Components & Inclusions

due by Monday, Sep 11, 2023
  1. Let $X$ be a topological space and $x_0$ be a point in $X$. Let $Z$ be the path-component of $x_0$ in $X$. Show that the inclusion $i: Z \hookrightarrow X$ induces an isomorphism on fundamental groups.

  2. Let $f: (X, x_0)\to (Y, y_0)$ be a map between based topological spaces and let $f_*: \pi_1(X, x_0)\to \pi_1(Y, y_0)$ be the induced map on fundamental groups. Prove or disprove the following statements:

    • (a) If $f$ is injective then so is $f_*$.
    • (b) If $f$ is surjective then so is $f_*$.
    • (c) If $f$ is a homeomorphism then $f_*$ is an isomorphism.