Orientability and Volume Forms

Let \(\omega =dx\wedge dy\) be the 2-form on \(\mathbb {R}^3\). Let \(h:\mathbb {R}^3\to \mathbb {R}\) be smooth so that 0 is a regular value. Let \(Z=h^{-1}(0)\). Assume that \(\frac {\partial h}{\partial z}\ne 0\) at all points of \(Z\). Show that the restriction of \(\omega \) to \(Z\) is a volume form on \(Z\).

For which values of \(n\) is the manifold \(\mathbb {R}P^n\) orientable? Prove that your answer is correct.

Let \(\omega \) be a closed \(n\)-form on a smooth manifold \(M\) . Let \(N\) be a compact, orientable \(n\)-manifold without boundary and \(f:N\to M\) a smooth map with

Show that deRham cohomology group \(H^n_{dR}(M)\) is not trivial.

Let \(M\) be a smooth manifold and \(TM\) its tangent bundle. Show that \(TM\) is orientable.

Let \(U\subset \mathbb {R}^n\) be open and \(f:U\to \mathbb {R}^n\) a smooth immersion. Then for each \(x\in U\) we have that \(T_xU\) is canonically isomorphic to \(\mathbb {R}^n\) so

we can define a smooth map \(F:U\times \mathbb {R}^n\to \mathbb {R}^n\times \mathbb {R}^n\) by \[ F(x,v)=(f(x),f_*(x)(v)). \] Show that the derivative \(F_*(x,v):T_{(x,v)}(U\times \mathbb {R}^n)\to T_{F(x,v)}(\mathbb {R}^n\times \mathbb {R}^n)\) has positive determinant as a linear map from \(\mathbb {R}^n\times \mathbb {R}^n= \mathbb {R}^{2n}\) to itself. Use this to show that for any smooth manifold \(M\) its tangent bundle \(TM\) is orientable.

1. Construct an explicit \(n\)-form \(\omega \) on \(\mathbb {R}^n\) with compact support such that \(\omega \) is closed but there does not exists an \((n-1)\)-form \(\eta \) with compact support and \(d\eta =\omega \). You can use the existence of bump functions in your construction but you need to prove that your construction works.
2. Let \(M\) be a smooth, compact orientable \(n\)-manifold without bound- ary and let \((U,\phi )\) be a chart for \(M\). Show that there exists a closed \(n\)-form on \(M\) with support in \(U\) that is not exact.

Let \(M\) and \(N\) be a smooth orientable manifolds. Show that the maps \(i_a:M\to \mathbb {R}\times M\) given by \(i_a(x)=(a,x)\) and \(\pi :\mathbb {R}\times M\to M\) given by \(\pi (t,x)=x\) induce inverse maps in de Rham Cohomology. Conclude that if smooth maps \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic, then they induce equal maps in de Rham Cohomology.

Suppose \(M\) is a closed connected orientable 5-manifold such that \(H_1(M)=\mathbb {Z}/2\mathbb {Z}\) and \(H_2(M)=\mathbb {Z}\oplus \mathbb {Z}/3\mathbb {Z}\). Compute \(H_i(M)\) and \(H^i(M)\) for all \(i\).

Prove that \(\mathbb {R}\mathrm {P}^2\) is not orientable.

Let \(M\) be a smooth compact manifold without boundary. Prove that if \(M\) has a smooth nowhere vanishing closed one form, then \(H^1_{\operatorname {dR}}(M)\ne \{0\}\). Deduce that if there is a smooth submersion of \(M\) onto \(S^1\), then \(H^1_{\operatorname {dR}}(M)\ne \{0\}\).

Prove that \(S^1\vee S^3\) is not homotopy equivalent to a compact, orientable 3-manifold without boundary.

Let \(M^m\) be a smooth compact orientable manifold and \(\omega \in \Omega ^m(M)\) a smooth, nonvanishing \(m\)-form that determines the orientation of \(M\). Show that \(\int _M\omega >0\).

