Transversality and Intersection Theory

For which \(a > 0\) do the surfaces

have nonempty transverse intersection?

Let \(R:S^2\to \mathbb {R}_+\) be a nonconstant \(C^\infty \) function. Show that \(F:S^2\to \mathbb {R}^3\) is defined by \(F(v)=R(v)v\), then there exists a value of \(r\in \mathbb {R}_+\) such that if \(S_r^2\) is the sphere of radius \(r\) centered at 0, then the intersection of \(S_r^2\) with the image of \(F\) is a nonempty finite union of circles.

Let \(M\) and \(N\) be submanifolds of \(\mathbb {R}^n\) and for all \(a\in \mathbb {R}^n\) let \(M_a=\) \(\{x+a\in \mathbb {R}^n\mid x\in M\}\). Show that \(M_a\) and \(N\) are transverse for almost all choices of \(a\in \mathbb {R}^n\).

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2=1\}\) be the 2-sphere and let \(M_a=f_a^{-1}(0)\) where \(a\in \mathbb {R}\) and \(f_a:\mathbb {R}^3\to \mathbb {R}\) is the function \(f_a(x,y,z)=x^2+y^2-z+a\). For what values of \(a\in \mathbb {R}\) are \(S^2\) and \(M_a\) transverse? Justify your answer.

Let \(f:\mathbb {R}^n\to \mathbb {R}^n\) be a smooth function such that \(f(0)=0\). Show that the graph of \(f\)

and the diagonal

intersect transversely at \((0,0)\) in \(\mathbb {R}^n\times \mathbb {R}^n\) if and only if the derivative \(f_*(0):T_0\mathbb {R}^n\to T_0\mathbb {R}^n\) does not have \(+1\) as an eigenvalue.

Let \(X\) and \(Y\) be closed sub-manifolds of \(\mathbb {R}^n\). Show that for almost all \(a\in \mathbb {R}^n\) the translate \(X+a=\{x+a\mid x\in X\}\) intersects \(Y\) transversally.

Let \(M\) be a compact, smooth two dimensional manifold without boundary embedded in \(\mathbb {R}^3\). Show that for almost every two-plane \(L\in \mathbb {R}^3\) such that \(L\cap M\) is nonempty, the intersection \(L\cap M\) consists of finitely many closed curves.

Let \(M\) be the smooth submanifold of \(\mathbb {R}^3\) that is the solution set of the equation \(x^2+y^2-z^2=1\). Let \(N\) be the smooth submanifold of \(\mathbb {R}^3\) that is the solution set of the equation \(x^2+y^2+z^2=1\). Prove that \(M\) and \(N\) do not intersect transversally in \(\mathbb {R}^3\).

Suppose \(f:M\to N\) is a smooth function of smooth manifolds. Suppose \(Q\) is an embedded submanifold of \(N\), and that \(f\) is transverse to \(Q\). Prove that if \(p\in f^{-1}(Q)\), then \[ T_p(f^{-1}(Q))=(D_pf)^{-1}[T_{f(p)}Q]. \] (The above line says that the tangent space for \(f^{-1}(Q)\) at \(p\) is the inverse image of the tangent space of \(Q\) at \(f(p)\) under the differential of \(f\) at \(p\).)

Let \(M,N,P\) be smooth manifolds without boundary and \(F:M\times P\to N\) a smooth map which is transverse to a submanifold \(Q\subset N\). Prove the Transversality Theorem: for almost every \(p\in P\) the map \(F_p:M\to N\) defined by \(F_p(x)=F(x,p)\) is transverse to \(Q\). If you’d like, you can consider the special case when \(M,Q\) are submanifolds of \(P=N=\mathbb {R}^n\) and \(F(x,p)=x+p\).

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(\mathrm {SL}(2,\mathbb {R})\subset \mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(\mathrm {SL}(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{id}\mathrm {SL}(2,\mathbb {R})\) as a subspace of \(T_{id}\mathcal {M}(2)=\mathbb {R}^4\). Is the line \[ \begin {pmatrix}t&0\\0&1\end {pmatrix},\quad t\in \mathbb {R} \] transverse to \(\mathrm {SL}(2,\mathbb {R})\)? Justify your answer.

