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 M n . 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 , M n ) of maps from a compactum X with dim X ⩾ n to an n-dimensional manifold M n ( n ⩾ 1 ). As applications, we also study the spaces of Bing maps, Lelek maps, k-dimensional maps and Krasinkiewicz maps in C ( X , M n ) .