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) ∈ Hm−kc(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) ∈Ȟm−kc(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.
