Abstract

For a proper continuous map f : M → N between smooth manifolds M and N with m = dimM < dimN = m + k, a homology class θ(f) in H_m−k^c(M;Z₂) 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 f_*θ(f) in Ȟ_m−k^c(f(M);Z₂) always vanishes, where f = 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.

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