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    A $\Delta$-complex structure and homology of $S^1$

    P-TOP-WORKSHOP-2020-WS4A-HW2

    Area
    Topology
    Topics
    Cell Complexes, Homology
    Status
    draft

    Give a \(\Delta\)-complex structure for \(S^1\), and use that to compute the homology groups for \(S^1\).

    Solution

    Give \(S^1\) the \(\Delta\)-complex structure with one vertex \(v\) and one \(1\)-simplex \(e\) whose two endpoints are both identified with \(v\). Then \[ 0\longrightarrow C_1\cong\mathbb Z \xrightarrow{\partial_1} C_0\cong\mathbb Z \longrightarrow0. \] Since both endpoints of \(e\) are \(v\), \[ \partial_1e=v-v=0. \] Therefore \[ H_1(S^1)\cong\ker\partial_1\cong\mathbb Z, \qquad H_0(S^1)\cong C_0/\operatorname{im}\partial_1\cong\mathbb Z, \] and \(H_n(S^1)=0\) for \(n\ge2\).

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