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Brown, A. (2019). Differential Topology and Applications. Wiley.

Miller, R. (2020). Stochastic Methods in Differential Topology. Springer.

Johnson, L. (2021). Applications of Differential Topology in Physics. Cambridge University Press.

Chen, H. (2022). Advanced Topics in Differential Topology: Morse Theory and Cobordism. MIT Press.

In each of the following examples, exactly one of the defining conditions of a topological manifold fails for $M$. State that condition and explain why it fails.
(i) $M=X / \sim$ where $X=\mathbb{R} \times{0} \cup \mathbb{R} \times{1} \subset \mathbb{R}^2 . \sim$ is the equivalence relation on $X$ generated by $(x, 0) \sim(x, 1)$ for all $x \neq 0$.
(1 point.)
(ii) $M=$ Disjoint union of uncountable copies of $\mathbb{R}$.
(1 point.)
(iii) $M=\left{(x, y) \in \mathbb{R}^2 \mid x y=0\right}$.
(1 point.)

Solution. For the first one Hausdorffness fails as one cannot separate $[(x, 0)]$ and $[(x, 1)]$. For the second one there are uncountably ( $\mathbb{R}$ many) disjoint open sets and hence it cannot have a countable basis. For the third, $M$ is not locally Euclidean around $(0,0)$.

Consider the mapping $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by
$$f(x, y, z)=\left(x^2+y, x^2+y^2+z^2+y\right) .$$
Show that $f^{-1}((0,1))$ is an embedded submanifold of $\mathbb{R}^3$.
(1 point.)
Show that $f^{-1}((0,1))$ is $C^1$-diffeomorphic to $S^1$.
(2 points.)

Solution. It is clear that $D f_{(a, b, c)}$ has rank two by considering the two cases when $a$ is zero (in which case for any point in $f^{-1}(0,1), c$ cannot be zero) and $a$ is nonzero in which it is evident that the two rows are linearly independent.

To construct the $C^1$ diffeomorphism just observe that $f^{-1}(0,1)$ is diffeomorphic to $\left{(x, y, z) \in \mathbb{R}^3 \mid x^4+z^2=1, y=-x^2\right}$. Then define $g(x)=\operatorname{sgn}(x) x^2$ and then look at the map $\Phi(x, y, z)=(g(x), z)$ and show that it is a $C^1$ diffeomorphism.

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