Let \(X\) and \(Y\) be closed submanifolds of \(\mathbb {R}^n\). Show that for almost all \(a\in \mathbb {R}^n\) the translate \(X+a=\{X+a\mid x\in X\}\) intersects \(Y\) transversally.

Suppose a space \(X\) can be written as the union \(\bigcup _{i=1}^n U_i\) of open subsets \(U_i\). Also assume that every \(U_i\), as well as every nonempty intersection \(U_{i_1}\cap U_{i_2}\cap \cdots \cap U_{i_k}\), is contractible. Prove that \(\widetilde {H}_i(X;\mathbb {Z})=0\) for \(i\ge n-1\). (Hint: For \(i=1,\ldots ,n-1\) let \(V_i=U_n\cap U_i\) and apply induction.)

Let \(S\subset \mathbb {R}^3\) be a (nonempty) smooth surface. Prove that there exists a plane \(ax+by+cz=d\) in \(\mathbb {R}^3\) whose intersection with \(S\) is a collection of circles and lines.

Let \(f:M\to N\) be a smooth map of smooth manifolds. If \(Q\subseteq N\) is an embedded submanifold, and \(f\) is transverse to \(Q\), prove that \(f^{-1}(Q)\) is a manifold.

1. What theorem from multivariable calculus is the proof of the regular value theorem dependent on?
2. What theorem from single variable calculus is the proof of Stokes’ theorem dependent on?
3. What theorem from differential topology is used to prove that transverse intersection of manifolds is a manifold?

For which values of \(a>0\) does the hyperboloid \[ x^2+y^2-z^2=1 \] intersect the sphere \(x^2+y^2+z^2=a\) transversally (in \(\mathbb {R}^3\))?

In this problem we identify \(\mathbb {C}\) with \(\mathbb {R}^2\) in the usual manner via \[ z\mapsto (\operatorname {Re}z,\operatorname {Im}z). \] Consider the following smooth submanifolds of \(\mathbb {C}^2-\{(0,0),(0,0.5)\}\): \[ M_1=\{(z,w)\in \mathbb {C}^2\mid z^2+w^2=1\} \] \[ M_2=\{(z,w)\in \mathbb {C}^2\mid z^2-w^2+w=1\} \] \[ M_3=\{(x_1,y_1,x_2,y_2)\in \mathbb {R}^4\mid 1-x_1+x_2^2+y_1^2+y_2^2=0\}. \] Which of the pairs \(M_i,M_j\) (\(i,j\in \{1,2,3\}\)) are transverse at \(p=(1,0)\in \mathbb {C}^2\)?

Show that the total space of the tangent bundle \(T(S^2)\) of the 2-sphere is not diffeomorphic to \(S^2\times \mathbb {R}^2\). Hint: One strategy is to find a 2-sphere in \(T(S^2)\) whose self-intersection number is nonzero and argue that there are no such 2-spheres in \(S^2\times \mathbb {R}^2\).

Let \(X\) and \(Y\) be two submanifolds of \(\mathbb {R}^n\). Show that for almost every \(a\in \mathbb {R}^n\) the translate \[ X+a=\{x+a\mid x\in X\} \] is transverse to \(Y\).

Let \(X\) and \(Y\) be closed submanifolds of \(\mathbb {R}^n\). Show that for almost all \(a\in \mathbb {R}^n\) the translate \(X+a=\{x+a\mid x\in X\}\) intersects \(Y\) transversally.

Let \(X\) and \(Y\) be two submanifolds of \(\mathbb {R}^n\). Show that there is \(a\in \mathbb {R}^n\) such that \(X+a=\{x+a\mid x\in X\}\) and \(Y\) are transverse.

Let \(M^m,N^n\subset P^p\) be smooth embedded submanifolds of a smooth manifold. \(M^m\) and \(N^n\) are said to intersect transversally if \(T_xM+T_xN=T_xP\) for every \(x\in M\cap N\). Prove that if \(M^m\) and \(N^n\) intersect transversally then \(M^m\cap N^n\) is a smooth submanifold of \(P^p\). What is its dimension?

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 \(p\)?

Section II. Pick four of the following eight questions to answer: