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 language | English |
|---|---|
| Pages (from-to) | 72-83 |
| Number of pages | 12 |
| Journal | Commentarii Mathematici Helvetici |
| Volume | 72 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Jan 1 1997 |
| Externally published | Yes |
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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 journal › Article
}
TY - JOUR
T1 - On the Betti number of the image of a generic map
AU - Biasi, Carlos
AU - Saeki, Osamu
PY - 1997/1/1
Y1 - 1997/1/1
N2 - 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.
AB - 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=0031473939&partnerID=8YFLogxK
U2 - 10.1007/PL00000368
DO - 10.1007/PL00000368
M3 - Article
AN - SCOPUS:0031473939
VL - 72
SP - 72
EP - 83
JO - Commentarii Mathematici Helvetici
JF - Commentarii Mathematici Helvetici
SN - 0010-2571
IS - 1
ER -