Diffeomorphisms and Isotopy Methods

Let \(M\) be a connected smooth manifold. Show that for all \(p,q\in M\) there is a diffeomorphism \(f:M\to M\) such that \(f(p)=q\).

Let \(f:M\to N\) and \(g:N\to P\) be diffeomorphisms of smooth manifolds. Let \(\sigma (f)\) equal 1 if \(f\) preserves orientation and \(-1\) otherwise. Defining \(\sigma (g)\) and \(\sigma (g\circ f)\) similarly, prove that \(\sigma (g\circ f)=\sigma (g)\sigma (f)\).

Suppose \(M\) is a closed, orientable, simply connected, smooth manifold of dimension 3. Let \(S_3\) be the symmetric group on 3 letters, and suppose \(S_3\) acts on \(M\) freely, by orientation preserving diffeomorphisms. Let \(N=S_3\backslash M\), and find \(H_n(N;\mathbb {Z})\) and \(H^n(N;\mathbb {Z})\) for all \(n\).

Let \(B\) be the open ball in \(\mathbb {R}^n\) that is centered at \(0\) and has radius \(1\). If \(y\in B\) and \(\lVert y\rVert <1/2\), define a vector field on \(B\) and use the resulting flow to find a diffeomorphism \(h:B\to B\) with \(h(0)=y\) such that \(h\) is homotopic to the identity map on \(B\). Conclude that if \(N\) is a smooth connected manifold and \(p,q\in N\), then there is a diffeomorphism \(H:N\to N\) such that \(H(p)=q\) and such that \(H\) is homotopic to the identity map on \(N\).

Let \(U\subset \mathbb {R}^n\) be a connected, open subset. For all \(p,q\in U\) show that there exists a diffeomorphism \(\phi :\mathbb {R}^n\to \mathbb {R}^n\) with \(\phi (p)=q\) and \(\phi \) the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

For \(t\in (0,1)\) define a map \(f_t:S^2\to S^2\) by \[ f_t(x,y,z)=\left ( \frac {x}{\sqrt {x^2+y^2+(z+t)^2}}, \frac {y}{\sqrt {x^2+y^2+(z+t)^2}}, \frac {z+t}{\sqrt {x^2+y^2+(z+t)^2}} \right ). \] Show that \(f_t\) is a Lefschetz map with isolated fixed points at the two antipodal points with \(z=\pm 1\) and the Lefschetz number is \(L(f)=2\). Conclude that every diffeomorphism of \(S^2\) that is homotopic to the identity has a fixed point. (Hint: One can simply compute the derivative of \(f_t\) to see that on the tangent spaces of the two fixed points it is a multiple of the identity. However, if you use that \(f_t\) is a composition of the map \((x,y,z)\mapsto (x,y,z+t)\) and the 3-dimensional version of the map \(r\) from (2) then the computation is easier.)

Let \(U\subset \mathbb {R}^n\) be a connected open set with \(p,q\in U\). Show that there exists a diffeomorphism \(\phi \) of \(\mathbb {R}^n\) such that \(\phi (p)=q\) and \(\phi \) is the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

Let \(\Gamma \) be a group of diffeomorphisms acting on a smooth manifold \(M\). When is \(\Gamma \backslash M\) a manifold?

Let \(M\) be a smooth compact manifold and let \(X\) be a smooth vector field with a corresponding 1-parameter flow group \(\{\theta _t^X:M\to M\}_{t\in \mathbb {R}}\). Let \(f\) be a diffeomorphism of \(M\) and let \(f_*X\) be its pushforward. Show that the flow group for \(f_*X\) is \(f\circ \theta _t^X\circ f^{-1}\).

Let \(f\) be the homeomorphism of the annulus \(S^1\times [0,1]\) given by \[ f(z,s)=(ze^{2\pi is},s) \] where we view \(S^1\) as the set of unit norm complex numbers.

1. Construct an explicit isotopy (i.e. homotopy through homeomorphisms) between \(f\) and the identity that does not move the points of \(S^1\times \{0\}\).
2. Prove that there is no homotopy between \(f\) and the identity that does not move points on both \(S^1\times \{0\}\) and \(S^1\times \{1\}\).

1. Let \(M\) be a smooth manifold. What’s the definition of an orientation on \(M\)?
2. Suppose \(M\) is a smooth oriented manifold with boundary. What is the induced orientation on \(\partial M\)?
3. State Stokes’ Theorem.
4. Let \(M\) be a smooth manifold without boundary that is oriented by the form \(\omega \in \Omega ^{\dim (M)}(M)\). Suppose that \([0,1]=\{x\in \mathbb {R}\mid 0\le x\le 1\}\) is oriented by the form \(dx\), and that \(M\times [0,1]\) has the orientation defined by \(\omega \wedge dx\). Endow \(\partial (M\times [0,1])=(M\times \{0\})\cup (M\times \{1\})\) with the orientation induced by \(M\times [0,1]\) and let \(i_0:M\to M\times \{0\}\) and \(i_1:M\to M\times \{1\}\) be the obvious diffeomorphisms. Prove that \(i_0\) preserves orientation if and only if \(i_1\) does not.
5. Let \(f:M\to N\) and \(g:M\to N\) be smooth maps between smooth manifolds without boundaries. What is the definition of \(f\) and \(g\) being smoothly homotopic to each other?
6. Suppose \(M\) is a smooth, compact, oriented manifold without boundary. Suppose \(N\) is a smooth manifold and that \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic smooth maps. Prove that if \(\theta \in \Omega ^k(N)\) for some \(k>\dim (M)\), then \[ \int _M f^*\theta =\int _M g^*\theta . \]

Given a Lie group \(G\) with Lie algebra \(\mathfrak {g}\) and some \(v\in \mathfrak {g}\), let \(\{\theta _t^v\}_{t\in \mathbb {R}}\) be the corresponding 1-paramater group of diffeomorphisms of \(G\). Prove that \(\theta _t^v(gh)=g\theta _t^v(h)\) for any \(t\in \mathbb {R}\) and \(g,h\in G\).

