I'm reading Introduction to Smooth Manifolds by John M. Lee, and I have a question about Proposition 2.12 (=Problem 2-2) on P.36.
Proposition 2.12. Suppose $M_1, \cdots, M_k$ and $N$ are smooth manifolds with or
without boundary, such that at most one of $M_1, \cdots, M_k$ has
nonempty boundary. For each $i$, let $\pi_i: M_1 \times \cdots \times M_k \rightarrow M_i$ denote the projection onto the $M_i$
factor. A map $F: N \rightarrow M_1 \times \cdots \times M_k$ is
smooth if and only if each of the component maps $F_i = \pi_i \circ F: N \rightarrow M_i$ is smooth.
The main idea of my solution is:
- Suppose $F$ is smooth. It is not hard to prove that each $\pi_i$ is smooth. By Proposition 2.10(d), the composition of smooth maps is smooth. Thus each $F_i$ is smooth.
- Suppose each $F_i$ is smooth. Let $p \in N$ be given. Then for each $i$, there exist charts $(U_i, \phi_i), (V_i, \psi_i)$ such that $p \in U_i, F_i(U_i) \subset V_i$ and $\psi_i \circ F_i \circ \phi_i^{-1}$ is smooth. Let $U = \cap U_i$. Then $(\phi_1\vert_U, U)$ and $(\psi_1 \times \cdots \times \psi_k, V_1 \times \cdots V_k)$ are charts that satisfy the properties in the definition of a smooth map.
The problem with my solution is that it does not use the condition that at most one of $M_1, \cdots, M_k$ has nonempty boundary. Which part of my proof would fail without that condition?
Thank you!
Best Answer
This question is actually answered in an earlier chapter in the textbook.
The first paragraph of P.29 of Chapter 1 mentions that "a finite product of smooth manifolds with boundary cannot generally be considered as a smooth manifold with boundary" with some explanation.
So, the condition was necessary to make sure that $M_1 \times \cdots \times M_k$ is a smooth manifold with boundary, but it was not a condition that I had to use directly in the solution.