One-half lives, one-half dies.

Suppose that $M$ is an oriented compact 3-dimensional manifold. Suppose that $\phi$ is the homomorphism on first homology, induced by the inclusion of the boundary of $M$ into the $M$. Then the image and kernel of $\phi$ have the same rank (namely, half the rank of the homology of the boundary).

One-half lives, one-half dies.

Suppose that $M$ is an oriented compact three-manifold. Suppose that $\phi$ is the homomorphism on first homology, induced by the inclusion of the boundary of $M$ into $M$. Then the image and kernel of $\phi$ have the same rank (namely, half the rank of the homology of the boundary).

One-half lives, one-half dies.

Suppose that $M$ is an oriented compact manifold. Suppose that $\phi$ is the homomorphism on first homology, induced by the inclusion of the boundary of $M$ into the $M$. Then the image and kernel of $\phi$ have the same rank (namely, half the rank of the homology of the boundary).

One-half lives, one-half dies.

Suppose that $M$ is an oriented compact 3-dimensional manifold. Suppose that $\phi$ is the homomorphism on first homology, induced by the inclusion of the boundary of $M$ into the $M$. Then the image and kernel of $\phi$ have the same rank (namely, half the rank of the homology of the boundary).