Prove or disprove that $D_n (U(n)) \simeq _H \{z \in \mathbb{C} :|Re(z)|= \sqrt{1/2}-|Im(z)|\}$

Question

We have that $U(n) = \{X \in GL_n(\mathbb{C}): X^*X= XX^* = I_n\}$, where $ X^* = \overline{X}^T $ and $D_n: GL_n \to \mathbb{C} /\{0\}, X\to \det (X)$, I proved that $D_n(U_n)=S^1$ where $S^1=\{z \in C : |z|=1\}$, the I have a question, if $S^1 \simeq_H \{z \in \mathbb{C} :|Re(z)|= \sqrt{1/2}-|Im(z)|\}$

Answer 1

If you know that $D_n(U(n)) = S^1$, it remains to show that $S^1 \approx S = \{z \in \mathbb{C} :|Re(z)|= \sqrt{1/2}-|Im(z)|\}$. Identifying $\mathbb R^2$ with $\mathbb C$ via $(x,y) \mapsto x + iy$ we see that $S = \{(x,y) \in \mathbb{R}^2 :|x| + |y| = \sqrt{1/2}\}$. But the expression $\lVert (x,y) \rVert_1 = |x| + |y|$ provides a norm on $\mathbb R^2$. Thus $S = \{\xi \in \mathbb{R}^2 :\lVert \xi \rVert_1 = \sqrt{1/2}\}$.

Define continuous maps $$h : S^1 \to S, h(\xi) = \sqrt{1/2}\frac{\xi}{\lVert \xi \rVert_1},$$ $$g : S \to S^1, g(\eta) = \frac{\eta}{\lVert \eta \rVert_2},$$ where $\lVert -\rVert_2$ is the Euclidean norm on $\mathbb{R}^2$. It is an easy exercise to show that $g \circ h = id$ and $h \circ g = id$. Thus $h$ and $g$ are homeomorphisms which are inverse to each other.