Stable Morse Number

The stable Morse number of a closed smooth manifold is the minimum number of critical points of a Morse function on , allowing any nonnegative integer , that is almost quadratic at infinity. This means that is uniformly bounded for a product Riemannian metric, where is a nondegenerate quadratic form in the auxiliary variables. The quadratic form may have either signature. Allowing shows that the stable Morse number is at most the ordinary Morse number. For a positive-dimensional sphere it is 2. The stabilization is a condition on actual smooth functions, beyond an algebraic minimization over chain complexes.