Algebraic Topology

Department of Mathematics, Indian Institute of Science, Aug-Dec 2014.

Assignment 4 (Due September 15, 2014)

The Mobius band $X$ is the quotient of $[0,1] \times [0,1]$ by the equivalence relation $\sim$ generated by $(0, y) \sim (1, 1 - y)$ for $y\in [0, 1]$.

  1. Show that the image $Y$ of the set $\{(x, 1/2) : x \in [0,1]\}$ is homeomorphic to a circle.
  2. Show that $X$ deformation retracts onto $Y$. Deduce $\pi_1(X, x_0)= \mathbb{Z}$ for $x_0 \in X$.
  3. Let $Z$ be the boundary of $X$, namely, the image of the set $\{(x, y) : y(1 -y) = 0\}$. Show that $Z$ is homeomorphic to the circle.
  4. Let $i: Z \to X$ be the inclusion map and $z\in Z$. Show that the induced homomorphism $i_*: \pi_1(Z, z)\to \pi_1(X, z)$ is an injection but not a surjection.
  5. Show that there is no retraction $r :X \to Z$.