The question is about the exercise 1.5.4 in Victor Guillemin & Alan Pollack's book Differential Topology. Let $X$ and $Z$ be transversal submanifolds of $Y$. Prove that if $y\in X\cap Z$, then $ T_y(X\cap Z)=T_y(X)\cap T_y(Z). $ ("The tangent space to the intersection is the intersection of the tangent spaces.'')
My attemption:
Firstly, $\forall y \in T_y(X\cap Z)$, the result that $y \in T_y(X)$ and $y \in T_y(Z)$ is directly gained. And then we have $y\in T_y(X) \cap T_y(Z)$. So $T_y(X \cap Z)\subseteq T_y(X) \cap T_y(Z)$.
We only need to show that $\mathrm{dim}\, T_y(X\cap Z)=\mathrm{dim}, \left(T_y(X)\cap T_y(Z)\right)$. Indeed, if the dimension of $T_y(X)$ is equal to $\mathrm{dim}\, X$, and similarly $\mathrm{dim}\, T_y(Z)=\mathrm{dim}\, Z$, we can have the identity as follows: \begin{equation*} \begin{aligned} \mathrm{dim}\, T_y(X\cap Z)&=\mathrm{dim}\, Y-\mathrm{codim}\, \left(T_y(X\cap Z)\right)=\mathrm{dim}\, Y-\mathrm{codim}\, \left(X\cap Z\right)\\& =\mathrm{dim}\, Y-\left(\mathrm{codim}\,X+\mathrm{codim}\,Z\right)\\ &=\mathrm{dim}\, Y-\left[\left(\mathrm{dim}\, Y-\mathrm{dim}\, X\right)+\left( \mathrm{dim}\, Y-\mathrm{dim}\, Z\right)\right]\\ &=\mathrm{dim}\, X+\mathrm{dim}\, Z-\mathrm{dim}\, Y=\mathrm{dim}\, T_y(X)+\mathrm{dim}\, T_y(Z)-\mathrm{dim}\, T_y(Y)\\ &=\mathrm{dim}\, \left(T_y(X)\cap T_y(Z)\right)+\mathrm{dim}\, \left(T_y(X)+T_y(Z)\right)-\mathrm{dim}\, T_y(Y)\\ &=\mathrm{dim}\, \left(T_y(X)\cap T_y(Z)\right). \end{aligned} \end{equation*} There are two identities using above: $\mathrm{codim}\, \left(X\cap Z\right)=\mathrm{codim}\,X+\mathrm{codim}\,Z$ is derived by the additivity of codimension of the intersection of two transversal submanifolds. $\mathrm{dim}\, T_y(X)+\mathrm{dim}\, T_y(Z)=\mathrm{dim}\, \left(T_y(X)\cap T_y(Z)\right)+\mathrm{dim}\, \left(T_y(X)+T_y(Z)\right)$ is the dimensional identity of the intersection of two linear subspace. We have proved this conclusion.
My question:
The most important question comfuses me is that, if we could gain the dimension of the underlying space(just is the manifold) by the dimension of the tangent space of a point in the manifold? Briefly, In the situation given by the exercise, given a point $p$ in $X\cap Z$, and $\mathrm{dim}\, T_p(X\cap Z)=n$, is the conclusion $\mathrm{dim}\, \left(X\cap Z\right)=n$ true? Or in more general, if $p$ is a point in a manifold $M$, and $T_p(M)$ the tangent space of $M$ at $p$ has dimension $n$, then could we we have $\mathrm{dim}\, M=n$?
