Proof . Start with a usual subdivision of a direct product of 2 t d -spheres, with the cells indexed by pairs ( x, y ) of t -tuples of d -symbols. For x = y , the set ( x, y ) ∪ ( y, x ) /Z2 is a cell in X/Z2 , which we label {x, y} . To do the same for x = y , we need to take a finer subdivision of ( x, x ). Let ( x, x, k ) + , resp. ( x, x, k ) − , be the set of all points ¯ α ∈ R 2 dim x , ¯ α = ( αi ) i ∈ [2 dim x] , such that αj = αj+dim x , for k +1 ≤ j ≤ dim x , and αk > αk+dim x , resp. αk < αk+dim x . Obviously, ( x, x, k ) + and ( x, x, k ) − are cells, which are mapped to each other by the Z2 -action. These cells are different for k ≥ 1, whereas ( x, x, 0) + = ( x, x, 0) − is fixed pointwise.