Surface Topology and Surface Groups

Carefully state Van Kampen’s theorem. The version for two subsets suffices. Use Van Kampen’s theorem to compute \(\pi _1(T^2 \vee T^2)\) where \(T^2\) is a torus.

1. Show that \(\mathbb {R}P^2\) has a 2-fold cover which is \(S^2\).
2. Show that the Klein bottle has a 2-fold cover which is a torus.

Let \(\Sigma _g\) be the closed, orientable, genus-\(g\) surface. Show that \(\pi _1(\Sigma _2)\) contains \(\pi _1(\Sigma _4)\) as a normal subgroup.

Prove or disprove:

1. For every \(d \in \mathbb {Z}\) there is a degree \(d\) map \(f_d : T^2 \to S^2\).

2. For every \(d \in \mathbb {Z}\) there is a degree \(d\) map \(g_d : S^2 \to T^2\).

   Here \(S^2\) is the 2-sphere and \(T^2 = S^1 \times S^1\) is the 2-dimensional torus.

Let \(\Sigma _2\) be the closed, orientable genus two surface. Show that \(\pi _1(\Sigma _2,x)\) has a normal subgroup \(G\) with the quotient group \(\pi _1(\Sigma _2)/G\) isomor- phic to \(\mathbb {Z}\). You can use a picture as part of your proof.

Let \(X\) be the compact surface obtained from the disk by deleting two disjoint open disks, as in the figure below. Fix orientations on two boundary components as in the figure. Identify the boundaries using each of the two possible orientations on the third component. Show that the two quotient spaces so obtained have non-isomorphic fundamental groups. Hint: abelianize.

Figure: a disk with two disjoint open disks removed; the two inner boundary components have the indicated orientations.

Let \(T^2\) be the real 2-dimensional torus and \(f:S^1\vee S^1\to T^2\) a continuous map. Prove or disprove the following assertions:

1. There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(f\circ g:T^2\to T^2\) is homotopy to the identity map.

2. There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(g\circ f\) is homotopy to the identity map.

   Justify your answers.

Recall that the mapping torus of a map \(f:X\to X\) is the quotient space \(T_f\) obtained from \(X\times [0,1]\) by identifying \((x,1)\) with \((f(x),0)\) for every \(x\in X\). Compute the homology groups of \(T_f\) when \(f:S^n\to S^n\) is a map of degree \(d\) and \(n > 1\).

Let \(S_g\) denote an orientable surface of genus \(g\). So for example \(S_0\) is the 2-sphere and \(S_{1}\) is the torus.

1. If \(g>0\) show that every map \(S_0\to S_g\) is null-homotopic.
2. If \(g>h>0\) show that every map \(S_h\to S_g\) has degree 0, but show by example that there exist maps that are not null-homotopic.

Suppose that \(M\) and \(N\) are smooth surfaces in \(\mathbb {R}^3\) that intersect at a point \(p\) and that do not have the same tangent plane at that point. Show that \(p\) is not an isolated point of \(M\cap N\).

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

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.

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 \(\Sigma \) be a closed, smooth surface. Show that if \(\Sigma \) admits a nonvanishing vector field, then the Euler characteristic of \(\Sigma \) equals 0.

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

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 \(\Sigma _g\) 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.

Give an example of an irregular (i.e. not normal) covering space of the Klein bottle (with a proof).

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 \(a:S^2\to S^2\) be the antipodal map on the 2-sphere: \(a(x)=-x\), 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 2-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 1-form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(\omega =d\eta \). You are allowed to use the fact that if \(\zeta \) is a 2-form on \(S^2\) and \(\int _{S^2}\zeta =0\) then \(\zeta \) is exact, but you are not allowed to use de Rham’s theorem.

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.

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).

Let \(M,N\) be two closed connected oriented \(n\)-manifolds and \(f:M\to N\) a degree 1 map. (Recall that this means that \(f_*[M]=[N]\) where \([M],[N]\) are the fundamental classes of \(M,N\) respectively.)

1. Show that \(f_*:H_k(M)\to H_k(N)\) is onto for every \(k\). Hint: \(f_*(\alpha \cap f^*(z))=f_*(\alpha )\cap z\). Explain.
2. Deduce that if \(M=S_g\), \(N=S_h\) are surfaces of genus \(g\), \(h\) respectively, then \(g\ge h\).

Let \(X\) be a connected surface with nonempty boundary \(\partial X\) and \(p\in X\). Show that inclusion \[ X\backslash \{p\}\hookrightarrow X \] induces an isomorphism in \(H_1\) if and only if \(p\in \partial X\). You may use the fact that \(H_2(X)=0\) without proof.

