Math 6510 • Differentiable Manifolds

Identify \(\mathbb{R}^4\) with \(\mathbb{C}^2\) so that
\(z=x_1+iy_1\), \(w=x_2+iy_2\). Let \(p=(1,0)\in\mathbb{C}^2\). Consider the following
smooth submanifolds of \(\mathbb{C}^2-\{(0,0),(0,0.5)\}\):
\[
  M_1:=\{(z,w)\mid z^2+w^2=1\},
\]
\[
  M_2:=\{(z,w)\mid z^2-w^2+w=1\},
\]
\[
  M_3:=\{(x_1,y_1,x_2,y_2)\mid 1-x_1+x_2^2+y_1^2+y_2^2=0\}.
\]
Which of the pairs of manifolds \((M_1,M_2)\), \((M_2,M_3)\), \((M_1,M_3)\) are transverse
at \(p\)?

\textbf{Section II. Pick four of the following eight questions to answer:}