It is quite obvious that if a map is null homotopic, then its mapping cylinder is contractible, but is the converse true: mapping cylinder contractible => the map is null homotopic? I am thinking about both the topological category and the category of chain complexes.

It is quite obvious that if a map is a homotopy equivalence, then its mapping cone is contractible, but is the converse true: mapping cone contractible => the map is a homotopy equivalence? I am thinking about both the topological category and the category of chain complexes.