Let \(X\) be a path connected, locally path connected, and semi-locally simply connected space. (Recall that \(X\) is semi-locally simply connected if every point in \(X\) has a neighborhood \(U\) such that every closed loop in \(U\) can be contracted to a point in \(X\).) Outline the construction of the universal cover \(p:\widetilde {X}\to X\). Specifically,

1. Define \(\widetilde {X}\) as a set and the projection \(p:\widetilde {X}\to X\),
2. Define the topology on \(\widetilde {X}\) (you don’t have to check that this is a topology),
3. Give an argument that \(\widetilde {X}\) is simply connected.

Indicate where each of the three assumptions on \(X\) is used.

Show that the degree of every map \(\mathbb {C}P^2\to \mathbb {C}P^2\) of the complex projective plane to itself is nonnegative.

Construct an explicit irregular (i.e. not normal) covering space of the Klein bottle.

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 \(S_g\) 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?

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?

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.

1. What is a subalgebra of a Lie algebra?
2. Let \(G\) be a Lie group, with Lie algebra \(\mathfrak {g}\). Prove that any Lie subgroup \(H\) of \(G\) corresponds to a subalgebra of \(\mathfrak {g}\).
3. What is a smooth plane field on \(G\)?
4. Give an example of a foliation on a 2-torus that has compact leaves and noncompact leaves.
5. State Frobenius’ theorem.

6. Prove that any subalgebra of \(\mathfrak {g}\) corresponds to a foliation of \(G\).

7. Given a subalgebra of \(\mathfrak {g}\), and the corresponding foliation of \(G\), prove that the leaf containing the identity is a group, and thus that there is a correspondence between connected Lie subgroups of \(G\) and subalgebras of \(\mathfrak {g}\).

Let \(f:S^1\to S^1\) be given by \(f(z)=z^2\), where we regard \(S^1\) as the unit circle in \(\mathbb {C}\). Find a presentation of the fundamental group of the mapping torus \[ M_f=S^1\times I/(z,1)\sim (f(z),0) \] of \(f\).

Let \(X\) denote the solid torus \(S^1\times D^2\). Using Mayer-Vietoris, compute the homology groups of the double of \(X\), obtained by taking two copies of \(X\) and gluing them along the boundary via the identity map. Also compute \(H_i(X,\partial X)\).

Prove that there are uncountably many 3-dimensional foliations on a 5-dimensional torus.

1. Define the concepts of chain morphisms and chain homotopies.
2. Show that every map \(S^2\to T^2\) from the 2-sphere to the 2-torus is null-homotopic.
3. Write down explicit cellular chain complexes \(C(S^2)\) and \(C(T^2)\) for \(S^2\) and \(T^2\) and a chain morphism \(\Phi :C(S^2)\to C(T^2)\) which is not chain homotopic to a chain morphism representing a constant map.

Prove that a closed orientable surface \(S_g\) of genus \(g\ge 1\) is not homotopy equivalent to the wedge \(X\vee Y\) of two finite cell complexes both of which have nontrivial \(H_1(\cdot ;\mathbb {Z})\).

Let \(V\) and \(W\) be vector fields on \(\mathbb {R}^3\) given by \[ V(x,y,z)=\frac {\partial }{\partial x} -y\frac {\partial }{\partial z} \quad \text {and}\quad W(x,y,z)=\frac {\partial }{\partial y} +x^2\frac {\partial }{\partial z}. \]

1. Find \([V,W](x,y,z)\).
2. What is the definition of an integrable, \(k\)-dimensional plane field on a manifold \(M\)?
3. Prove or disprove: \(\Delta (x,y,z)=\operatorname {Span}\{V(x,y,z),W(x,y,z)\}\) is an integrable plane field.
4. Give an example of three foliations on a two dimensional torus: one with every leaf compact; one with every leaf noncompact; and one with some compact and some noncompact leaves.

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 \(S^2\) be the 2-sphere and \(T\) the torus.

1. Show that every map from \(S^2\) to \(T\) is null-homotopic.
2. Show that there are infinitely many homotopy classes of maps from \(T\) to \(S^2\).

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

Let \(T=S^1\times S^1\subset \mathbb {C}\times \mathbb {C}\) be the torus. Let \(X\) be the vector field on \(T\) defined by \(X(z_1,z_2)=(iz_1,aiz_2)\), where \(a\in \mathbb {R}\). Compute the flow \(\Phi (t,p)\), \(p=(p_1,p_2)\) of \(X\). For \(a\) rational, show that \(t\to \Phi (t,p)\) is periodic.

