Perturbing embeddings in codimension two

Stanley Ocken

Vol. 81 (1979), No. 1, 197–205

DOI: 10.2140/pjm.1979.81.197

Abstract

Let f : Xⁿ → Wⁿ⁺² be a fixed embedding of manifolds, assume X compact, and let g : X → W be an embedding close to f in the C⁰ topology. In general, g and f will not be concordant. What small perturbation of g will yield an embedding concordant to f?

In this paper, our goal is to replace g by a new embedding while altering the image manifold g(X) as little as possible. In case X is simply connected, the problem was solved by Cappell and Shaneson as follows: if n is odd, g(X) is already concordant to f [5]. If n is even and f has trivial normal bundle, g(X) may be replaced by its connected sum with a knot in W [4]. The current paper applies previous work of the author [9, 10] to study the nonsimply connected case.

Mathematical Subject Classification 2000

Primary: 57Q35

Secondary: 57Q45

Milestones

Received: 26 July 1977

Published: 1 March 1979