Pullbacks and Wedge Products

Let \(G\) be an \(n\)-dimensional Lie group. Show that the tangent bundle \(TG\) is isomorphic as a bundle to the product bundle \(G\times \mathbb {R}^n\).

Let \(G_0\) and \(G_1\) be Lie groups. Then the product \(G_0\times G_1\) is a Lie group and we let \(p_i:G_0\times G_1\to G_i\) be the projection to each coordinate. Let \(X\) be a vector field on \(G_0\times G_1\) and \(Y_0\) and \(Y_1\) vector fields on \(G_0\) and \(G_1\) such that \(X\) and the \(Y_i\) are \(p_i\)-related. Show that \(X\) if left-invariant if and only if both \(Y_{0}\) and \(Y_{1}\) are left-invariant.

In this problem you are allowed to use the fact that every closed 1-form on the 2-sphere \(S^2\) is exact and every 2-form \(\omega \) on \(S^2\) such that \(\int _{S^2}\omega =0\) is exact. You are also allowed to use that \(\int _{S^2} a^*(\omega )=-\int _{S^2}\omega \) for every 2-form \(\omega \) on \(S^2\), where \(a:S^2\to S^2\) is the antipodal map. Consider the map \(p:S^2\to \mathbb {R}P^2\) that identifies the antipodal points. Show that all closed 1- and 2-forms on \(\mathbb {R}P^2\) are exact by considering their pullbacks to \(S^{2}\) and applying the above facts.

Let \(\omega \) and \(\eta \) be two differential forms on a manifold \(M\) .

1. Define the wedge product \(\omega \wedge \eta \).
2. Prove that if \(\omega \) and \(\eta \) are both closed then so is \(\omega \wedge \eta \).
3. Prove that the de Rham cohomology class of \(\omega \wedge \eta \) is unchanged if \(\omega \) and \(\eta \) are replaced by different representatives of the same cohomology classes. You are allowed to use the identity relating exterior differentiation and wedge product, but you should state it carefully.

Let \(f:\mathbb {R}^3\to \mathbb {R}^2\) be the function \(f(x,y,z)=(xy,z)\) and let \(\omega =dx\wedge dy\) be the area form on \(\mathbb {R}^2\). Compute the pull-back \(f^*(\omega )\).

Construct a \(CW\)-complex whose fundamental group is the free product \(\mathbb {Z}*\mathbb {Z}_2\).

Recall that if \(X\) is a topological space its suspension \(SX\) is the product \(X\times [0,1]\) with both \(X\times \{0\}\) and \(X\times \{1\}\) collapsed to points. Show that \(\operatorname {H}_{n+1}(SX)=\operatorname {H}_n(X)\) for \(n\ge 0\).

Construct a \(\Delta \)-complex structure for \(\mathbb {R}\mathrm {P}^2\) and use it to calculate the homology, cohomology and cup product structure for \(\mathbb {R}\mathrm {P}^2\) with \(\mathbb {Z}_2\)-coefficients.

Let \(X\) be the wedge product of a circle and a projective plane. Find all connected three sheeted covers of \(X\).

Let \(A:\mathbb {R}^n\to \mathbb {R}^n\) be linear and let \(\phi \in \Lambda ^n(\mathbb {R}^n)\). Prove \(A^*\phi =\det (A)\phi \).

1. Define the DeRham cohomology groups \(H^k(M)\) for a smooth manifold \(M\).
2. Let \(N\) be a compact, smooth, \(k\)-dimensional manifold without boundary and \(f:N\to M\) a smooth map. Let \(\alpha \in \Omega ^k(M)\) be a closed \(k\)-form on \(M\) with \(\int _N f^*\alpha \ne 0\). Either show that \(\alpha \) represents a non-zero element of \(H^k(M)\) or find a counterexample.
3. The wedge product defines a map \(\Omega ^k(M)\times \Omega ^l(M):\Omega ^{k+l}(M)\) by \((\alpha ,\beta )\mapsto \alpha \wedge \beta \). Show that this descends to a map from \(H^k(M)\times H^l(M)\) to \(H^{k+l}(M)\).

