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

Fingerprint

Betti numbers
Immersion
Brouwer's theorem
Stable Bundle
Normal Bundle
Converse
Codimension
Differentiable
Homology
Corollary
If and only if
Closed
Generalise
Generalization

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

On the Betti number of the image of a generic map. / Biasi, Carlos; Saeki, Osamu.

In: Commentarii Mathematici Helvetici, Vol. 72, No. 1, 01.01.1997, p. 72-83.

Research output: Contribution to journalArticle

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