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

Let \(M\) be an open 3-manifold such that every compact set of \(M\) is contained in a solid torus.

1. Show that \(\pi _1(M)\) is abelian.
2. Assume that \(\pi _1(M)\) is finitely generated. Show that \(\pi _1(M)\) is trivial or \(\mathbb {Z}\).

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.

Suppose \(X\) is a smooth submanifold of \(\mathbb {R}^3\) and a nonempty closed surface. Prove that there is a plane \(P\subset \mathbb {R}^3\) such that \(X\cap P\) is a nonempty collection of circles.

Give an example of an irregular (i.e. not normal) covering space of the Klein bottle (with a proof).

Compute de Rham cohomology of the 2-torus. You may use the Poincare lemma and the Mayer-Vietoris sequence. For example, you may want to compute \(H^*(S^1)\) first.

Consider the vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad \text {and}\qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

Define a \(\Delta \)-complex structure on the Klein bottle and compute its homology with \(\mathbb {Z}\) and \(\mathbb {Z}/2\) coefficients.

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point.

Define a \(\Delta \)-complex structure on the Klein bottle \(K\) and use it to compute \(H_*(K;\mathbb {Z})\), \(H_*(K;\mathbb {Z}_2)\), and \(H_*(K;\mathbb {Q})\).

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 \(>n/2\).

Two covering spaces \(p_0:X_0\to X\) and \(p_1:X_1\to X\) of a space \(X\) are isomorphic if there is a homeomorphism \(h:X_0\to X_1\) such that \(p_0=p_1\circ h\). In this question all spaces are path-connected, locally path-connected and semilocally path-connected.

1. Prove the following piece of the classification theorem for covering spaces: Let \(x\in X\), \(x_0\in X_0\) and \(x_1\in X_1\) be basepoints such that \(p_0(x_0)=p_1(x_1)=x\). The covers are isomorphic if and only if \((p_0)_*(\pi _1(X_0,x_0))\) and \((p_1)_*(\pi _1(X_1,x_1))\) are conjugate subgroups of \(\pi _1(X,x)\).
2. Show that there are non-isomorphic covers of a genus two surface of every degree.

Let \(M\) be an open 3-manifold such that every compact set of \(M\) is contained in a solid torus.

1. Show that \(\pi _1(M)\) is abelian.
2. Assume that \(\pi _1(M)\) is finitely generated. Show that \(\pi _1(M)\) is trivial or \(\mathbb {Z}\).

Let \(c\) be a separating simple closed curve on a compact surface \(S\) without boundary. Show that \(S\) doesn’t retract onto \(c\).

Construct a \(\Delta \)-complex structure for a genus two surface. Use the \(\Delta \)-complex to calculate the cohomology and cup product structure for the surface.

Let \(X_d\) be the CW-complex obtained by attaching a disk with a degree \(d\) map to the circle \(S^1\times \{x_0\}\) on the torus \(S^1\times S^1\) (and \(x_0\in S^1\) is a basepoint). Calculate the homology and cohomology of \(X_d\) with \(\mathbb {Z}\), \(\mathbb {R}\) and \(\mathbb {Z}_p\) coefficients where \(p\) is prime.

Let \(V,W:\mathbb {R}^4\to \mathbb {R}^4\) be vector fields on \(\mathbb {R}^4\) defined by \[ V(x,y,z,w)=(y,-x,w,-z) \] and \[ W(x,y,z,w)=(w,z,-y,-x). \] Is there a nonempty surface \(\Sigma \subset \mathbb {R}^4\) such that for every \(p\in \Sigma \), we have \(V(p),W(p)\in T_p\Sigma \)? Find such a surface or prove that it does not exist.

Let \(T^2=S^1\times S^1\) be the standard 2-torus and \(p,q\in T^2\) be two distinct points. Let \(X=T^2/{\sim }\) be the identification space where the two points are identified \(p\sim q\). Find \(\pi _1(X)\).

Let \(X\) be the space obtained from two tori, \(S^1\times S^1\), by gluing \(S^1\times \{pt\}\) in one torus to \(S^1\times \{pt\}\) in the other torus with a degree two map. Calculate \(\pi _1(X)\).

