Does the Poincare duality ($H^1=H_1$) hold for $2$-dimensional manifolds with punctured points?
Example:
Take $g=2$ surface with $2$ points punctured. It permits for $4+1=5$ linearly independent (modulo boundary) cycles. This gives $\operatorname{rank} H_1 =5$. From this, can I deduce that $\operatorname{rank} H^1=5$? (and that one can define $5$ linearly independent (up to a total derivative) $1$-forms on this surface?)