Lemma A.1. Let X 1 , . . . X n be manifolds without boundary, let g i : X i → R be Morse functions, and let V : X 1 × • • • × X n → R be the additive function over a product space defined by V (x 1 , . . . , x n ) = ∑ n i=1 g i (x i ). Then V is a Morse function. Further, the point (x 1 , x 2 , . . . , x n ) is a critical point of V if and only if each x i is a critical point of g i . Finally, the index of the critical point (x 1 , x 2 , . . . , x n ) is equal to the sum of all indices of the building block functions: