Singular homology

The homology of a topological space $ X $ defined using singular simplices in the same way as ordinary (simplicial) homology (and cohomology) of a polyhedron is defined using linear simplices. By a singular simplex $ \sigma ^ {n} $ one means a continuous mapping of an $ n $- dimensional standard simplex $ \Delta ^ {n} $ into $ X $; the image of $ \sigma ^ {n} $ is usually called the support of $ \sigma ^ {n} $ and is denoted by $ | \sigma ^ {n} | $. Singular chains are formal linear combinations of singular simplices with coefficients in an Abelian group $ G $. They form a group $ S _ {n} ( X; G) $ which is isomorphic to the direct sum (over all $ \sigma ^ {n} $) of the groups $ G _ {\sigma ^ {n} } = G $. Taken together, the chain groups form a singular chain complex $ S _ {*} ( X; G) $ with boundary homomorphism $ \partial : S _ {n} ( X; G) \rightarrow S _ {n - 1 } ( X; G) $ given by

$$ \partial \sigma ^ {n} = \sum _ { i } (- 1) ^ {i} \sigma _ {i} ^ {n - 1 } , $$

where $ \sigma _ {i} ^ {n - 1 } $ is the composite with $ \sigma ^ {n} $ of the standard covering by $ \Delta ^ {n - 1 } $ of the $ i $- th face of $ \Delta ^ {n} $. As usual, cycles and boundaries are those chains which belong to the kernel and image of $ \partial ^ {n} $, respectively. The $ n $- dimensional singular homology group $ H _ {n} ^ {s} ( X; G) $ is defined as the quotient group of the group of $ n $- dimensional cycles by the subgroup of boundaries.

If $ A \subset X $, then the groups $ H _ {n} ^ {s} ( A; G) $ are defined using the subcomplex of $ S _ {*} ( X; G) $ consisting of all chains with supports in $ A $, and groups of a pair, $ H _ {n} ^ {s} ( X, A; G) $, by using the corresponding quotient complex. There is an exact homology sequence

$$ {} \dots \rightarrow H _ {n} ^ {s} ( A; G) \rightarrow \ H _ {n} ^ {s} ( X; G) \rightarrow \ H _ {n} ^ {s} ( X, A; G) \mathop \rightarrow \limits ^ \delta $$

$$ \mathop \rightarrow \limits ^ \delta H _ {n - 1 } ^ {s} ( A; G) \rightarrow \dots , $$

which is a covariant functor on the category of pairs $ ( X, A) $ of topological spaces and their continuous mappings.

The homomorphism $ \delta $ is defined as the boundary in $ X $ of a cycle of $ ( X, A) $ representing the corresponding element of $ H _ {n} ^ {s} ( X, A; G) $. Singular homology is homology with compact supports, in the sense that the groups associated with $ X $ are equal to the direct limits of the homology groups of the compact sets $ C \subset X $.

Singular cohomology is defined in a dual way. The cochain complex $ S ^ {*} ( X; G) $ is defined as the complex of homomorphisms into $ G $ of the integral singular chain complex $ S _ {*} ( X; \mathbf Z ) $. Less formally, cochains are functions $ \xi $ defined on singular simplices and taking values in $ G $, and the co-boundary homomorphism $ d $ is given by

$$ ( d \xi ) ( \sigma ^ {n + 1 } ) = \ \sum _ { i } (- 1) ^ {i} \xi ( \sigma _ {i} ^ {n+} 1 ). $$

The singular cohomology group $ H _ {s} ^ {n} ( X; G) $ is the quotient group of the group of $ n $- dimensional cocycles (the kernel of $ d $) by the subgroup of coboundaries (the image of $ d $). The cochain groups of a subspace $ A $ are defined by using the restriction of the cochain groups $ S ^ {*} ( X; G) $ to $ A $, while the cohomology groups $ H _ {s} ^ {n} ( X, A; G) $ of a pair are defined by using the cohomology of the subcomplex of $ S ^ {*} ( X; G) $ consisting of all cochains which vanish on the singular simplices of $ A $. There is an exact sequence

$$ {} \dots \rightarrow H _ {s} ^ {n} ( X, A; G) \rightarrow \ H _ {s} ^ {n} ( X; G) \rightarrow \ H _ {s} ^ {n} ( A; G) \mathop \rightarrow \limits ^ \delta \ $$

$$ \mathop \rightarrow \limits ^ \delta H _ {s} ^ {n + 1 } ( X, A; G) \rightarrow \dots , $$

which is a contravariant functor of $ ( X, A) $. The mapping $ \delta $ is defined as the coboundary in $ X $ of a cocycle of $ A $ representing the corresponding element of $ H _ {s} ^ {n} ( A; G) $.

Homology and cohomology groups with coefficients in an arbitrary group $ G $ can be expressed in terms of integral homology and cohomology groups using universal coefficient formulas. In the case of cohomology, this formula is valid only when $ G $ is finitely generated.

In the category of polyhedra, the singular theory is equivalent to the simplicial (and also the cellular) theory. This fact is commonly used to establish the topological invariance of the latter. However, the importance of the singular homology groups is not limited to this. Being defined in a simple way, they are applicable to a fairly wide category of topological spaces and are homotopy invariants. The natural connections with homotopy theory make the singular theory indispensable to homotopical topology.

However, although singular homology groups are defined for all topological spaces without any restriction, their application is only justified under such restrictions as local contractibility or homological local connectedness. Singular chains, being by their nature "too" linearly connected, do not carry information about "continuous" cycles if the latter are not "sufficiently" linearly connected. There are other possible "anomalies" (for example, the homology groups of a compact subspace of Euclidean space can differ from zero in arbitrarily high dimensions, the homology and cohomology groups of a pair $ ( X, A) $ can be mapped non-isomorphically under the natural mapping from $ X $ onto the quotient space $ X/A $ corresponding to a closed subset $ A \subset X $, etc.). Therefore, in general categories of topological spaces one also uses the corresponding Aleksandrov–Čech cohomology and its associated homology (cf. Aleksandrov–Čech homology and cohomology). These theories are free from the above deficiencies and coincide with the singular one, at least in every case when its application does not raise problems.

References

[1]	A. Dold, "Lectures on algebraic topology" , Springer (1972)
[2]	W.S. Massey, "Homology and cohomology theory" , M. Dekker (1978) pp. Chapts. 8; 9
[3]	E.G. Sklyarenko, "On homology theory associated with the Aleksandrov–Čech cohomology" Russian Math. Surveys , 34 : 6 (1979) pp. 103–137 Uspekhi Mat. Nauk. , 34 : 6 (1979) pp. 90–118
[4]	W.S. Massey, "Singular homology theory" , Springer (1980)
[5]	E.G. Sklyarenko, "Homology and cohomology of general spaces" , Springer (Forthcoming) (Translated from Russian)

References

[a1]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapt. 10
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Sects. 4.4; 5.4