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I was reading this proof by Henry T Horton for the problem below:

Let $X$ and $Z$ be transversal submanifolds of $Y$. Prove that if $y \in X \cap Z$, then $$T_y(X \cap Z) = T_y(X) \cap T_y(Z).$$

Linked here The tangent space to the intersection is the intersection of the tangent spaces..

My question is how did he go from $$\mathrm{codim}(X \cap Z) = \mathrm{codim}(X) + \mathrm{codim}(Z),$$ to $$\dim(X \cap Z) = \dim(X) + \dim(Z) - n,$$.

I was under the impression that for a sub manifold $X \subset Y$; the codim(X) = dim(Y) -dim(X).

This would then imply that $$\mathrm{codim}(X \cap Z) = \mathrm{codim}(X) + \mathrm{codim}(Z) = \dim(Y) -\dim(X) + \dim(Y) - \dim(Z)$$

Can some help elaborate where the $n$ comes from and why its being subtracted? Please and thank you!

1 Answers1

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For $M = X, Z, X \cap Z$ we have $\dim (M) + \operatorname{codim}( M) = n$, where $n = \dim Y$. Therefore the $ \operatorname{codim}$-equation implies $$\dim (X \cap Z) = n - \operatorname{codim}(X \cap Z) = n - \operatorname{codim} (X) - \operatorname{codim} (Z) \\= n - \operatorname{codim} (X) + n - \operatorname{codim} (Z) - n = \dim (X) + \dim (Z) - n .$$

Paul Frost
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