calculate $\pi_1(\mathbb D-\{(0,0)\})$

Question

I'm interested in calculating $\pi_1(\mathbb D-\{(0,0)\})$.

My guess would be that $\mathbb D-\{(0,0)\}$ is homotpic to $S^1$ and so the fundamental group would be $\mathbb Z$. Am I right? How would one show that the spaces are homotopic?

Thanks

Answer 1

Hint: try to construct a deformation retract of $\mathbb{D}-\{ (0,0) \}$ onto the unit circle.

Answer 2

We wish to define maps \begin{align*} f&: \Bbb S^1\to\Bbb D^\prime & g&:\Bbb D^\prime\to \Bbb S^1 \end{align*} such that $f\circ g\simeq 1_{\Bbb D^\prime}$ and $g\circ f\simeq 1_{\Bbb S^1}$.

Using our intuition, we may define $f$ and $g$ by the formulas \begin{align*} f(x) &= x & g(x) &= \frac{x}{\lVert x\rVert} \end{align*} Now, we wish to construct maps \begin{align*} F &: I\times\Bbb D^\prime\to\Bbb D^\prime & G&:I\times\Bbb S^1\to\Bbb S^1 \end{align*} where $I=[0,1]$ such that \begin{align*} F(0,x) &= (f\circ g)(x) & G(0,x) &= (g\circ f)(x) \\ &= \frac{x}{\lVert x\rVert} & &= \frac{x}{\lVert x\rVert} \\ &&&=x \\ F(1,x) &= x & G(1,x) &= x \end{align*} Clearly $G(\lambda,x)=x$ works. Can you define $F(\lambda,x)$?