1. Let \(M\) be a closed, connected, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_2. \] Calculate the remaining homology and cohomology groups with \(\mathbb {Z}\) coefficients.
2. Show that the Euler characteristic of any closed 3-manifold is zero.
3. Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that \(H_1(M;\mathbb {Z})\) is infinite.

Let \(a:S^2\to S^2\) be the antipodal map \(a(x)=-x\) and let \[ p:S^2\to \mathbb {R}\mathrm {P}^2=S^2/x\sim -x,\qquad p(x)=[x] \] be the projection to the projective plane. Let \(\omega \) be a two form on \(\mathbb {R}\mathrm {P}^2\).

1. Prove that \(\int _{S^2}p^*\omega =0\). [Hint: \(p^*\omega \) is \(a\)-invariant.]
2. Prove that there is a one form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(d\eta =\omega \). You are allowed to use the fact that if \(\beta \) is a two-form on \(S^2\) and \(\int _{S^2}\beta =0\) then \(\beta \) is exact, but you are not allowed to use the de Rham Theorem.
3. Prove that \(\mathbb {R}\mathrm {P}^2\) is not orientable.

Let \(X\) be a connected compact orientable manifold of dimension \(n\). Explain carefully how Poincaré duality implies the existence of a bilinear pairing \[ H^k(X;\mathbb {Z})\times H^{n-k}(X;\mathbb {Z})\to \mathbb {Z} \] that modulo torsion becomes a perfect (i.e., non-singular) pairing. Deduce from this the structure of the cohomology ring \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\).

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 \(\sigma (g\circ f)=\sigma (g)\sigma (f)\).

1. State the van Kampen theorem for fundamental groups. Use it to compute, \(\pi _1(K,*)\), where \(K\) is the Klein bottle, and \(\pi _1(S_g,*)\), where \(S_g\) is the orientable surface of genus \(g\). (Here \(*\) is any base-point.)
2. Show that \(S_g\) is indeed orientable; carefully state the results you are using.

Prove that \(S^n\) is orientable.

Let \(\Sigma _2\) be a closed, orientable surface of genus 2. Prove that \(\pi _1(\Sigma _2)\) has a normal subgroup \(G\) such that \(\pi _1(\Sigma _2)/G\cong \mathbb {Z}\).

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\ge 1\). Prove that \[ \pi _1(\Sigma _g)= \left \langle a_1,b_1,a_2,b_2,\ldots ,a_g,b_g \,\middle |\, \prod _{i=1}^{g}[a_i,b_i] \right \rangle . \]

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 \(\Sigma _g\) be a closed, orientable surface of genus \(g\), and let \(T\Sigma _g\) be its tangent bundle. Prove that \(T\Sigma _g\) is trivial if and only if \(g=1\).

Let \(G\) be a Lie group. Prove that \(G\) is orientable, and that if \(L_g\in \operatorname {Diff}(G)\) is left multiplication by \(g\in G\), then \(L_g\) preserves orientation.

Let \(\Sigma _g\) be a closed, orientable surface of genus \(g\). Given \(p\ge 1\) distinct points \(\{x_1,\ldots ,x_p\}\subseteq \Sigma _g\), let \(\Sigma _{g,p}=\Sigma _g-\{x_1,\ldots ,x_p\}\). Find the cohomology of \(\Sigma _{g,p}\) with integer coefficients, including its cup product structure.

Let \(M\) and \(N\) be smooth, compact, oriented manifolds without boundary, both of dimension \(n\). Suppose that \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic. Prove that for any \(\omega \in \Omega ^n(N)\), we have \[ \int _M f^*\omega =\int _M g^*\omega . \]

Let \(M\) be a closed, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_3. \] Calculate the remaining homology and cohomology groups for \(M\) with \(\mathbb {Z}\) coefficients.

