Uniform and Lipschitz homotopy classes of maps

Abstract:

If $X$ is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line $R$ into $X$ an element of $H_{1} (X, U/U_{0}),$ where $U$ is the space of uniformly continuous functions from $R$ to $R$ and $U_{0}$ is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to $H_{1}(X,U/U_{0})$ is surjective. If $X$ is the $n$-dimensional torus, it is bijective, while if $X$ is a compact orientable surface of genus $>1$, it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds $P$ to compact smooth manifolds $X.$ With each such Lipschitz homotopy class we associate an element of $H_{n} (X, B^+/B_{0}^+),$ where $n$ is the dimension of $P,$ $B$ is the space of bounded continuous functions from the positive real axis to $R,$ and $B_{0}^+$ is the set of all $f\in B^+$ such that $\lim _{t \rightarrow \infty } f(t) = 0.$