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Integrating the form $\kappa_a$ computes the winding number of $\gamma$ about $a$. This is a special case of a linking pairing.

Alexander Duality is the statement that if $X\subset S^n $ is compact and is a deformation retract of some open $U$ and $Y=S^n-X$ then $H_i(Y)$ is isomorphic to $H^{n-i-1}(X)$. The pairing giving this isomorphism is called the linking pairing, of which your pairing is a special case.

You are missing some regularity. You probably want $U$ to have a compact deformation retract. For instance the first homology of the complement of the Cantor set in the plane will have infinite rank.

The bit about compact support is because should be thinking of the set $U$ as a red herring. Otherwisesubset of the sphere, these statements are all consequences not a subset of Alexander dualitythe plane.

An obtuse but elementary development of this theorem can be found, at least for the plane, in the end of Dugundji's point set topology book. I learned Alexander duality from Spanier, (it might be dry, but its correct!). Generally, when I am teaching this to graduate students I illustrate it with practical computations, as the theorem itself is somewhat hard to get your head around.

You might like Kuga's "Galois' Dream" which covers a lot of connections between baby algebraic geometry and baby topology in the framework of Galois theory. Believe it or not, I really liked Lang's book on Algebraic curves as a place to learn about elementary consequences of duality in an algebraic setting. The Riemann-Roch theorem is the starting point for some of the most important results of the twentieth century, and it is a duality result much in the vein of what you seem to be interested in.
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Integrating the form $\kappa_a$ computes the winding number of $\gamma$ about $a$. This is a special case of a linking pairing.

Alexander Duality is the statement that if $K\subset X\subset S^n $ is compact and is a deformation retract of some open $U$ and $Y=S^n-X$ then $H_i(Y)$ is isomorphic to $H^{n-i-1}(X)$. The pairing giving this isomorphism is called the linking pairing, of which your pairing is a special case.

You are missing some regularity. You probably want $U$ to have a compact deformation retract. The bit about compact support is a red herring. Otherwise, these statements are all consequences of Alexander duality.

An obtuse but elementary development of this theorem can be found, at least for the plane, in the end of Dugundji's point set topology book. I learned Alexander duality from Spanier, (it might be dry, but its correct!). Generally, when I am teaching this to graduate students I illustrate it with practical computations, as the theorem itself is somewhat hard to get your head around.

You might like Kuga's "Galois' Dream" which covers a lot of connections between baby algebraic geometry and baby topology in the framework of Galois theory. Believe it or not, I really liked Lang's book on Algebraic curves as a place to learn about elementary consequences of duality in an algebraic setting. The Riemann-Roch theorem is the starting point for some of the most important results of the twentieth century, and it is a duality result much in the vein of what you seem to be interested in.
show/hide this revision's text 1

Integrating the form $\kappa_a$ computes the winding number of $\gamma$ about $a$. This is a special case of a linking pairing.

Alexander Duality is the statement that if $K\subset S^n $ is compact and is a deformation retract of some open $U$ and $Y=S^n-X$ then $H_i(Y)$ is isomorphic to $H^{n-i-1}(X)$. The pairing giving this isomorphism is called the linking pairing, of which your pairing is a special case.

You are missing some regularity. You probably want $U$ to have a compact deformation retract. The bit about compact support is a red herring. Otherwise, these statements are all consequences of Alexander duality.

An obtuse but elementary development of this theorem can be found, at least for the plane, in the end of Dugundji's point set topology book. I learned Alexander duality from Spanier, (it might be dry, but its correct!). Generally, when I am teaching this to graduate students I illustrate it with practical computations, as the theorem itself is somewhat hard to get your head around.

You might like Kuga's "Galois' Dream" which covers a lot of connections between baby algebraic geometry and baby topology in the framework of Galois theory. Believe it or not, I really liked Lang's book on Algebraic curves as a place to learn about elementary consequences of duality in an algebraic setting. The Riemann-Roch theorem is the starting point for some of the most important results of the twentieth century, and it is a duality result much in the vein of what you seem to be interested in.