非常感谢您对我们微分拓扑专家团队的信任！以下是一些关于微分拓扑的类似文字，可能会对您有所帮助。请注意，以下文献的引用信息是虚构的。
Smith, J. (2018). Introduction to Differential Topology. Oxford University Press.
这本教材是一本关于微分拓扑基础概念的入门书籍，涵盖了流形、切空间、微分形式和Stokes定理等内容。
Brown, A. (2019). Differential Topology and Applications. Wiley.
该书提供了更深入的微分拓扑内容，包括向量场、黎曼几何、纤维丛和同伦理论等。
Miller, R. (2020). Stochastic Methods in Differential Topology. Springer.
这本书将随机分析与微分拓扑相结合，探讨了随机微分流形、随机同伦和随机同调等内容。
Johnson, L. (2021). Applications of Differential Topology in Physics. Cambridge University Press.
该书介绍了微分拓扑在物理领域的应用，包括量子场论、广义相对论和弦理论等方面。
Chen, H. (2022). Advanced Topics in Differential Topology: Morse Theory and Cobordism. MIT Press.
这本书深入研究了微分拓扑的高级主题，如莫尔斯理论、边界理论和特殊全纯性等。
请注意，这些文献只是提供了一些关于微分拓扑的参考，您可能需要根据具体的研究或学习需求来选择适合您的资料。另外，如果您需要更具体的文献推荐或对特定主题的深入解释，请随时告诉我们，我们将竭诚为您提供帮助！
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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