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I'm working on two exercises from Guillemin-Pollack which have the same flavor:

(General Position Lemma) Let $X$ and $Y$ be submanifolds of $\mathbb R^N$. Show that for almost every $a\in \mathbb R^N$, the translate $X+a$ intersects $Y$ transversally.

Suppose that $X$ is a submanifold of $\mathbb R^N$. Show that almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ transversally. [HINT: The set $S\subset (\mathbb R^N)^l$ consisting of all linearly independent $l$-tuples if vectors in $\mathbb R^N$ is open in $\mathbb R^{Nl}$, and the map $\mathbb R^l\times S\rightarrow \mathbb R^N$ defined by $[(t_1,\dots,t_l),v_1,\dots,v_l]\mapsto t_1v_1+\dots t_lv_l$ is a submersion.]

In both cases, I believe I need to apply the following version of the Transversality Theorem (see this answer):

Theorem: Suppose that $F:X\times S\to Y$ is a smooth map of manifolds and $Z$ is a submanifold of $Y$, all manifolds without boundary. If $F$ is transverse to $Z$ then for almost every $s\in S$ the map $f_s : x\mapsto F(x,s)$ is transverse to $Z$.

I have the same problem in both exercises.

  • In the first exercise, the theorem guarantees that for almost every $a\in \mathbb R^N$, the map $f_a: X\rightarrow \mathbb R^N$ given by $x\mapsto x+a$ is transversal to $Y$. Note that the image of this map is $X+a$. I need to show that the image is transversal to $Y$. How does it follow?
  • In the second exercise, the theorem guarantees that for almost every $v=(v_1,\dots,v_l)\in S$, the map $f_v: \mathbb R^l\rightarrow \mathbb R^N$ given by $(t_1,\dots,t_l)\mapsto t_1v_1+\dots+t_lv_l$ is transversal to $X$. Note that the image of this map is an $l$-dimensional subspace of $\mathbb R^N$. I need to show that the image is transversal to $X$. How does it follow?
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  • $\begingroup$ The second one is false, as stated. You need an affine $l$-dimensional subspace, not a vector subspace. Can you see why? By the way, you can find a list of errata in Guillemin and Pollack here. $\endgroup$
    – Ted Shifrin
    Feb 28, 2018 at 17:54
  • $\begingroup$ @TedShifrin For the points $v_i$'s they may not have linear combination satisfies $0$. About affine spaces, how to prove second question? $\endgroup$
    – Jale'de jaled
    Jan 29, 2021 at 16:50
  • $\begingroup$ I have shown this is a submersion but I am having problem with showing the intersection is transversal, intuitively it is good but formally I cannot write. $\endgroup$
    – Jale'de jaled
    Jan 29, 2021 at 16:51
  • $\begingroup$ @Jale'dejaleuffnejale Of course, $0$ is in the span of any vectors. So a linear subspace cannot be transverse to $X$ if $0\in X$ and dimensions are too small. For the second, a submersion is transverse to any submanifold! $\endgroup$
    – Ted Shifrin
    Jan 29, 2021 at 17:59

1 Answer 1

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Let $i_1 : M_1 \to X$ and $i_2 : M_2 \to X$ be embeddings. The following statements are equivalent:

  • $i_1$ and $i_2$ are transverse,
  • $i_1$ and $M_2$ are transverse,
  • $i_2$ and $M_1$ are transverse,
  • $M_1$ and $M_2$ are transverse.

This follows from the fact that for an embedding $f : M \to X$, $f_*(T_pM) = T_{f(p)}(f(M))$.

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  • $\begingroup$ I'm not sure whether I understand what are $i_1$ and $i_2$ in the context of my question and also what the transversality of two maps means. $\endgroup$
    – user557
    Feb 28, 2018 at 18:26
  • $\begingroup$ In your first example, you've shown that the embedding $f_a$ is transversal to the submanifold $Y$ (this is the case of the second dot point I listed), and you want to conclude that the image of $f_a$ is transversal to $Y$ (this is the fourth dot point I listed). For transversality of two maps, the definition can be found here. $\endgroup$
    – Michael Albanese
    Mar 1, 2018 at 14:11

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