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    Fundamental groups of the solid torus and the torus-sphere product

    E-C22V5

    Area
    Topology
    Topics
    Fundamental Group
    Status
    draft

    Compute the fundamental groups of the “solid torus” \(S^1 \times B^2\) and the product space \(S^1 \times S^2\).

    Solution

    The ball \(B^2\) is contractible, so \[ S^1\times B^2\simeq S^1\times\{0\}\cong S^1. \] Hence \[ \pi_1(S^1\times B^2)\cong\pi_1(S^1)\cong\mathbb Z. \]

    For the second product, the product formula for fundamental groups gives \[ \pi_1(S^1\times S^2)\cong\pi_1(S^1)\times\pi_1(S^2). \] Since \(S^2\) is simply connected, \[ \pi_1(S^1\times S^2)\cong\mathbb Z\times0\cong\mathbb Z. \] Thus both spaces have infinite cyclic fundamental group.

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