Singular homology and cohomology

Remark 3.3.2 (Singular (co)homology). Given a topological space \(X\), one can write down an explicit chain complex of abelian groups \(C_*^{\mathrm {sing}}(X;\Z )\) called the singular chain complex of \(X\). The \(n\)-th term of this chain complex is given by \[ C_n^{\mathrm {sing}}(X;\Z ) := \Z [\Sing _n(X)], \] the free abelian group generated by the set \(\Sing _n(X) = \Hom _{\Top }(\abs {\Delta ^n}, X)\) of continuous maps from the geometric \(n\)-simplex to \(X\). The boundary map \(d_n\colon C_n^{\mathrm {sing}}(X;\Z ) \to C_{n-1}^{\mathrm {sing}}(X;\Z )\) sends a generating \(n\)-simplex \(\sigma \) to the alternating sum \(\sum _{i=0}^n (-1)^i d_i(\sigma )\) of its faces.

For an abelian group \(A\), tensoring with \(A\) and taking hom into \(A\) produces chain and cochain complexes \[ C_*^{\mathrm {sing}}(X;A) := C_*^{\mathrm {sing}}(X;\Z ) \otimes A \qquadtext {and} C^*_{\mathrm {sing}}(X;A) := \Hom _{\Ab }(C_*^{\mathrm {sing}}(X;\Z ),A). \] Their homology groups are the classical singular homology and cohomology groups of \(X\) with coefficients in \(A\). This is the traditional construction of ordinary (co)homology. The fact that this satisfies the axioms is non-trivial.

In this book, we will not take singular chains as the definition, because it depends on choosing a topological presentation of an anima. Instead, in Section 6.2 we will construct ordinary homology and cohomology intrinsically using the derived category \(\D (\Z )\). The comparison there shows that, for a topological space \(X\), the intrinsic ordinary (co)homology of the underlying anima \(\Pi _{\infty }(X)\) agrees with the singular (co)homology of \(X\).