Extra Problems: Algebraic Topology

Algebraic Topology

Fundamental Group

  • Compute \(\pi_1(X)\) where \(X \coloneqq S^2/\sim\), where \(x\sim -x\) only for \(x\) on the equator \(S^1 \hookrightarrow S^2\).
    • Hint: try cellular homology. Should yield \([{\mathbf{Z}}, {\mathbf{Z}}/2{\mathbf{Z}}, {\mathbf{Z}}, 0, \cdots]\).
  • Show that if \(X = S^2 {\textstyle\coprod}_{\operatorname{id}} S^2\) is a pushout along the equators, then \(H_n(X) = [{\mathbf{Z}}, 0, {\mathbf{Z}}^3, 0, \cdots]\).

Covering Spaces

  • Describe all connected covering spaces of \({\mathbb{RP}}^2 \vee {\mathbb{RP}}^2\).

Homology

  • Compute the homology of the Klein bottle using the Mayer-Vietoris sequence and a decomposition \(K = M {\textstyle\coprod}_f M\)
  • Use the Kunneth formula to compute \(H^*(S^2\times S^2; {\mathbf{Z}})\).
    • Known to be \([{\mathbf{Z}}, 0, {\mathbf{Z}}^2, 0, {\mathbf{Z}}, 0, 0, \cdots]\).
  • Compute \(H^*(S^2 \vee S^2 \vee S^4)\)
    • Known to be \([{\mathbf{Z}}, 0, {\mathbf{Z}}^2, 0, {\mathbf{Z}}, 0, 0, \cdots]\).
  • Show that \(\chi(\Sigma_g + \Sigma_h) = \chi(\Sigma_g) + \chi(\Sigma_h) - 2\).