Assume that $M$ is a smooth manifold of dimension $m$ and $A$, $B$ are its smooth submanifolds of dimensions $a$ and $b$ respectively and $a+b=m$. Is it true that one can find $A'$ and $B'$ of dimensions $a$ and $b$ so that $A'$ and $B'$ are transverse and $$ H_0(A\cap B)=H_0(A'\cap B')? $$
2026-03-28 12:33:32.1774701212
Is it possible to change intersecting submanifolds of complementary dimension so that the homology class of their intersection won't change?
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No. Consider $A$ as the graph of $y = x^2$ and $B$ as the graph of $y = -x^2$. Then any transverse intersection of the two has an even number of points (because $A' = x^2 + c$, $B' = B$, for $c > 0$, does). $A \cap B$ has a single point, the origin.