(May 2013)
Describe how to construct a cell structure on the \(2\)-sphere \(S^2\) consisting of one \(0\)-cell, one \(1\)-cell, and two \(2\)-cells, and explain how to use this cell structure to calculate the simplicial homology groups of \(S^2\).
Explain how a long exact sequence may be used to calculate all of the (singular) homology groups \(H_i(S^n)\) of the \(n\)-sphere \(S^n\) (and calculate these groups for all \(i\) and \(n\)).
Solution
Use the equator as the \(1\)-skeleton. Give that circle one \(0\)-cell \(v\) and one \(1\)-cell \(e\). The northern and southern open hemispheres are two \(2\)-cells \(N,S\), each attached along the equator. Orient them so their attaching maps have degrees \(+1\) and \(-1\). The cellular chain complex is \[ 0\to\mathbb Z^2\langle N,S\rangle \xrightarrow{\partial_2} \mathbb Z\langle e\rangle \xrightarrow{0} \mathbb Z\langle v\rangle\to0, \] with \[ \partial_2(m,n)=m-n. \] Therefore \[ H_2(S^2)=\ker\partial_2\cong\mathbb Z, \qquad H_1(S^2)=\mathbb Z/\operatorname{im}\partial_2=0, \qquad H_0(S^2)=\mathbb Z. \]
Decompose \(S^n\) into open neighborhoods of its upper and lower hemispheres. Each is contractible and their intersection deformation retracts onto \(S^{n-1}\). The reduced Mayer–Vietoris long exact sequence therefore yields isomorphisms \[ \widetilde H_i(S^n)\cong\widetilde H_{i-1}(S^{n-1}) \] for \(n\ge1\). Starting from \[ \widetilde H_0(S^0)\cong\mathbb Z, \] induction gives \[ \widetilde H_i(S^n)\cong \begin{cases} \mathbb Z,&i=n,\\ 0,&i\ne n. \end{cases} \] Hence for \(n\ge1\), \[ H_i(S^n)\cong \begin{cases} \mathbb Z,&i=0,n,\\ 0,&\text{otherwise}. \end{cases} \] For \(n=0\), \(S^0\) has two points, so \(H_0(S^0)\cong\mathbb Z^2\) and all higher groups vanish.