and show that twisting the loop bundle by such an operator satisfies the criterion and admits a Fourier decomposition.
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