Peter J. Eccles

Codimension one immersions and the Kervaire invariant one problem

Let i : M ↬ℝ n +1 be a self-transverse immersion of a compact closed smooth n -dimensional manifold in ( n + 1)-dimensional Euclidean space. A point of ℝ n +1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self-transversality of i implies that the set of r -fold intersection points is the image of an immersion of a manifold of dimension n +1- r (the empty set if r > n + 1). In particular, the set of ( n + l)-fold intersection points is finite of order, say, θ( i ). In this paper we are concerned with the set of values of θ( i ) for (self-transverse) immersions of all (compact closed smooth) manifolds of given dimension n .

Double point self-intersection surfaces of immersions

A self-transverse immersion of a smooth manifold M k+2 in R 2k+2 has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k 1m od 4 or k+1 is a power of 2. This corrects a previously published result by Andr as Sz} ucs [22].

Determining the Characteristic Numbers of Self-Intersection Manifolds

The bordism class of a self-transverse immersion f: M n?k # R n corresponds to an element of the homotopy group n 1 1 MO(k). We explain how the Z=2 Hurewicz image h() 2 H n ((1 1 MO(k); Z=2) may be used to determine the characteristic numbers of the self-intersection manifolds r (f) of the immersion f.

Bordism groups of immersions and classes represented by self-intersections

A well-known formula of R J Herbert’s relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert’s formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert’s formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the formula to other homology theories. The proof is based on Herbert’s but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions. 57R42; 57R90, 55N22

Bordism classes represented by multiple point manifolds of immersed manifolds

We present a geometrical version of Herbert’s theorem determining the homology classes represented by the multiple point manifolds of a self-transverse immersion. Herbert’s theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert’s paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.

Double Point Manifolds of Immersions of Spheres in Euclidean Space

Anyone who has been intrigued by the relationship between homotopy theory and diierential 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 self-transverse immersion S n # R n+k is either a boundary or bordant to the real projective space RP n?k. The values of n and k for which non-trivial double point manifolds arise are determined.