In Lee‘s Introduction to smooth manifolds he states the Transversality Homotopy Theorem as follows:
Suppose $M,N$ are smooth manifolds and $X \subset M$ is an embedded submanifold. Every smooth map $f:N\to M$ is homotopic to a smooth map $g: N \to M$ that is transverse to $X.$
I think that this is not true without any further conditions, because if $\dim X + \dim N< \dim M,$ then $\dim T_{g(x)}X+\dim dg_x(T_x N)<\dim T_{g(x)}M$ should hold for all smooth maps $g:N\to M$ and all $x\in N,$ i.e. there should not be any map $g:N \to M$ transverse to $X.$
Is this correct or am I missing something? Thanks in advance.
The theorem is true. It just means that if $\dim(X)+\dim N<\dim M$ that $g$ will not intersect $X$. So two circles in $\mathbb R^3$ in general position will not intersect. The same for a circle and a point in $\mathbb R^2$.