# Local homology

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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