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This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds.

Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ for some smooth $F:N\times S\to M.$ A class $\mathcal F$ of maps is called stable if whenever $\{F_s:N\to M \;|\; s\in S\}$ is a smooth family of maps and $F_{s_0}\in\mathcal F$ for some $s_0\in S$, then there is some neighborhood $U$ of $s_0$ in $S$ so that $F_s\in \mathcal F$ for each $s\in U.$

The problem I am asked to do is show that, under the assumption that $N$ is compact, the following classes of maps are stable: immersions, submersions, embeddings, diffeomorphisms, local diffeomorphisms, and maps which are transverse to a given properly embedded submanifold $X\subseteq M.$

I have solved the problem for immersions and submersions using the fact that maps of this form are maps which have maximal rank, and rank is lower semicontinuous. I am stuck when it comes to the other classes of maps, however. For example, when $F_{s_0}$ is a diffeomorphism, it is clear to me that under small perturbations of $s,$ the matrix for $d(F_s)_x$ will be invertible for each $x,$ since $GL_n(\mathbb R)$ is open in $M_{n\times n}(\mathbb R).$ What is not clear to me is why injectivity and surjectivity should be preserved under small perturbations. I'm having similar troubles with the other parts.

Something that I'm sure of is that compactness of $N$ is a necessary assumption for all of these classes of maps to be stable. I used this crucially in my solutions for the immersions and submersions, and I have done the next exercise demonstrating that none of these classes are stable when $N$ is not compact.

Any help or hints are appreciated.

EDIT: I have reduced the parts for the class of embeddings and diffeomorphisms to showing that injectivity is preserved under perturbations. I am unsure of how to show this.

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    $\begingroup$ Does Lee use a definition where diffeomorphisms are not immersions? Every definition I know of a diffeomorphism yields an isomorphism of the differential; in particular, the differential is injective so this should follow from the immersion case? $\endgroup$ Commented Feb 24, 2019 at 20:18
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    $\begingroup$ Yes diffeomorphisms are immersions, but it's not clear to me why it should follow from the immersion case. A priori it seems that any perturbation of $s$ could yield immersions which are not diffeomorphisms. In particular, these perturbations could kill surjectivity or injectivity. $\endgroup$
    – D. Brogan
    Commented Feb 24, 2019 at 20:34
  • $\begingroup$ Would you mind adding details regarding how the diffeomorphism case reduces to injectivity? I can't figure out the surjectivity... $\endgroup$
    – The Ledge
    Commented Nov 5, 2020 at 5:49
  • $\begingroup$ As to showing injectivity, the smooth family of $F_s$'s mapping $N$ to $M$ implies a function $F : N \times S \longrightarrow M$ mapping $(x,s)$ to $F_s(x),$ with $F$ smooth. If you let $x \neq y \in N,$ then you know that $F_0(x) \neq F_0(y).$ I believe this implies a neighborhood $T \ni 0$ such that $F_s(x) \neq F_y(y)$ whenever $s \in T,$ since otherwise you could have a sequence $(s_i),\, s_i \longrightarrow 0,$ such that $F_{s_i}(x) = F_{s_i}(y),$ which would break continuity of $F,$ and therefore would be a contradiction (if I'm not mistaken). $\endgroup$
    – The Ledge
    Commented Nov 5, 2020 at 5:54
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    $\begingroup$ Most of these cases are proven in Guillemin & Pollacks's excellent book Differential Topology. $\endgroup$
    – Laz
    Commented Apr 6, 2023 at 20:22

1 Answer 1

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I found solution on the page 13 of

http://math.berkeley.edu/~kozai/m141f14/m141_notes.pdf

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