On manifolds containing a submanifold whose complement is contractible

Extract

The problem discussed here arose in the course of some reflections on the critical point theory of Lusternik and Schnirelmann (4). In (4) it is shown how it is possible to associate, with a suitably differentifiable real-valued function f defined on a compact manifold M, a set of real numbers λ₁ ≤ λ2 ≤ … λ_c, which are critical levels of f and which in certain respects are analogous to, and indeed generalizations of, the eigenvalues of a quadratic form. The number c depends on M and is called the category of M. If Rⁿ is Euclidean n-space, Sⁿ the unit sphere of Rⁿ⁺¹, and Pⁿ the real projective n-space obtained from Sⁿ by identifying opposite points, then a quadratic form φ in the (n + 1) coordinates of Rⁿ⁺¹ defines a real function on Sⁿ and, by passage to the quotient, on Pⁿ. Pⁿ has category n + 1, and the numbers λ in this case are just the eigenvalues of the quadratic form.