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Note on the Codimension Two Splitting Problem

  • Matsumoto, Yukio (Department of Mathematics, Gakushuin University)
  • Received : 2016.06.23
  • Accepted : 2016.11.11
  • Published : 2019.09.23

Abstract

Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let $f:W{\rightarrow}V$ be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence $g:W{\rightarrow}V$ such that g is transverse regular along M and such that $g{\mid}g^{-1}(M):g^{-1}(M){\rightarrow}M$ is a simple homotopy equivalence? $L{\acute{o}}pez$ 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 $$m{\geq_-}5$$.

Keywords

codimension two splitting problem;weak h-regularity problem;codimension two surgery;surgery obstruction;relatively non-singular Hermitian K-theory

Acknowledgement

Supported by : JSPS KAKENHI

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