Homology with compact support

A partially exact homology theory (cf. Homology theory) which satisfies the following axiom of compact support: For each element $ h $ of the $ r $- dimensional group $ H _ {r} ( X, A) $ of an arbitrary pair of spaces $ ( X, A) $ there exists a compact pair $ ( X ^ \prime , A ^ \prime ) \subset ( X, A) $ such that $ h $ is contained in the image of the homomorphism

$$ \mu : H _ {r} ( X ^ \prime , A ^ \prime ) \rightarrow H _ {r} ( X, A) $$

which is induced by the inclusion. If the homology theory $ H $ is exact and has compact support, the following theorem is valid: For any element $ h \in H _ {r} ( X ^ \prime , A ^ \prime ) $ which belongs to the kernel of $ \mu $ there exists a compact pair $ ( X ^ {\prime\prime} , A ^ {\prime\prime} ) $ such that

$$ ( X ^ \prime , A ^ \prime ) \subset ( X ^ {\prime\prime} , A ^ {\prime\prime} ) \subset ( X,\ A) , $$

and $ h $ belongs to the kernel of the homomorphism

$$ H _ {r} ( X ^ \prime , A ^ \prime ) \rightarrow H _ {r} ( X ^ {\prime\prime} , A ^ {\prime\prime} ). $$

An exact theory has compact support if and only if for any pair $ ( X, A) $ the group $ H _ {r} ( X, A) $ is the direct limit $ \lim\limits _ \rightarrow \{ H _ {r} ( X ^ \prime , A ^ \prime ) \} $, where $ ( X ^ \prime , A ^ \prime ) $ runs through the compact pairs contained in $ ( X, A) $. An exact homology theory with compact support is unique on the category of arbitrary (non-compact) polyhedral pairs for a given coefficient group and is equivalent to the singular theory. For a general homology theory, in addition to the group $ H _ {r} ( X, A) $ there is also the group

$$ H _ {r} ^ {c} ( X, A) = \ \lim\limits _ \rightarrow \{ H _ {r} ( X ^ \prime , A ^ \prime ) \} , $$

where $ ( X ^ \prime , A ^ \prime ) $ are compact subpairs in $ ( X, A) $. The singular homology group, having compact support, is isomorphic to $ H _ {r} ^ {c} ( X, A) $. In spectral theory one also considers — in addition to the Aleksandrov–Čech homology groups $ H _ {r} ( X, A) $ and the groups $ H _ {r} ^ {c} ( X, A) $— the group which is the image of the natural homomorphism

$$ H _ {r} ^ {c} ( X, A) \rightarrow H _ {r} ( X, A) . $$

This group, like the group $ H _ {r} ^ {c} ( X, A) $, satisfies the axiom of compact support, but in spectral theory it is the latter group which has the name of homology with compact support. In spectral theory these three groups differ from one another, and each of them is the object of a duality theorem, both for a discrete and for a compact group of coefficients (cf. Duality in topology).

References

[1]	N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[2]	P.S. Aleksandrov, "General duality theorems for non-closed sets in -dimensional space" Mat. Sb. , 21 (63) : 2 (1947) pp. 161–232 (In Russian)
[3]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)