Apparent contradiction in computing $H^{0}(M,\mathbb{Z}_{2})$ for a compact manifold with boundary.

Question

I am trying to show that, for a compact manifold with boundary $M$ that $H_{n}(M;\mathbb{Z}_2)=0$.

I am running into apparent contradictions.

By Lefschetz duality, $H_{n}(M;\mathbb{Z}_2)=H^{0}(M,\mathbb{Z}_2)$, but $H^{0}(M;\mathbb{Z}_{2})$ should count the path components of $M$ so should be some product of $\mathbb{Z}_{2}$

What have I done wrong?

Answer 1

The duality statement you have used is only valid for a closed manifold. For a compact manifold with boundary, the statement involves (co)homology relative to the boundary, and in particular would give $H_n(M;\mathbb{Z}_2)\cong H^0(M,\partial M;\mathbb{Z}_2)$.