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph that is a retract of \(\Sigma \). Prove that: \[ \operatorname {rank}(H_1(X))\le g. \]

Let \(M\) be a closed, orientable manifold of dimension \(2k\) and assume that \(H_{k-1}(M,\mathbb {Z})\) is torsion-free. Show that \(H_k(M,\mathbb {Z})\) is torsion free.

Describe the cohomology ring \(H^*(\Sigma _g,\mathbb {Z})\) (i.e. the groups together with the cup product) of a compact oriented surface \(\Sigma _g\) of genus \(g\).

1. Prove that if \(g<h\), then every map \(f:\Sigma _g\to \Sigma _h\) has degree zero.
2. Show that for any \(g\ge h\), there is a map \(f:\Sigma _g\to \Sigma _h\) of degree one.

Find all cohomology and homology groups (with \(\mathbb {Z}\) coefficients) of a closed connected orientable 3-manifold \(M\) with \(\pi _1(M,m_0)=\mathbb {Z}^r\times G\) where \(G\) is a finite group.

Let \(M\) be a smooth manifold. Show that its tangent bundle \(TM\) is orientable.

Show that \(\pi _1(S^1)=\mathbb {Z}\); that \(\pi _1(S^n)=1\) if \(n\ge 2\); that \(\pi _1(\Sigma _1-\{x\})=F_2\) (where \(\Sigma _1-\{x\}\) is a closed orientable genus 1 surface with a single point removed and \(F_2\) is a free group on two generators); and find \(\pi _1(\Sigma _2)\) (where \(\Sigma _2\) is a closed orientable surface of genus 2).

Prove that \(S^1\vee S^3\) is not homotopy equivalent to a compact, orientable, 3-manifold without boundary.

Give two definitions of a differentiable manifold being orientable; one via charts and the other via forms. Show that the two definitions are equivalent.

Let \(M\) be a smooth manifold. Show that the tangent bundle \(TM\) is orientable.

Let \(S_g\) denote the closed oriented surface of genus \(g\). Show that if \(h<g\) then every map \(S_h\to S_g\) has degree 0. Does every such map have to be null-homotopic?

For \((x,y)\in \mathbb {R}^2-\{(0,0)\}\), let \[ \omega (x,y)=\frac {-y}{x^2+y^2}\,dx+\frac {x}{x^2+y^2}\,dy. \] Find \(\int _{C(r)}\omega \) where \(C(r)\) is the circle of vectors in \(\mathbb {R}^2\) of norm equal to \(r>0\) oriented counterclockwise. Find a smooth function \(f:\mathbb {R}^2-\{(0,0)\}\to \mathbb {R}\) such that \(\omega =df\) or explain why there is no such \(f\).

Let \(M\) be a closed connected orientable 3-manifold with \(H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}\). Compute \(H_i(M;\mathbb {Z})\) and \(H^i(M;\mathbb {Z})\) for all \(i\).

Let \(S_g\) be the closed connected oriented surface of genus \(g\).

1. Prove that for \(g<h\) every map \(S_g\to S_h\) has degree 0.
2. Prove or disprove: for \(g<h\) every map \(S_g\to S_h\) is nullhomotopic?

Consider the disk with 2 holes \[ P=\{(x,y)\in \mathbb {R}^2\mid \lVert (x,y)\rVert \le 4,\, \lVert (x,y)-(2,0)\rVert \ge 1,\, \lVert (x,y)+(2,0)\rVert \ge 1\}. \] Orient each boundary component counterclockwise. Let \(X\) be the space obtained from \(P\) by identifying all boundary components via orientation preserving homeomorphisms. Find a presentation of \(\pi _1(X)\).

Prove that when \(g<h\) every map \(S_g\to S_h\) between orientable closed surfaces of genus \(g\) and \(h\) respectively is equal to 0.

Prove that a closed, orientable, genus 2 surface does not admit a 1-dimensional foliation.

