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    $T\cup_f D^2$ with $f$ attaching inside the torus

    P-ALVYE

    Area
    Topology
    Topics
    Cell Complexes, Homology
    Status
    draft
    1. \(T\union_f D^2\), where \(f\) attaches inside the torus
    Solution

    The homotopy class of the attaching map \(f:S^1\to T\) is not specified by the phrase “attaches inside the torus.” Different choices give non-homotopy-equivalent spaces. For example, attaching along a null-homotopic loop gives \(T^2\vee S^2\), while attaching along a primitive essential loop kills one generator of \(\pi_1(T^2)\) and yields a different homotopy type. Thus the homology depends on the attaching curve.

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