Let \(f:S^2\to S^2\) be smooth and surjective. Prove that there is an open set \(U\subseteq S^2\) such that \(f:U\to f(U)\) is a diffeomorphism.

1. Let \(X\) be a vector field on a smooth manifold \(M\) and let \(f:M\to N\) be a diffeomorphism. Define \(f_*(X)\) – the pushforward of \(X\) on \(N\).
2. Let \(f:\mathbb {R}^3\to \mathbb {R}\times (-\frac {\pi }{2},\frac {\pi }{2})\times \mathbb {R}\) be defined by \(f(x,y,z)=(e^x,\tan ^{-1}(y),-2z)\). Calculate the bracket \[ \left [ f_*\left (\frac {\partial }{\partial x}+\frac {\partial }{\partial z}\right ), f_*\left (\frac {\partial }{\partial y}-\frac {\partial }{\partial z}\right ) \right ]. \]

Show that the group of diffeomorphisms of a compact manifold \(M\) acts transitively on \(M\). That is, if \(p,q\in M\) there exists a diffeomorphism \(\phi :M\to M\) such that \(\phi (p)=q\). Hint: First assume \(p\) and \(q\) are in the same coordinate neighborhood \(U\) and construct \(\phi \) as the flow of a vector field supported in \(U\).

Let \(f:X\to Y\) be a smooth map between smooth manifolds that is one-to-one on a compact submanifold \(Z\) of \(X\). Suppose that for all \(x\in Z\) \[ df_x:T_x(X)\to T_{f(x)}(Y) \] is an isomorphism. Prove that there is a neighborhood \(U\) of \(Z\) such that \[ f|_U:U\to f(U) \] is a diffeomorphism.

Prove that for any \(x,y\in \mathbb {R}^n\) there a compactly supported isotopy taking \(x\) to \(y\) (i.e. there is an isotopy \(f_t:\mathbb {R}^n\to \mathbb {R}^n\) such that \(f_0=\id \), \(f_1(x)=y\) and \(f_t=\id \) outside some ball).

Show that for any two points \(x,y\in \mathbb {R}^n\) there is a compactly supported isotopy \(\phi _t\) (i.e. it is identity outside a compact set) such that \(\phi _0=id\) and \(\phi _1(x)=y\).

1. State the tubular neighborhood theorem for a compact submanifold \(X\subset \mathbb {R}^n\).
2. Give an outline of the proof with details of the step that shows that the constructed map is a diffeomorphism.

Let \(f:X\to Y\) be a smooth map between smooth manifolds that is one-to-one on a compact submanifold \(Z\) of \(X\). Suppose that for all \(x\in Z\) \[ df_x:T_x(X)\to T_{f(x)}(Y) \] is an isomorphism. Prove that there is a neighborhood \(U\) of \(Z\) such that \[ f|_U:U\to f(U) \] is a diffeomorphism.

Let \(X\) be a closed manifold, \(\Delta =\{(x,x)\in X\times X\}\) the diagonal, and \[ N(\Delta ,X\times X)=\{((x,x),(v,-v))\mid x\in X,\ v\in T_x(X)\} \] the normal bundle of \(\Delta \). Show that there is a diffeomorphism of a neighborhood of \(X\) (viewed as the zero section) in the tangent bundle \(T(X)\) with a neighborhood of \(\Delta \) in \(X\times X\), extending the usual diffeomorphism \(X\to \Delta \), \(x\mapsto (x,x)\).

1. Show that \(\Phi :\mathbb {R}^2\to \mathbb {R}^2\) given by \[ (p,q)=\Phi (x,y)=(x+y^2+2y,x+y^2-2y) \] is a diffeomorphism. Compute \(d\Phi (\partial /\partial y)\) and the pullback \(\Phi ^*(p^2\,dp\wedge dq)\).
2. Let \(X\) be a smooth vector field and \(\omega \) a smooth \(n\)-form on the smooth compact boundaryless manifold \(M^n\). Let \(M\times \mathbb {R}\ni (x,t)\mapsto \phi _t(x)\in M\) be the flow generated by \(X\). Let \[ A(t)=\int _M\phi _t^*\omega \] be the integral of the pulled back \(n\)-form. Find \(dA/dt\).

Let \(f:M\to N\) and \(g:N\to M\) be smooth maps, where \(M\) and \(N\) are compact connected smooth oriented manifolds of the same dimension. If \(g\circ f\) is a diffeomorphism, either prove that both \(f\) and \(g\) are diffeomorphisms, or find a counterexample.

Let \(f\in C^\infty (S^n)\) be a smooth function and let \(x_1,x_2\in S^n\) be two points such that \(f(x_1)<0<f(x_2)\). Let \(\omega \) be a smooth nonvanishing \(n\)-form. Then there is a diffeomorphism \(\varphi :S^n\to S^n\) such that \[ \int _{S^n}(\varphi ^*f)\,\omega =0. \]

II. Ten point questions.