Alexander duality

Let ${\displaystyle X}$ be a non-empty compact, locally contractible subspace of the sphere ${\displaystyle S^{n}}$ of dimension n. Let ${\displaystyle S^{n}\setminus X}$ be the complement of ${\displaystyle X}$ in ${\displaystyle S^{n}}$. Then if ${\displaystyle {\tilde {H}}}$ stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism

${\displaystyle {\tilde {H}}_{q}(S^{n}\setminus X)\cong {\tilde {H}}^{n-q-1}(X)}$

for all ${\displaystyle q\geq 0}$. Note that we can drop local contractibility as part of the hypothesis if we use Čech cohomology, which is designed to deal with local pathologies.

This is useful for computing the cohomology of knot and link complements in ${\displaystyle S^{3}}$. Recall that a knot is an embedding ${\displaystyle K\colon S^{1}\hookrightarrow S^{3}}$ and a link is a disjoint union of knots, such as the Borromean rings. Then, if we write the link/knot as ${\displaystyle L}$, we have

${\displaystyle {\tilde {H}}_{q}(S^{3}\setminus L)\cong {\tilde {H}}^{3-q-1}(L)}$,

giving a method for computing the cohomology groups. In particular the dimension of the cohomology of the complement only depends on the homology of ${\displaystyle L}$, that is the number of its connected components (or number of strands). For example it is impossible to distinguish a borromean link from a trivial link (i.e. three separated circles). However it is possible to differentiate between different links using multiplicative operations on these cohomology spaces, like cup-products and the Massey products.^[1] For example, for the Borromean rings ${\displaystyle L}$, the homology groups are

${\displaystyle {\begin{aligned}{\tilde {H}}_{0}(S^{3}\setminus L)&\cong {\tilde {H}}^{2}(L)=0\\{\tilde {H}}_{1}(S^{3}\setminus L)&\cong {\tilde {H}}^{1}(L)=\mathbb {Z} ^{\oplus 3}\\{\tilde {H}}_{2}(S^{3}\setminus L)&\cong {\tilde {H}}^{0}(L)=\mathbb {Z} ^{\oplus 3}\\{\tilde {H}}_{3}(S^{3}\setminus L)&\cong 0\\\end{aligned}}}$

Let ${\displaystyle X}$ be an abstract simplicial complex on a vertex set ${\displaystyle V}$ of size ${\displaystyle n}$. The Alexander dual ${\displaystyle X^{*}}$ of ${\displaystyle X}$ is defined as the simplicial complex on ${\displaystyle V}$ whose faces are complements of non-faces of ${\displaystyle X}$. That is

${\displaystyle X^{*}=\{\sigma \ \colon \ V\setminus \sigma \not \in X\}}$.

Note that ${\displaystyle (X^{*})^{*}=X}$.

Alexander duality implies the following combinatorial analog (for reduced homology and cohomology, with coefficients in any given abelian group):

${\displaystyle {\tilde {H}}_{q}(X^{*})\cong {\tilde {H}}^{n-q-3}(X)}$

for all ${\displaystyle q\geq 0}$. Indeed, this can be deduced by letting ${\displaystyle Y\simeq S^{n-2}}$ be the ${\displaystyle (n-2)}$-skeleton of the full simplex on ${\displaystyle V}$ (that is, ${\displaystyle Y}$ is the family of all subsets of size at most ${\displaystyle n-1}$) and showing that the geometric realization ${\displaystyle |X^{*}|}$ is homotopy equivalent to ${\displaystyle |Y|\setminus |X|}$. Björner and Tancer presented an elementary combinatorial proof and summarized a few generalizations.^[2]

For smooth manifolds, Alexander duality is a formal consequence of Verdier duality for sheaves of abelian groups. More precisely, if we let ${\displaystyle X}$ denote a smooth manifold and we let ${\displaystyle Y\subset X}$ be a closed subspace (such as a subspace representing a cycle, or a submanifold) represented by the inclusion ${\displaystyle i\colon Y\hookrightarrow X}$, and if ${\displaystyle k}$ is a field, then if ${\displaystyle {\mathcal {F}}\in \operatorname {Sh} _{k}(Y)}$ is a sheaf of ${\displaystyle k}$-vector spaces we have the following isomorphism^[3]: 307

${\displaystyle H_{c}^{s}(Y,{\mathcal {F}})^{\vee }\cong \operatorname {Ext} _{k}^{n-s}(i_{*}{\mathcal {F}},\omega _{X}[n-s])}$,

where the cohomology group on the left is compactly supported cohomology. We can unpack this statement further to get a better understanding of what it means. First, if ${\displaystyle {\mathcal {F}}={\underline {k}}}$ is the constant sheaf and ${\displaystyle Y}$ is a smooth submanifold, then we get

${\displaystyle \operatorname {Ext} _{k}^{n-s}(i_{*}{\mathcal {F}},\omega _{X}[n-r])\cong H_{Y}^{n-s}(X,\omega _{X})}$,

where the cohomology group on the right is local cohomology with support in ${\displaystyle Y}$. Through further reductions, it is possible to identify the homology of ${\displaystyle X\setminus Y}$ with the cohomology of ${\displaystyle Y}$. This is useful in algebraic geometry for computing the cohomology groups of projective varieties, and is exploited for constructing a basis of the Hodge structure of hypersurfaces of degree ${\displaystyle d}$ using the Jacobian ring.