Property of non-transversal manifold In Guillemin and Pollack they ask: > Suppose that X and Z do not intersect transversally in $Y$. May $X \cap Z$ still be a manifold? If so, must its codimensions be $\text{codim}\space{X} + \text{codim}\space{Z}$? **(Can it be?)** Answer with drawings. The example I was able to think of is let $X = Z = R^1$ embedded in $R^2$. The codimension is not $\text{codim}\space{X} + \text{codim}\space{Z}$ since it is just $\text{codim}\space{X} = 1$. I can't see any particular reason why it should be $\text{codim}\space{X} + \text{codim}\space{Z}$ but I can't see a reason why it shouldn't be either. I have strong suspicion that it can't be though, but I am not sure.

Take a cylinder of radius $1$, and a sphere of radius $1$.

Put the sphere inside. Now put it outside, and touch the cylinder with it.