Suppose that \(a\) is a smooth, real valued, compactly supported function on \(\mathbb {R}^n\) so that \(\theta =a\,dx_1\wedge \cdots \wedge dx_n\) is an \(n\)-form on \(\mathbb {R}^n\). What is the definition of \(\int _{\mathbb {R}^n}\theta \)? What’s the definition of \(\int _M\omega \) where \(\omega \) is a compactly supported \(k\)-form on an oriented \(k\)-dimensional smooth manifold \(M\)? Show that this definition is well-defined.

Let \(f:M\to N\) be a smooth map between smooth manifolds, and let \(y\) be a regular value of this map. If \(M\) and \(N\) are compact, oriented, and have the same dimension, then what is the definition of \(\deg (f;y)\), the degree of \(f\) measured at the value \(y\)? We have seen that the degree is independent of the regular value chosen, and an ingredient of that proof is the following lemma:

Suppose \(Q\) is a compact smooth manifold and that \(\partial Q=M\). If \(F:Q\to N\) is smooth, and \(z\) is a regular value of \(F|_{\partial Q}\), then \(\deg (F|_{\partial Q};z)=0\).

Prove the lemma.

Let \(M\) be a closed orientable connected 4-manifold with \(H^1(M)=H^3(M)=0\) and \(H^2(M)=H^4(M)=\mathbb {Z}\). What are the possible cup product structures on \(H^*(M)\)? State all theorems you are using. All cohomology groups are with integral coefficients.

Let \(M\) be a connected topological \(n\)-manifold. Define what it means for \(M\) to be orientable. Then state the Poincaré duality theorem (including a careful definition of the map that is claimed to be an isomorphism).

Is there a covering map \(p:X\to Y\) where

1. \(X=S_3\), \(Y=S_2\) where \(S_g\) is a closed orientable surface of genus \(g\)?
2. \(X\) is \(S_3\) with a small open disk removed and \(Y\) is \(S_2\) with a small open disk removed.

When \(p\) exists, what is the number of sheets?

Let \(M\) and \(N\) be manifolds. Show that \(M\times N\) is orientable if and only if \(M\) and \(N\) are orientable.

1. Let \(M\) be a closed, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_2. \] Calculate the remaining homology and cohomology groups for \(M\) with \(\mathbb {Z}\) coefficients.
2. Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that \(H_1(M;\mathbb {Z})\) is infinite.

Let \(M\) and \(N\) be compact orientable \(n\)-dimensional manifolds and \(f:M\to N\) a smooth map. Suppose that \[ \int _M f^*\Omega \ne 0 \] for some \(n\)-form \(\Omega \). Prove that \(f\) is surjective.

Let \(M\) be a compact oriented Riemannian manifold with boundary. Suppose that \(u\) and \(v\) are harmonic functions that are equal on \(\partial M\). Show that \(u=v\) on \(M\). Recall that \(u\) is harmonic on \(M\) if \(\Delta u=\operatorname {div}(\nabla u)=0\) on \(M\).

Let \(M\) be a closed, orientable manifold of dimension \(2k\) and assume that \(H_{k-1}(M)\) is torsion free. Show that \(H_k(M)\) is torsion free.

Let \(M\) be a compact orientable manifold with boundary. Use differential forms to show that there is no smooth retraction \(r:M\to \partial M\).

Let \(M\) be an orientable manifold of dimension \(n\). Prove that the top dimensional de Rham cohomology with compact supports \(H_c^n(m)\) is nontrivial.

Let \(S_g\) be the closed, orientable surface of genus \(g\). Calculate \(\pi _2(S_g)\) for all \(g\).

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph which is a retract of \(\Sigma \). Prove that \[ \operatorname {rank} H_1(X)\le g \]

Let \(\Sigma _g\) denote the closed oriented surface of genus \(g\). Show that every map \(f:\Sigma _2\to \Sigma _3\) has degree 0.

