Higher-dimensional Bruckner–Garg type theorem

In this paper, we investigate several properties of maps from a compactum X to an n-dimensional (combinatorial) manifold Mn. We introduce the notions of stable point and locally extreme point of map, and we prove a higher-dimensional Bruckner–Garg type theorem for the fiber structure of a generic map in the space C(X,Mn) of maps from a compactum X with dimX⩾n to an n-dimensional manifold Mn (n⩾1). As applications, we also study the spaces of Bing maps, Lelek maps, k-dimensional maps and Krasinkiewicz maps in C(X,Mn).