Double Point Manifolds of Immersions of Spheres in Euclidean Space
Peter John Eccles · Princeton University Press eBooks · 1996
Anyone who has been intrigued by the relationship between homotopy theory and differential topology will have been inspired by the work of Bill Browder. This note contains an example of the power of these interconnections. We prove that, in the metastable range, the double point manifold a selftransverse immersion S n # R n+k is either a boundary or bordant to the real projective space RP n\\Gammak . The values of n and k for which non-trivial double point manifolds arise are determined. 1 Introduction Given a self-transverse immersion f : S n # R n+k , the r-fold intersection set I r (f) is defined as follows: I r (f) = f f(x 1 ) = f(x 2 ) = : : : = f(x r ) j x i 2 S n ; i 6= j ) x i 6= x j g: The self-transversality of f implies that this subset of R n+k is itself the image of an immersion (not necessarily self-transverse) ` r (f ): L n\\Gammak(r\\Gamma1) # R n+k of a manifold L of dimension n \\Gamma k(r \\Gamma 1) called the r-fold intersection manifold of f . It is...