Give a proof that \(\mathbb {R}P^2\) is nonorientable.

Let \(f:M\to N\) be a map of degree 1 between two smooth, closed, connected, oriented \(n\)-manifolds. Prove that \(f_*:\pi _1(M)\to \pi _1(N)\) is surjective.

Show that a \(p\)-sheeted covering map \(M\to N\) between closed connected oriented smooth manifolds has degree \(\pm p\).

Show that if a closed orientable manifold \(M\) of dimension \(2k\) has \(H_{k-1}(M;\mathbb {Z})\) torsionfree, then \(H_k(M;\mathbb {Z})\) is also torsionfree.

Show that if the closed orientable surface \(M_g\) of genus \(g\) retracts onto a graph \(X\subset M_g\), then \(H_1(X)\) has rank at most \(g\). You may use the following algebraic fact: a nonsingular skew-symmetric bilinear pairing over the rationals \(\mathbb {Q}\), of the form \(\mathbb {Q}^n\times \mathbb {Q}^n\to \mathbb {Q}\), cannot be identically 0 when restricted to \(V\times V\) for any \(\mathbb {Q}\)-linear subspace \(V\subset \mathbb {Q}^n\) of dimension \(>n/2\).

Let \(M_0\) and \(M_1\) be orientable closed \(n\)-manifolds and \(f:M_0\to M_1\) a map such that \(f^*:H^n(M_1;\mathbb {R})\to H^n(M_0;\mathbb {R})\) is an isomorphism.

1. Assume that \(H^i(M_0;\mathbb {R})\cong H^i(M_1;\mathbb {R})\) for all \(i\). Show that \(f^*\) is an isomorphism on cohomology with \(\mathbb {R}\) coefficients in all dimensions.
2. Assume that there exists a map \(g:M_1\to M_0\) such that \(g^*:H^n(M_0;\mathbb {R})\to H^n(M_1;\mathbb {R})\) is an isomorphism. Show that \(f^*\) and \(g^*\) are isomorphisms on cohomology with \(\mathbb {R}\) coefficients in all dimensions.

Is there a smooth embedding of \(\mathbb {R}P^2\) into an orientable 3-manifold? Provide a proof or an example.

Prove that the tangent bundle \(TM\) of a smooth manifold \(M\), viewed as a smooth manifold, is orientable.

Let \(f:M\to N\) be a map between two closed connected oriented manifolds. If \(f\) has degree 1, show that \(f_*:\pi _1(M)\to \pi _1(N)\) is an epimorphism.

Let \(X\) and \(Y\) be two closed oriented manifolds. Prove that Euler characteristic is multiplicative, i.e. that \[ \chi (X\times Y)=\chi (X)\times \chi (Y). \] You may not appeal to the Kunneth formula.

Let \(M\) be a closed, orientable manifold \(M\) of dimension \(2k\). If \(H_{k-1}(M)\) is torsion free show that \(H_k(M)\) is torsion free.

Let \(S_g\) be the closed, orientable surface of genus \(g\). Show that there exists a map \(f:S_g\to S_h\) with \[ f_*:H_2(S_g)\to H_2(S_h) \] non-trivial if and only if \(g\ge h\).

1. Let \(S_g\) be the closed, orientable surface of genus \(g\). Calculate \(\pi _2(S_g)\) for all \(g\).
2. Show that \(\pi _3(S_0)\) is non-trivial. (\(S_0\) is the 2-sphere.)

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

1. Let \(M\) be a manifold with boundary. Show that if \(M\) is orientable then \(\partial M\) is orientable.
2. Let \(M\) be an \(n\)-dimensional manifold and \(N\subset M\) an \(n-1\) dimensional submanifold such that \(N\) has a neighborhood \(U\subset M\) with \(U\) homeomorphic to \(N\times (0,1)\). Show that if \(M\) is orientable then \(N\) is orientable.

