On the Betti number of the image of a generic map

Carlos Biasi, Osamu Saeki

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let f : M → N be a differentiable map of a closed m-dimensional manifold into an (m + k)-dimerisional manifold with k > 0. VVe show, assuming that f is generic in a certain sense, that f is an embedding if and only if the (m - k + 1)-th Betti numbers with respect to the Čech homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings [BS1]. As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of [BR, BMS1, BMS2, Sae1] for immersions with normal crossings.

Original languageEnglish
Pages (from-to)72-83
Number of pages12
JournalCommentarii Mathematici Helvetici
Volume72
Issue number1
DOIs
Publication statusPublished - Jan 1 1997
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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