Homologically trivial immersion Are there any examples when some manifold $N$ maps in other manifold $M$ as codimension 1 submanifold, its fundamental class is zero in the homology of M, but still this map $i\colon N\to M$ induces a non-trivial homomorphism of homology groups. I'm very interested in case when homology is taken with coefficients in some field, but any counter example would be much appreciated.

Consider an embedding of a torus $N$ inside $M=(\mathbb R^3 \text{with a line removed})$, such that the torus is the boundary of a regular neighborhood of a curve that goes around the line once. Since $M$ is homotopy equivalent to a circle, the induced map on $H_2(N)\to H_2(M)$ is trivial. Yet on the level of $H_1$ we have a nontrivial map $\mathbb Z\oplus \mathbb Z\to \mathbb Z$.