Let \(S\) be a closed, orientable surface and \(\gamma \) a simple closed curve on \(S\). Show that \(S\) retracts onto \(\gamma \) if and only if \(\gamma \) represents a non-trivial element in homology.

1. Let \(M\) be a closed, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}. \] Calculate the remaining homology and cohomology groups for \(M\) with \(\mathbb {Z}\) coefficients.
2. Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that \(H_1(M;\mathbb {Z})\) is infinite.

Give a proof of the Boundary Theorem: If a smooth map \(f:X\to Y\) between two closed oriented \(n\)-manifolds extends to a smooth map \(F:W\to Y\) of a compact oriented \((n+1)\)-manifold \(W\) with \(\partial W=X\) then \(\deg f=0\).

Let \(\Sigma \) be an oriented genus 2 surface smoothly embedded in \(\mathbb {R}^3\).

1. Describe the Gauss map \(g:\Sigma \to S^2\) and define the Gaussian curvature at a point of \(\Sigma \) in terms of \(g\).
2. State the Gauss-Bonnet theorem for this situation.
3. Prove that \(\Sigma \) has (many) saddle points, i.e. points of negative curvature.

Let \(\Sigma \) be a closed oriented surface of genus \(g\) and suppose that \(X\subset \Sigma \) is a graph which is a retract of \(\Sigma \). Prove that \[ \operatorname {rank}H_1(X)\le g. \]

Using methods of differential topology prove that for each compact orientable differentiable manifold (without boundary) \(M\) of dimension \(m\), \(H^m(M,\mathbb {R})\ne 0\), where \(H^m(M,\mathbb {R})\) is the de Rham cohomology group.

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

Prove that \(M\#N\) is orientable if and only if both \(M\) and \(N\) are orientable.

1. Give a definition of the tangent bundle \(TM\) of a manifold \(M\).
2. Give a definition of what it means for a manifold to be orientable.
3. Show that \(TM\), viewed as a differentiable manifold, is always orientable.

Let \(\Sigma \) be a smooth, compact, orientable surface (2 dimensional manifold) of genus \(g\). Prove that \(\Sigma \) admits a smooth non-vanishing vectorfield if and only if \(g=1\).

Let \(M\) be a smooth, orientable \(n\)-dimensional manifold. Prove that the \(n\) dimensional DeRham cohomology with compact supports, \(H_c^n(M)\) is nonzero.

A. Let \(F_g\) be a closed orientable surface of genus \(g\). Prove that \(\pi _1(F_g)\) is infinite, and non-abelian, provided \(g>1\).

B. Let \(G\) be the one point union of two circles (a figure eight). Draw a picture of an explicit irregular covering map \(\widetilde {G}\to G\).

True or false (give a short explanation or a counterexample): Let \(\Sigma \) be a smooth, closed, oriented surface. Assume that \(\Sigma \) has a Riemannian metric of constant curvature \(K\). Suppose that \(\gamma \subset \Sigma \) is a closed embedded geodesic loop which separates \(\Sigma \). (That is, \(\Sigma -\gamma \) has two components.)

1. Then \(K\le 0\).
2. Then \(K\ne 0\).

Let \(M^n\) be a smooth oriented manifold. Suppose \(f:M\to \mathbb {R}\) is a smooth function such that \(df\ne 0\) at each point of \(M\). Show that there is a smooth vector field \(V\) on \(M\) so that \(df(V)\le -1\). Show also, that if \(\phi _t\) is the flow generated by \(V\), then \(f(\phi _1(x))\le f(x)-1\) for all \(x\in M\) such that \(\phi _1(x)\) exists.

Find the Euler characteristic \(\chi (\mathbb {R}P^3\#(S^1\times S^2))\) of the connected sum.

In #14 and #15 by \(\Sigma _g\) we denote a closed oriented surface of genus \(g\).

