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    A 2-complex with fundamental group $\langle x,y\mid x^2yxy\rangle$

    P-TOP-WORKSHOP-D7-GW1

    Area
    Topology
    Topics
    Cell Complexes, Fundamental Group, Groups
    Status
    draft

    Let \(G=\langle x,y\mid x^2yxy=1\rangle\). Describe a space \(X\) that has \(\pi_1(X)=G\).

    Solution

    Start with the wedge of two oriented circles \[ R=S^1_x\vee S^1_y, \] whose fundamental group is the free group \(F(x,y)\). Attach one \(2\)-cell to \(R\) by a map \(S^1\to R\) representing the word \[ x^2yxy. \] If \(X\) denotes the resulting CW complex, Seifert–van Kampen gives \[ \pi_1(X)\cong F(x,y)/\langle\!\langle x^2yxy\rangle\!\rangle \cong\langle x,y\mid x^2yxy=1\rangle=G. \] Thus a two-petal rose with one \(2\)-cell attached along the indicated word realizes the presentation.

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