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 \[ l_2^Q = \{ ({x_i}) \in {l_2}|\sup |i \cdot {x_i}| < \infty \} {\text { and }}l_2^f = \{ ({x_i}) \in {l_2}|{x_i} = 0{\text { except for finitely many }}i\} .\] We show that the triple $(H(X),{H^{{\text {LIP}}}}(X),{H^{{\text {PL}}}}(X))$ 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 $(E(I,X),{E^{{\text {LIP}}}}(I,X),{E^{{\text {PL}}}}(I,X))$ 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.