Extra Problems: Algebraic Topology

Homotopy 101

  • Show that if \(X\xrightarrow{f} X^n\) is not surjective, then \(f\) is nullhomotopic.

\(\pi_1\)

  • Compute \(\pi_1(S^1 \vee S^1)\)
  • Compute \(\pi_1(S^1 \times S^1)\)

Surfaces

  • Show that if \(M^\text{orientable} \xrightarrow{\pi_k} M^\text{non-orientable}\) is a \(k{\hbox{-}}\)fold cover, then \(k\) is even or \(\infty\).
  • Show that \(M\) is orientable if \(\pi_1(M)\) has no subgroup of index 2.