Q1: Which of the following maps can be a transition map for some manifold?

A1: $ \omega: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}, ~ x \mapsto x$

A2: $ \omega: [0,1] \rightarrow \mathbb{R}^{2}, ~ x \mapsto \binom{x}{1}$

A3: $ \omega: \mathbb{R}^n \rightarrow \mathbb{R}^{n}, ~ x \mapsto x$

A4: $ \omega: \mathbb{R}^2 \rightarrow \mathbb{R}^{2}, ~ x \mapsto 0$

Q2: Let $\omega: \mathbb{R}^n \rightarrow \mathbb{R}^{n}$ be a $C^k$-diffeomorphism for $k\geq1$. What is not correct in general?

A1: $\omega$ is a $C^{k-1}$-diffeomorphism.

A2: $\omega$ is a homeomorphism.

A3: $\omega$ is invertible.

A4: $\omega$ is continuous.

A5: $\omega$ is $(k+1)$-times continuously differentiable.

Q3: A pair $(M,\mathcal{A})$ is called a $C^k$-smooth manifold if

A1: $M$ is topological manifold and $\mathcal{A}$ is an atlas.

A2: $M$ is topological manifold and $\mathcal{A}$ is a maximal $C^k$-atlas.

A3: $M$ is topological manifold and $\mathcal{A}$ is a $C^k$-atlas.