Manifolds of Piecewise Linear Maps and a Related Normed Linear Space

1. Spaces of piecewise linear maps. Let X and Y be separable polyhedra, X compact and Y locally compact; for the moment let them be connected and of dimension > 0 . Denoting the separable hilbert space of square-summable sequences by h, a space is an kmanifold if separable, metrizable and locally homeomorphic to k. In [4] the author showed that the space C(X, Y) of all continuous maps from X to Y with compact-open topology is an /2-manifold. I t is natural to ask whether the dense subspace PL(X, Y) consisting of all piecewise linear (p.l.) maps lies inside C(X, Y) in some "nice" way. For example, is PL(X, Y) an infinite-dimensional manifold? and if so what is its model? and how are PL(X, Y) and C(X, Y) related as manifolds? To answer, let l{ be the (dense, incomplete) linear subspace of h consisting of those sequences having only finitely many nonzero entries. Then we claim that PL(X, Y) is an /^-manifold. A pair {M, N) is an (Z2, l{)-manifold pair, if M is an ^-manifold for which there is an open cover 01 and open embeddings {ƒ17: U—>k\ t / G ^ } such that for each î/GOl, fu(UC\N) =fu(U)rM2. We claim that the pair (C(X, F), PL(X, Y)) is an (/2, /^-manifold pair. Among other things, it follows that PL(X, Y) has a (metric) triangulation, and that if PL(X, Y) is contractible, then it is homeomorphic to l2.