Local homology

The homology groups (cf. Homology group)

$$ H _ {p} ^ {x} = H _ {p} ^ {c} ( X , X \setminus x ; G ) , $$

defined at points $ x \in X $, where $ H _ {p} ^ {c} $ is homology with compact support. These groups coincide with the direct limits

$$ \lim\limits _ \rightarrow H _ {p} ^ {c} ( X , X \setminus U ; G ) $$

over open neighbourhoods $ U $ of $ x $, and for homologically locally connected $ X $ they also coincide with the inverse limits

$$ \lim\limits _ \leftarrow H _ {p-} 1 ^ {c} ( U \setminus x ; G ) . $$

The homological dimension of a finite-dimensional metrizable locally compact space $ X $ over $ G $( cf. Homological dimension of a space) coincides with the largest value of $ n $ for which $ H _ {n} ^ {x} \neq 0 $, and the set of such points $ x \in X $ has dimension $ n $.

Let $ {\mathcal C} _ {*} $ be the differential sheaf over $ X $ defined by associating with each open set $ U \subset X $ the chain complex $ C _ {*} ( X , X \setminus U ; G ) $. The groups $ H _ {p} ^ {x} $ are the fibres of the derived sheaves $ {\mathcal H} _ {p} = H _ {p} ( {\mathcal C} _ {*} ) $. For generalized manifolds, $ H _ {p} ^ {x} = 0 $ for $ p \neq n = \mathop{\rm dim} X $. In this case the homology sequence of the pair $ ( X , A ) $ with coefficients in $ G $ coincides with the cohomology of the pair $ ( X , X \setminus A ) $ with coefficients in the sheaf $ {\mathcal H} _ {n} $( Poincaré–Lefschetz duality). The similar facts for the local cohomology of locally compact spaces do not hold.

References

[1]	E.G. Sklyarenko, "On the theory of generalized manifolds" Math. USSR Izv. , 5 : 4 (1971) pp. 845–858 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 831–843
[2]	A.E. [A.E. Kharlap] Harlap, "Local homology and cohomology, homology dimension and generalized manifolds" Math. USSR Sb. , 25 : 3 (1975) pp. 323–349 Mat. Sb. , 96 (1975) pp. 347–373

References

[a1]	A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Sect. IV.3