Let \[ f:S^1\times S^1\to S^1\times S^1 \] be map of the torus to itself. Since \(H_1(S^1\times S^1)=\mathbb {Z}\oplus \mathbb {Z}\), after choosing generators for \(H_1(S^1\times S^1)\) we can represent \[ f_*:H_1(S^1\times S^1)\to H_1(S^1\times S^1) \] as a \(2\times 2\) matrix \(A\) with integer entries. If \(f\) does not have fixed points what are the possibilities for the trace and determinant of \(A\) and the degree of \(f\)? For each possible triple give an example of a fixed point free map.

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point. (Hint: Work in local coordinates, in standard charts \(U_0,U_1,U_2\) where \(U_0=\{x\ne 0\}\) etc.)

Compute de Rham cohomology of the 3-torus \(S^1\times S^1\times S^1\).

Construct a \(\Delta \)-complex for the Klein bottle and calculate its simplicial homology for both \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

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 \(g\ge h\).

Let \(X=S(S^1\times S^1)\) be the suspension of the torus and \(Y=SX\) the suspension of \(X\). Show that \(X\) and \(Y\) aren’t manifolds.

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.)

Consider vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

Let \(f:\Sigma \to S^1\) be a smooth map from the surface of genus 2 to the circle. Suppose that \(y\in S^1\) is a regular value of \(f\) and that \(f^{-1}(y)\) is a nonseparating circle in \(\Sigma \) (i.e. \(\Sigma \setminus f^{-1}(y)\) is connected). Prove that \(f\) is not nullhomotopic.

Let \(M\) be a 3-manifold such that for every compact set \(K\subset M\) there is a solid torus \(W\subset M\) with \(K\subset W\). Show that \(\pi _1(M)=\mathbb {Z}\) or \(0\).

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.

Let \(M\) be a closed 4-manifold, \(S\) a surface with \(\chi (S)\le 0\) and \(p:M\to S\) a fiber bundle. Show that there exists a closed surface \(F\) with \(\pi _n(M)=\pi _n(F)\) for \(n\ge 2\).

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point.

Consider the following vector fields in \(\mathbb {R}^3\): \[ X=\frac {\partial }{\partial x},\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Here \(x,y,z\) are the coordinates on \(\mathbb {R}^3\) and \(\frac {\partial }{\partial x}\), \(\frac {\partial }{\partial y}\), \(\frac {\partial }{\partial z}\), are the three standard coordinate vector fields. Show that there is no (nonempty) surface \(M\subset \mathbb {R}^3\) which is tangent to both vector fields at each of its points.

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.

Compute the fundamental group of the space obtained from two tori \(S^1\times S^1\) by identifying a circle \(S^1\times \{x_0\}\) in one torus with the corresponding circle \(S^1\times \{x_0\}\) in the other torus.

Compute the homology groups of the space obtained from a torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle of the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus.

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 \[ \operatorname {rank}H_1(X)\le g. \]

Consider the vector fields \[ X=\frac {\partial }{\partial x}\qquad \text {and}\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z} \] on \(\mathbb {R}^3\). Prove or disprove: There is a nonempty surface \(M^2\subset \mathbb {R}^3\) such that for every \(p\in M\) the tangent space \(T_pM\) is spanned by \(X_p\) and \(Y_p\).

Give a definition of (Gaussian) curvature for surfaces in \(\mathbb {R}^3\) and state the Gauss-Bonnet theorem.

The surface \(M_g\) of genus \(g\), embedded in \(\mathbb {R}^3\) in the standard way, bounds a compact region \(R\). Two copies of \(R\), glued together by the identity map between their boundary surfaces \(M_g\), form a closed 3-manifold \(X\). Compute the homology groups of \(X\) via the Mayer-Vietoris sequence for this decomposition of \(X\) into two copies of \(R\).

Diagram: a standard embedded genus \(g\) surface, drawn schematically as a handled surface bounding a compact region \(R\).

1. Show that the Klein bottle cannot be embedded (as a smooth submanifold) into \(\mathbb {R}^3\).
2. It is a fact that the Klein bottle can be embedded into \(\mathbb {R}^4\). Show that no such embedding \(K\subset \mathbb {R}^4\) can be given by a system of equations \[ f_1=0,\qquad f_2=0 \] with independent smooth functions \(f_1,f_2\) (i.e. where \(df_1,df_2\) are linearly independent at each point \(x\in K\)).

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 \(g=1\).

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 \(\widetilde {G}\to G\).