Prove that there is no map \(h:\mathbb {R}\mathrm {P}^3\to \mathbb {R}\mathrm {P}^2\) that induces an isomorphism between fundamental groups. Hint: use cup products with \(\mathbb {Z}/2\mathbb {Z}\) coefficients. State all theorems you are using.

The cup product for the de Rham cohomology of a smooth manifold \(M\) is defined as \[ \cup :H^k(M)\times H^\ell (M)\to H^{k+\ell }(M) \] where \([\omega ]\cup [\theta ]=[\omega \wedge \theta ]\). Show that the cup product is well-defined.

Let \(f:\mathbb {R}^3\to \mathbb {R}^2\) be the function \(f(x,y,z)=(xy,z)\). Find the pullback form \(f^*(dy\wedge dz+x^2\,dx\wedge dy)\).

Let \(f:\mathbb {R}^3\times \mathbb {R}^3\to \mathbb {R}^2\) be defined by \(f(x,y)=(x\cdot y, |x|^2-|y|^2)\), where \(x\cdot y\) is the dot product of \(x\) and \(y\) in \(\mathbb {R}^3\) and \(|x|\) is the Euclidean norm of \(x\). Suppose that \((a,b)\ne (0,0)\). Show that \[ \{(x,y)\in \mathbb {R}^3\times \mathbb {R}^3:f(x,y)=(a,b)\} \] is a smooth submanifold, and compute it’s dimension.

Construct a \(\Delta \)-complex structure for the torus and use it to calculate the homology, cohomology and cup product structure for the torus.

Let \(f:\mathbb {R}^2\to \mathbb {R}^3\) be the function given by \(f(x,y)=(x^2-y,x+y^2,\sin (x))\). Let \(\omega \) be the 2-form on \(\mathbb {R}^3\) given by \(\omega (u,v,w)=u\,dv\wedge dw\). Compute the pull-back \(f^*\omega \).

Show that the product \(M\times N\) of two manifolds is orientable if and only if both \(M\) and \(N\) are orientable.

Give definition of a (differentiable) orientable manifold. Prove that product of two orientable manifolds is again orientable.

Compute the pull-back of the differential form \[ \omega =x\,dx\wedge dy+z\,dy\wedge dz \] in \(\mathbb {R}^3\) under the map \(f:\mathbb {R}^2\to \mathbb {R}^3\) given by \[ f(u,v)=(u^2,u+v,\sin (u+v)). \]

Recall that an \(n\times n\) real matrix \(M\) is orthogonal if \[ \langle Mv,Mw\rangle =\langle v,w\rangle \] for all \(v,w\in \mathbb {R}^n\) (and \(\langle \cdot ,\cdot \rangle \) is the inner product). Let \(O(n)\) denote the set of all orthogonal \(n\times n\) matrices, viewed as a subset of \(\mathbb {R}^{n^2}\). Show that \(O(n)\) is a manifold, and compute its dimension.

Compute the pullback \(f^*\omega \) of the differential form \[ \omega =dx\wedge dy \] by the map \(f:\mathbb {R}^3\to \mathbb {R}^3\), \(f(x,y,z)=(xyz,y+x,z)\).

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 \(F:\mathbb {R}^2\to \mathbb {R}^3\) be a mapping given by the formula \[ F(u,v)=(e^u\cos (v),e^u\sin (v),0). \] Compute the pull-back \(F^*(\omega )\) of the differential form \[ \omega =xy\,dx\wedge dy+x\,dy\wedge dz\in \Omega ^*(\mathbb {R}^3). \]

In the product \(S^1\times S^1\) consider the subset \[ A:=S^1\times \{p\}\cup \{p\}\times S^1 \] where \(p\) is a point on \(S^1\). Calculate \(H_j(A,\mathbb {Z})\), \(j=0,1,2\).