Let \(M\) and \(N\) be two connected smooth manifolds. Show that \(M\times N\) is orientable if and only if both \(M\) and \(N\) are orientable.

Let \(X\) be the set of points \((x,y,z)\in \mathbb {R}^3\) satisfying the equation \[ (1+z)x^2-(1-z)y^2=2z(1-z^2). \] Show that \(X\) is a smooth oriented surface embedded in \(\mathbb {R}^3\).

Let \(X\) be the space obtained from a disk by identifying the boundary according to the following diagram. Find the fundamental group \(\pi _1(X)\). Is \(X\) homeomorphic to a closed surface? Explain.

Diagram description: a hexagonal disk with paired boundary labels. Reading clockwise around the boundary, the labeled sides are \(a^{-1},c,b,a,c^{-1},b^{-1}\); paired sides have matching labels with inverse orientations.

True or false: There is no orientable smooth 3-dimensional manifold \(M\) and a smooth embedding \(\mathbb {R}P^2\to M\). Justify your answer.

Let \(X\) be the topological space described on the Figure below: it is obtained from the 2-dimensional square \(I^2\) by identifying segments with arrows (the identification respects the orientation of edges determined by the arrows). Compute \(\pi _1(X)\).

Diagram description: the square \(I^2\) has arrows on the left and right vertical sides pointing upward, on the top and bottom horizontal sides pointing to the right, and on the diagonal from the lower-left corner to the upper-right corner pointing toward the upper-right corner. The arrowed segments are identified respecting these orientations.

Let \(S^4\) be the 4-dimensional sphere and \(F\subset S^4\) be a smoothly embedded surface of genus \(g\) (this surface is orientable, compact and has no boundary). Compute the homology groups \[ H_k(S^4-F,\mathbb {R}) \] for \(k=2,3\). (You may assume that \(F\) is contained in an open subset \(U\) of \(\mathbb {R}^4\), which is homeomorphic to \(F\times \mathbb {R}^2\) so that \(F\subset U\) corresponds to \(F\times \{0\}\).)

a. Let \(P:M\to N\) be a regular covering map between connected differentiable manifolds \(M,N\) and let \(M\) be orientable. Prove that \(N\) is orientable if and only if every deck transformation (covering space automorphism) preserves the orientation of \(M\).

b. Prove that \(\mathbb {R}P^n\) is orientable if and only if \(n\) is odd.

Let \((M^n,g)\) be a Riemannian manifold. Show that the form given locally by \[ \omega =\sqrt {g}\,dx^1\wedge dx^2\wedge \cdots \wedge dx^n \] represents a globally-defined, nowhere vanishing \(n\)-form. (Here \(\sqrt {g}\) is the standard shorthand for the square root of the determinant of the matrix \([g_{ij}]=[\langle \frac {\partial }{\partial x^i},\frac {\partial }{\partial x^j}\rangle ]\) which gives the metric \(g\) in local coordinates.)

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.

Suppose a smooth oriented manifold \(M^n\) has a Riemannian metric which is given in any local coordinate system by \[ ds^2=\sum _{i,j=1}^{n}g_{ij}(x)\,dx^i\otimes dx^j. \] Then the form given locally in the same coordinate systems by \[ \omega =\det (g_{ij}(x))\,dx^1\cdots dx^n \] represents a globally defined nowhere vanishing \(n\)-form on \(M\).

Let \(M^2\) be a smooth closed connected orientable surface with Gauss curvature \(K(x)<0\) and genus \(g\). Let \(\{\gamma _1,\ldots ,\gamma _\ell \}\) be a system of pairwise nonintersecting closed geodesics of \(M\). (A closed geodesic is a smoothly embedded loop \(\gamma _i:S^1\to M\) whose geodesic curvature vanishes identically \(\kappa _i=0\).) Show that \(\ell \le 3g-3\). (Hint: can two nonintersecting closed geodesics be freely homotopic?)