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KYUNGPOOK Math. J. 2019; 59(3): 563-589

Published online September 23, 2019

Copyright © Kyungpook Mathematical Journal.

Note on the Codimension Two Splitting Problem

Yukio Matsumoto

Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan
e-mail : yukiomat@math.gakushuin.ac.jp

Received: June 23, 2016; Accepted: November 11, 2016

Abstract

Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let f : WV be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence g : WV such that g is transverse regular along M and such that g|g−1(M) : g−1(M) → M is a simple homotopy equivalence? López de Medrano (1970) called this problem the weak h-regularity problem. We solve this problem applying the codimension two surgery theory developed by the author (1973). We will work in higher dimensions, assuming that m ≧ 5.

Keywords: codimension two splitting problem, weak h-regularity problem, codimension two surgery, surgery obstruction, relatively non-singular Hermitian K-theory.