Topological Shift Spaces

Abstract

Abstract Two of the problems listed in [14, 74.17] ask to prove or disprove the following statements: A) For each differentiable planar map ƒ : IR2 → IR2 the set of all differentials defines a spread of IR4. B) If the differentials of a differentiable map ƒ : IR2 → IR2 define a spread of IR4 then the map ƒ is planar. By restricting to vertical 3-dimensional subspaces, we get the notion of a 3-dimensional shift space, and for differentiable shift spaces we may formulate analogous problems A′ and B′. Under the additional assumption that there exists a 1-dimensional group of shears, we prove A′ and B′ for 3-dimensional shift spaces and—as a corollary—also A and B for 4-dimensional shift planes.