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Separation properties of codimension-1 maps between generalized manifolds

Edivaldo L. dos Santos, Telmo I. Acosta Vellozo

Abstract

In this work, we obtained separation results via codimension-1 maps to generalized manifolds. More specifically, we proved results that allow us to estimate the number of connected components of the complement of the image of such maps.

Separation properties of codimension-1 maps between generalized manifolds

Abstract

In this work, we obtained separation results via codimension-1 maps to generalized manifolds. More specifically, we proved results that allow us to estimate the number of connected components of the complement of the image of such maps.

Paper Structure

This paper contains 4 sections, 15 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Generalized Manifold
  3. Separation by codimension-1 map depending on the self-intersection set
  4. Separation by codimension-1 map depending on the primary obstruction to topological embedding

Key Result

Theorem 2.2

The duality map $D_M : H_c^k(M;R)\rightarrow H_{n-k}(M;R)$ given by $D_M(\alpha)= \alpha \frown [M]$, is an isomorphism for all $k$ whenever $M$ is an $R$-oriented generalized $n$-manifold. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (26)

  • Definition 2.1
  • Theorem 2.2: Poincaré's Duality Theorem for Generalized Manifold
  • Theorem 2.3: Alexander's Duality Theorem for Generalized Manifold
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Edwards's approximation theorem
  • Corollary 2.7
  • Theorem 2.8: Edwards-Quinn
  • Theorem 2.9: Bryant, Ferry, Mio, Weinberger
  • Lemma 3.1
  • ...and 16 more