TY - JOUR
T1 - On the Betti number of the union of two generic map images
AU - Biasi, Carlos
AU - Libardi, Alice K.M.
AU - Saeki, Osamu
N1 - Funding Information:
∗Corresponding author. E-mail: saeki@top2.math.sci.hiroshima-u.ac.jp. 1The third author has been partly supported by CNPq, Brazil, and by the Anglo-Japanese Scientific Exchange Programme, run by the Japan Society for the Promotion of Science and the Royal Society. 2A manifold is said to be closed if it is compact and has no boundary.
PY - 1999
Y1 - 1999
N2 - Let f : M → N and g : K → N be generic differentiable maps of compact manifolds without boundary into a manifold such that their intersection satisfies a certain transversality condition. We show, under a certain cohomological condition, that if the images f(M) and g(K) intersect, then the (ν + 1)th Betti number of their union is strictly greater than the sum of their (ν + 1)th Betti numbers, where ν = dim M + dim K - dim N. This result is applied to the study of coincidence sets and fixed point sets.
AB - Let f : M → N and g : K → N be generic differentiable maps of compact manifolds without boundary into a manifold such that their intersection satisfies a certain transversality condition. We show, under a certain cohomological condition, that if the images f(M) and g(K) intersect, then the (ν + 1)th Betti number of their union is strictly greater than the sum of their (ν + 1)th Betti numbers, where ν = dim M + dim K - dim N. This result is applied to the study of coincidence sets and fixed point sets.
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U2 - 10.1016/s0166-8641(97)00273-3
DO - 10.1016/s0166-8641(97)00273-3
M3 - Article
AN - SCOPUS:15944390989
SN - 0166-8641
VL - 95
SP - 31
EP - 46
JO - Topology and its Applications
JF - Topology and its Applications
IS - 1
ER -