Spectral homology

The inverse limit

$$ \check{H} _ {n} (X; G) = \mathop{\rm lim} _ \leftarrow H _ {n} ( \alpha ; G) $$

of homology groups with coefficients in the Abelian group $ G $ of nerves of open coverings $ \alpha $ of a topological space $ X $ (also called Čech homology, or Aleksandrov–Čech homology). For a closed set $ A\subset X $, the groups $ \check{H} _ {n} (A; G) $ can be defined in a similar way using the subsystems $ \alpha ^ \prime \subset \alpha $ of all those subsets of $ \alpha $ having non-empty intersection with $ A $. The inverse limit of the groups $ H _ {n} ( \alpha , \alpha ^ \prime ; G) $ is called the spectral homology group $ \check{H} _ {n} (X, A; G) $ of the pair $ (X, A) $.

Since the inverse limit functor does not preserve exactness, the homology sequence of the pair $ (X, A) $ is, in general, not exact. It is semi-exact, in the sense that the composite of any two mappings in the sequence is equal to zero. For a compact space $ X $ the sequence turns out to be exact in the case when $ G $ is a compact group or field (or, more generally, if $ G $ is algebraically compact). The spectral homology of compact spaces is continuous in the sense that

$$ \check{H} _ {n} \left ( \mathop{\rm lim} _ \leftarrow X _ \lambda ; G \right ) = \mathop{\rm lim} _ \leftarrow \check{H} _ {n} (X _ \lambda ; G). $$

Lack of exactness is not the only deficiency of spectral homology. The groups $ \check{H} _ {n} $ turn out to be non-additive, in the sense that the homology of a discrete union $ X = \cup _ \lambda X _ \lambda $ can be different from the direct sum $ \sum _ \lambda \check{H} _ {n} (X _ \lambda ; G) $. This deficiency disappears if one considers the spectral homology groups $ H _ {n} ^ {c} (X; G) $ with compact support, defined as the direct limit $ {\mathop{\rm lim}\nolimits} \check{H} _ {n} (C; G) $ taken over all compact subsets $ C\subset X $. It is natural to consider the functor $ \check{H} _ {n} ^ {c} $, in view of the fact that all the usual homologies (simplicial, cellular and singular) are homologies with compact support.

The difference between the functors $ \check{H} _ {n} $ and $ \check{H} _ {n} ^ {c} $ is one of the examples of how homology groups react to small changes in their initial definition (on the other hand, cohomology groups exhibit significant stability in this respect). Among the logically possible variants of the definition of homology groups in general categories of topological spaces, the correct one was not the first to be selected. The theory of the homology groups $ H _ {*} ^ {c} $ associated with the Aleksandrov–Čech cohomology achieved great recognition only in the 1960's (although the first definitions were given in the 1940's and 1950's). The theory of $ H _ {*} ^ {c} $ satisfies all the Steenrod–Eilenberg axioms (and is a theory with compact supports). For compact spaces $ X $ the following sequence is exact:

$$ 0 \mathop \rightarrow \limits \mathop{\rm lim} _ \leftarrow {} ^ {1} H _ {n+1} ( \alpha ; G) \mathop \rightarrow \limits H _ {n} (X; G) \mathop \rightarrow \limits \check{H} (X; G) \mathop \rightarrow \limits 0 , $$

where $ {\mathop{\rm lim}\nolimits} _ \leftarrow ^ {1} $ is the derived inverse limit functor. In general there is an epimorphism $ H _ {n} ^ {c} (X; G) \mathop \rightarrow \limits \check{H} _ {n} ^ {c} (X; G) $ whose kernel is zero for any algebraically compact group $ G $. For any locally compact space that is also homologically locally connected (with respect to $ H _ {*} ^ {c} $), the functors $ \check{H} _ {n} $, $ \check{H} _ {n} ^ {c} $, $ H _ {n} ^ {c} $ are isomorphic.

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