Alexander Duality

Alexander duality relates the homology of a subspace of the sphere to the cohomology of its complement, providing a powerful tool for computing the topology of complements of embedded submanifolds.

The Theorem

Theorem8.9Alexander Duality

Let $K$ be a compact, locally contractible, nonempty, proper subspace of $S^n$ . Then for all $q$ , there is an isomorphism $\tilde{H}^q(K; \mathbb{Z}) \cong \tilde{H}_{n-q-1}(S^n \setminus K; \mathbb{Z})$ where $\tilde{H}$ denotes reduced (co)homology.

The proof combines Poincare-Lefschetz duality with excision. Consider the compact manifold with boundary obtained by removing a small open neighborhood of $K$ from $S^n$ ; Lefschetz duality for this manifold, combined with a limit argument, yields Alexander duality.

Classical Applications

ExampleJordan Curve Theorem

Let $K \cong S^1$ be a simple closed curve in $S^2$ . Alexander duality gives $\tilde{H}_0(S^2 \setminus K) \cong \tilde{H}^1(S^1) \cong \mathbb{Z}$ Since $\tilde{H}_0$ measures one fewer than the number of path components, $S^2 \setminus K$ has exactly two path components. This is the Jordan Curve Theorem.

ExampleKnot complements

For a knot $K \cong S^1$ embedded in $S^3$ , Alexander duality gives:

$\tilde{H}_0(S^3 \setminus K) \cong \tilde{H}^2(S^1) = 0$ , so the complement is connected
$\tilde{H}_1(S^3 \setminus K) \cong \tilde{H}^1(S^1) \cong \mathbb{Z}$
$\tilde{H}_2(S^3 \setminus K) \cong \tilde{H}^0(S^1) \cong \mathbb{Z}$

All knot complements in $S^3$ have the same homology groups! The fundamental group $\pi_1(S^3 \setminus K)$ , the knot group, provides the finer invariant needed to distinguish knots.

Generalization

Theorem8.10Alexander Duality for Manifolds

More generally, if $M$ is a closed oriented $n$ -manifold and $K \subset M$ is a compact subspace, then $H^q(K; \mathbb{Z}) \cong H_{n-q}(M, M \setminus K; \mathbb{Z})$ When $M = S^n$ and we use the long exact sequence of the pair $(S^n, S^n \setminus K)$ together with $\tilde{H}_*(S^n) = 0$ for $* \neq n$ , this reduces to the classical form.

RemarkLinking numbers

Alexander duality provides the algebraic framework for defining linking numbers. If $K$ and $L$ are disjoint oriented closed curves in $S^3$ , the linking number $\operatorname{lk}(K, L)$ is the image of $[L]$ under the composition $H_1(S^3 \setminus K) \cong H^1(K) \cong \mathbb{Z}$ . This integer measures how many times $L$ winds around $K$ and is a fundamental invariant in knot theory.

Alexander duality reveals a deep symmetry: the topology of a subspace and its complement in a sphere are intimately linked, with information about one completely determining the other at the level of homology.