TY - JOUR

T1 - On the self-intersection set and the image of a generic map

AU - Biasi, Carlos

AU - Saeki, Osamu

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 1997

Y1 - 1997

N2 - Let f:M→N be a continuous map of a closed m-dimensional manifold into an n-dimensional manifold with k=n-m>0. We define a primary obstruction to the existence of a homotopy between f and a smooth embedding which is related to the self-intersection set of a generic map homotopic to f. When f is a smooth generic map in the sense of [9], we show that f is a smooth embedding if and only if the primary obstruction vanishes and the (m-k+1)th Betti numbers of M and the image f(M) coincide, generalizing the authors' previous result [4] for immersions with normal crossings. As a corollary we obtain a converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results in [3, 1, 2, 20] for immersions with normal crossings. Using generic maps, we show the vanishing of the primary obstruction for injective maps. Furthermore, for non-generic smooth maps, we find a homology class in the closure of the self-intersection set which corresponds to the primary obstruction.

AB - Let f:M→N be a continuous map of a closed m-dimensional manifold into an n-dimensional manifold with k=n-m>0. We define a primary obstruction to the existence of a homotopy between f and a smooth embedding which is related to the self-intersection set of a generic map homotopic to f. When f is a smooth generic map in the sense of [9], we show that f is a smooth embedding if and only if the primary obstruction vanishes and the (m-k+1)th Betti numbers of M and the image f(M) coincide, generalizing the authors' previous result [4] for immersions with normal crossings. As a corollary we obtain a converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results in [3, 1, 2, 20] for immersions with normal crossings. Using generic maps, we show the vanishing of the primary obstruction for injective maps. Furthermore, for non-generic smooth maps, we find a homology class in the closure of the self-intersection set which corresponds to the primary obstruction.

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U2 - 10.7146/math.scand.a-12609

DO - 10.7146/math.scand.a-12609

M3 - Article

AN - SCOPUS:0031285786

SN - 0025-5521

VL - 80

SP - 5

EP - 24

JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

IS - 1

ER -