A PRIMARY OBSTRUCTION TO TOPOLOGICAL EMBEDDINGS FOR MAPS BETWEEN GENERALIZED MANIFOLDS

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) ∈ Hcm−k(M; Z2) has been defined and studied by the first and the third authors, where Hc ∗ 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) ∈ Ȟcm−k(f(M); Z2) always vanishes, where f ̄ = f: M → f(M) and Ȟc ∗ denotes the Čech homology with closed support. As a corollary, we show that if f is properly homotopic to a topo-logical 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 applica-tion to the study of maps of the real projective plane into 3-dimensional generalized manifolds