The exponential law for partial, local and proper maps and its application to otopy theory (Q2923452)

``The main goal of [the] paper is to introduce a topology on the set of local maps from \(X \rightarrow Y\)'', denoted here by \(Loc \, ( Y^I)\), so that the paths on the space correspond to otopies. Local mapping spaces gave rise to homotopies between two local maps, by means of the exponential law: \(Z^{X^I}\) is homeomorphic to \(Z^{(X\times I)}\). The local map \({(X\times I ) \rightarrow} Y\) looks like it should be a homotopy, but the domains at 0 and 1 have no connection with each other. An otopy is a special kind of homotopy so that the domains of the functions on \(0 \times X\) are connected to the domains on \(0 \times X\) by a sequence of connected domains on \( t \times X\). This is ratified by the fact that the sum of the local degrees of a proper gradient map is the same at each end of the otopy if and only if the top and bottom gradient maps have the same sum of degrees.NEWLINENEWLINEFirst the authors define the set of \textit{partial maps}, \(Par(Y^X)\) by \(f:D_f \rightarrow Y\) where \(f:D_f \) is an open subset of \(X\). The exponential law for sets holds, and so also for spaces where \( Y\) is a topological space and \(Z\) is a locally compact Hausdorff space and \(X\) is Hausdorff and \(Y^X \) is the space of maps with the compact-open topology.NEWLINENEWLINENow \(\kappa: Par(Y^{(X)} ) \rightarrow (Y^+)^X : x \mapsto f(x) x\) if \(x \in D_f\) and \(x\mapsto +\), if \(x \notin D_f\) for any \(f\), and where \(Y^+\) is \(Y\) with an outside point +. Now \(\kappa\) is well-defined and a homeomorphism. And since \( Par(Y^{(X)} )\) is homeomorphic to a mapping space, the exponential law follows.NEWLINENEWLINEThe authors introduce a subbasis for any space X with a different topology \(\tau(X)\). They call it the \textit{compact-open topology} for \(\tau(X)\). ``For any compact set \(C\) in \(X\), let \(H(C)=\{ U\in \tau(X)\mid C \Subset U \}\). Then the collection \(\{H(C)=|C\Subset X\}\) forms a subbasis for a topology on \(X: U \rightarrow \).''NEWLINENEWLINEThe compact-open topology in \(\tau(X)\) is given by the homeomorphism \(\chi : \tau(X) \rightarrow Par(\mathfrak{p}^X)\) where \(\mathfrak{p}\) is a point space.NEWLINENEWLINEThe \textit{compactly generated topology} was used by Steenrod to describe a change of topologies so that the exponential law was true for the spaces involved, cf. \textit{N. E. Steenrod} [Mich. Math. J. 14, 133--152 (1967; Zbl 0145.43002)]. The \textit{compactly generated topology} is quite different from the \textit{compact-open topology} which is not \(T_1 \), hence not Hausdorff.NEWLINENEWLINENext, local maps are defined by \(Loc(Y^X,y)= \{f \in Par(Y^X)|f^{-1}(y) \Subset D_f\}\). This also satisfies an exponential law if \(X\) and \(Z\) are locally compact Hausdorff: \(Loc(Y^Z\times X, y)=((LocY^X,y)^{Z^*})_*\) where \(Z^*\) is the one-point compactification and the maps preserve base points, the empty map being the base point of \(Loc\) and \(*\) the base point of \(Z^*\).NEWLINENEWLINENext the space of proper local maps, \(Prop(Y^X)\), is defined such that every partial \(f\) inverse image of a compact set \(K\) is contained in \(D_f\). Thus every \(f\) is proper. \(Prop(Y^X)\) also satisfies an exponential law. If \(X\) is locally compact Hausdorff then \(Prop(Y^X)\) = \(Par(Y^X)\) where every \(f\) is a proper map. Thus \(Prop(X,Y) = Map_*(X^*,Y^*) \). So base points are fixed.NEWLINENEWLINENow \(Prop\) and \(Loc\) are compared in Euclidean space \(\mathbb{R}^{n+ k}\). Any element of \(Loc(I\times X,Y)\) is called an\textit{otopy} if \(X = {\mathbb R}^{n+ k}\) and \(Y={\mathbb R}^{n}\). Similarly, any element of \(Prop\) is called a \textit{proper otopy}.NEWLINENEWLINEThen it follows that the otopy components of \(Loc\) and \(Prop\) in the Euclidean space situation are weakly homotopy equivalent. Similarly the subspaces of \textit{gradient} and \textit{proper gradient maps} are weakly homotopy equivalent.NEWLINENEWLINEFinally a concrete example of an otopy is given with formulas in a one dimensional Euclidean space.