An embedding space triple of the unit interval into a graph and its bundle structure

Abstract

Let l 2 denote a Hilbert space, and let l 2 Q ={(x i )∈l 2 |sup|i.x i |<∞} and l 2 f ={(x i )∈l 2 |x i =0 except for finitely many i}. We show that the triple (H(X), H LIP (X), H PL (X)) of spaces of homeomorphisms, of Lipschitz homeomorphisms, and of PL homeomorphisms of a finite graph X onto itself is an (l 2 ,L 2 Q , l 2 f )-manifold triple, and that the triple (E(I,X), E LIP (I,X), E PL 5I,X)) of spaces of embeddings, of Lipschitz embeddings, and of PL embeddings of I=[0,1] into a graph X is an (l 2 ,l 2 Q ,l 2 4 )-manifold triple