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Sam Nead
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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).

Sam Nead
  • 28.2k
  • 5
  • 72
  • 131