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    Nonhomeomorphic spaces with isomorphic nontrivial fundamental groups

    P-TOP-WORKSHOP-D7-GW2

    Area
    Topology
    Topics
    Counterexamples, Fundamental Group, Homeomorphisms
    Status
    draft

    Find two spaces \(X\) and \(Y\) such that \(\pi_1(X)\cong\pi_1(Y)\) and are non-trivial, and are NOT homeomorphic.

    Solution

    Take \[ X=S^1,\qquad Y=S^1\vee S^2. \] By van Kampen, since \(S^2\) is simply connected, \[ \pi_1(Y)\cong\pi_1(S^1)*\pi_1(S^2)\cong\mathbb Z\cong\pi_1(X), \] so the common fundamental group is nontrivial.

    The spaces are not homeomorphic. Indeed, \[ H_2(S^1)=0, \qquad H_2(S^1\vee S^2)\cong H_2(S^2)\cong\mathbb Z. \] Homeomorphic spaces have isomorphic singular homology groups, so \(X\not\cong Y\).

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