Alexander duality

The connection between the homological properties of complementary subsets of a topological space, owing to which the homological properties of a set can be defined by certain properties of its complement. The first theorems of this kind were formulated in terms of set-theoretic rather than algebraic topology. It was shown by C. Jordan in 1892 that a simple closed curve divides the plane into two domains and forms the common boundary of the two (the Jordan theorem). This theorem was extended in 1911 by H. Lebesgue and L.E.J. Brouwer (independently of each other) to the case of an $n$-dimensional manifold in an $(n+1)$-dimensional spherical (or Euclidean) space; a connection was also established between this fact and the property of an $r$-dimensional manifold (in an $n$-dimensional space) being linked with an $(n-r-1)$-dimensional manifold (Lebesgue). It was shown by Brouwer in 1913 that the number of domains into which a plane is subdivided by a closed subset depends only on the topological properties of this set. This type of duality was first expressed by J.W. Alexander in 1922 [1] in purely homological terms. Alexander's theorem [2], [3], [4] states that the $r$-dimensional Betti number mod 2 of a (finite) polyhedron $\Phi$ in an $n$-dimensional spherical space is equal to the $(n-r-1)$-dimensional Betti number mod 2 of its complement. P.S. Aleksandrov in 1927 extended this theorem to the case of any closed set $\Phi$. The duality expressed by this theorem is known as Alexander duality.

The next important step in the development of duality of this kind was the theorem of Pontryagin [2], [3], [4] (1934), which states that the $r$-dimensional homology group $H_r(A,X)$ of a closed set $A$ in an $n$-dimensional spherical manifold $M^n$ with a compact group $X$ of coefficients, and the $(n-r-1)$-dimensional homology group $H_{n-r-1}(B,Y)$ of the complement $B=M^n\setminus A$ with the (discrete) group $Y$ of coefficients which is dual to $X$ in the sense of the theory of characters, are duals, and their scalar product defines the linking coefficient of arbitrary cycles from the homology classes scalarly multiplied. This theorem is known as the Alexander–Pontryagin theorem; the duality formulated in it is known as Alexander–Pontryagin duality or as Pontryagin duality. A series of subsequent publications led to Aleksandrov's theorem [5], [7], the formulation of which differs from that of Pontryagin's theorem in that $A$ may be an arbitrary subset of $M^n$, the group $X$ may be both compact or discrete, and $H_r(A,X)$ and $H_{n-r-1}(B,Y)$ are understood to mean the Aleksandrov–Čech homology groups (cf. Aleksandrov–Čech homology and cohomology), one of which has compact support, while the other is of spectral type. The forms of Alexander duality for arbitrary sets are obtained by replacing the latter type of groups by their dual cohomology groups of the same dimension over the dual group of coefficients.

In subsequent generalizations, spherical manifolds were replaced by more general manifolds (homology manifolds which are acyclic in certain dimensions, cf. Homology manifold), the Aleksandrov–Čech groups were replaced by Sitnikov–Steenrod groups (groups of projective type) and other groups; the groups of coefficients were replaced by modules, sheaves, etc.

References

[1]	J.W. Alexander, II, "A proof of the invariance of certain constants of analysis situs" Trans. Amer. Math. Soc. , 16 (1915) pp. 148–154
[2]	P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)
[3]	L.S. Pontryagin, "The general topological theorem of duality for closed sets" Ann. of Math. , 35 : 4 (1934) pp. 904–914
[4]	S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1955)
[5]	P.S. Aleksandrov, "Fundamental duality theorems for non-closed sets of an $n$-dimensional space" Mat. Sb. , 21 (63) : 2 (1947) pp. 161–232 (In Russian)
[6a]	P.S. Aleksandrov, "Topological duality theorems I. Closed sets" Trudy Mat. Inst. Steklov. , 48 (1955) (In Russian)
[6b]	P.S. Aleksandrov, "Topological duality theorems II. Non-closed sets" Trudy Mat. Inst. Steklov. , 54 (1959) (In Russian)
[7]	G.S. Chogoshvili, "Homology theory of compact spaces" , Proc. 4-th. All-Union Math. Congress , 2 , Moscow (1964) pp. 57–62 (In Russian)
[8]	S. Kaplan, "Homology properties of arbitrary subsets of Euclidean spaces" Trans. Amer. Math. Soc. , 62 (1947) pp. 248–271
[9]	D.G. Bourgin, "Modern algebraic topology" , Macmillan (1963)
[10]	G.S. Chogoshvili, "On homology theory of non-closed sets" , General topology and its relations to modern analysis and algebra (Proc. Symp. Prague) , Acad. Press (1961) pp. 123–132