Codimension one immersions and the Kervaire invariant one problem

Abstract

IntroductionLet i : M<\ > U n+1 be a self-transverse immersion of a compact closed smootĥ -dimensional manifold in (n+ 1)-dimensional Euclidean space. A point of U 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+l-r (the empty set if r > n+ 1). In particular, the set of (n+ l)-fold intersection points is finite of order, say, 6(i). In this paper we are concerned with the set of values of 6(i) for (selftransverse) immersions of all (compact closed smooth) manifolds of given dimension n.Placing n +1 copies of the ^-sphere S n in general position in U n+1 provides an immersion i 0 : \jS nc \ > U n+1 with 6(i 0 ) = 2. Thus given an immersion i x : M3 > U n+1 disjoint union with i 0 provides another immersion i % with 6(i 2 ) = 0(ij) + 2. On the other hand, given an immersion i x : M9 > U n+1 with 0(i x ) ^ 2 handles may be attached to M to eliminate two of the (n + 1 )-fold points giving an immersion i 2 with 6(i 2 ) = d(i x ) -2 (the argument for n = 2 is given in detail by T. F. Banchoff (2)). This leaves the problem of whether d(i) can be odd.In a previous paper (11) it was shown that if M is orientable then 6(i) can be odd if and only if n = 0,1 or 3 (see also (18)). The method of proof was to observe that the parity of 6 is a bordism invariant and so to translate the problem into homotopy theory where it could be solved using standard techniques of algebraic topology. The same method can be applied in the general case, as explained in § 2, but so far has not led to a complete solution. However it does show that for certain values of n the problem is equivalent to well known problems in homotopy theory. The main result of this paper is as follows.

Stable Homotopy Around the Arf-Kervaire Invariant

“…The main result which is claimed in the preprints [16], [17] and [18] is that the stable homotopy classes with Arf-Kervaire invariant one (modulo 2) exist only in a finite, unspecified range of dimensions. The immersion interpretation dates back to work in the 1980's by Peter Eccles ([74], [75], [76]; see also [19] and [20]). The method of attack adopted in ( [16], [17], [18]) interprets the Arf-Kervaire invariant in terms of counting certain multiple points of immersions and features an invariant given by a formula similar to that of Theorem 2.2.3.…”

Geometric approach to stable homotopy groups of spheres. Kervaire invariants. II

“…We consider an approach to a solution of this problem based on results of P. J. Eccles (see [8]). For a geometrical approach, see also [5,6].…”

Triple Points of Immersed Orientable 2n -Manifolds in 3n -Space

“…In order to state the main result it is necessary to review some of these ideas and to explain how the number of triple points of an immersion can be viewed as a bordism invariant. Further details may be found in [5,6,7].…”

“…The method is homotopy theoretic; explicit manifolds and immersions are not constructed.In order to state the main result it is necessary to review some of these ideas and to explain how the number of triple points of an immersion can be viewed as a bordism invariant. Further details may be found in [5,6,7].Given a self-transverse immersion /: M 2 "^ <*+ |R 3n of a closed oriented 2w-manifold, compactness ensures that the number of triple points is finite. If n is even a sign may be attached to each triple point by comparing the standard orientation of IR 3n with that provided by the orientation of the three normal planes at that point.…”