A more sensitive requirement is that the map $f: N \to M$ is null bordant in the oriented bordism of $M$. I.e., there is a manifold $W$ with boundary equal to $N$ and a map $F: W \to M$ extending the given map $f$ on $N$. This would imply your conditions 1 and 2. Up to isotopy you can always assume that If $F$ is homotopic to an immersion , and (see Oscar's answer) then the question is now whether it can be made into an embedding. , and I believe there is some surgery theory machinery designed to answer this question.
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A more sensitive requirement is that the map $f: N \to M$ is null bordant in the oriented bordism of $M$. I.e., there is a manifold $W$ with boundary equal to $N$ and a map $F: W \to M$ extending the given map $f$ on $N$. This would imply your conditions 1 and 2. Up to isotopy you can always assume that $F$ is an immersion, and the question is now whether it can be made into an embedding. I believe there is some surgery theory machinery designed to answer this question. |
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