### Abstract

For a proper continuous map f : M → N between smooth manifolds M and N with m = dim M < dim N = m + k, a homology class θ(f) ∈ H^{c}_{m-k}(M;Z_{2}) has been defined and studied by the first and the third authors, where H^{c}_{*} denotes the singular homology with closed support. In this paper, we define θ(f) for maps between generalized manifolds. Then, using algebraic topological methods, we show that f̄_{*}θ(f) ∈ Ȟ^{c}_{m-k}(f(M); Z_{2}) always vanishes, where f̄ = f : M → f(M) and Ȟ^{c}_{*} denotes the Čech homology with closed support. As a corollary, we show that if f is properly homotopic to a topological embedding, then θ(f) vanishes: In other words, the homology class can be regarded as a primary obstruction to topological embeddings. Furthermore, we give an application to the study of maps of the real projective plane into 3-dimensional generalized manifolds.

Original language | English |
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Pages (from-to) | 275-289 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 197 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2001 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Pacific Journal of Mathematics*,

*197*(2), 275-289. https://doi.org/10.2140/pjm.2001.197.275