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    Fundamental groups, universal covers, and simplicial homology of standard spaces

    P-WESTOP08-II2

    Area
    Topology
    Seen at
    WESLEYAN
    Years
    2008
    Status
    draft
    Contents
    1. Determine the fundamental group of each space:
      • \(S^1\);
      • the torus;
      • the infinite cylinder \(S^1\times\mathbb R\);
      • \(S^1\cup(\mathbb R\times\{0\})\subset\mathbb R^2\);
      • \(\mathbb R^2\setminus(\mathbb R\times\{0\})\).
    2. Find the universal covering space of each:
      • \(S^1\);
      • \(S^2\);
      • \(\mathbb{RP}^2\);
      • the torus.
    3. Compute the simplicial homology groups of each:
      • \(S^1\);
      • \(S^2\);
      • \(\mathbb{RP}^2\).

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