Consider the two vector fields in \(\mathbb {R}^3\) given by \(V(x,y,z)=(x^2+y^2,0,-y)\) and \(W(x,y,z)=(0,x^2+y^2,x)\). Find a two dimensional surface \(M^2\subset \mathbb {R}^3\) containing the point \(p=(1,0,0)\) which is tangent to the vector fields \(V\) and \(W\) in a neighborhood of \(p\), or show that such a surface \(M\) doesn’t exist.

Let \(\Sigma ^2\) be a (nonempty) smooth closed (compact without boundary) surface smoothly embedded in \(\mathbb {R}^3\). Show that there is a two-plane \(P\) in \(\mathbb {R}^3\) which intersects \(\Sigma \) such that \(P\cap \Sigma \) consists of finitely many disjoint smooth closed loops.

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 \(T^2=S^1\times S^1\) be the standard 2-torus and \(p,q\in T^2\) be two distinct points. Let \(X=T^2/{\sim }\) be the identification space where the two points are identified \(p\sim q\). Find \(\pi _1(X)\).

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 \(g\).

True or false (give a short explanation or a counterexample): The Klein bottle cannot be embedded in \(\mathbb {R}^3\) as a smooth submanifold.

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 \(\mathbb {R}^3\).

Suppose the three sphere \[ S^3=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2+z^2+w^2=1\} \] is embedded into four space the standard way. Two independent vector fields on \(S^3\) are given by \(V=(y,-x,w,-z)\), \(W=(w,z,-y,-x)\). Determine whether it is possible to embed a closed smooth two dimensional surface \(N\subset S^3\) so that the vector fields are tangent to \(N\) at every point of \(N\), i.e. \(V(p),W(p)\in T_pN\) for every \(p\in N\).

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.

Determine all covering spaces of the two-torus \(T^2\) up to equivalence. Explain.

Let \(t^2\subset S^5\) be a homeomorph of the two torus \(T^2\). Find the singular homology \(H_*(S^5-t^2;\mathbb {Z})\).

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 \(S^3\) be the 3-sphere, and let \(T^2=S^1\times S^1\) be the 2-torus. Compute the singular homology of \(S^3\times T^2\) using its CW-complex structure.

b. Recompute the singular homology of \(S^3\times T^2\) using the Eilenberg-Zilber and Künneth Theorems.

Construct a nontrivial circle bundle over the Klein bottle.

Let \[ X(s,\theta )=(\varphi (s)\cos (\theta ),\varphi (s)\sin (\theta ),\gamma (s)) \] be a surface of revolution such that \(\varphi '(s)^2+\gamma '(s)^2=1\). Prove that the meridians \(\theta =\mathrm {const}\) are geodesics. What condition on the lattitude \(s=\mathrm {const}\) is necessary and sufficient to made it geodesic?

Prove that torus can not be immersed in \(\mathbb {R}^3\) in such a way that the Gauss curvature of the induced metric is everywhere non-negative.

Let \(S_1\) and \(S_2\) be two (2-dimensional) surfaces. Derive the relationship between the Euler characteristic of the connected sum \(S_1\#S_2\) and the Euler characteristics of \(S_1\) and \(S_2\).

Find a cell-complex structure for the Klein bottle.

Let \(\Delta \) be the 2-dimensional distribution in \(\mathbb {R}^3\) defined by \[ \Delta =\operatorname {span}\left \{ \frac {\partial }{\partial x^1}+x^2\frac {\partial }{\partial x^3}, \frac {\partial }{\partial x^3}+x^3\frac {\partial }{\partial x^2} \right \}. \] Determine whether this distribution is tangent to a family of two-dimensional surfaces.

Let the curve \((f(s),g(s))\) be parameterized by arclength in \(\mathbb {R}^2\) (\(f>0\)). Consider the surface of revolution \(S\) in \(\mathbb {R}^3\) given parametrically by \[ X(s,\theta )=(f(s)\cos \theta ,f(s)\sin \theta ,g(s)). \]

(a) Compute the mean curvature and Gauss curvature for points on \(S\).

(b) Find necessary and sufficient conditions for the “latitudes” obtained when \(s\) is constant to be geodesics.

Let \(\Sigma \) denote the surface obtained from an annulus by identifying antipodal points of the inner circle and by identifying antipodal points of the outer circle. Compute the fundamental group of \(\Sigma \). Can you identify \(\Sigma \) as a connected sum of tori and projective planes?

The surface parameterized by \(X:(u,v)\mapsto (\cos u,\sin u,2\cos v,2\sin v)\) is a torus in \(\mathbb {R}^4\) with the induced metric \(ds^2=X^*ds^2_{\mathbb {R}^4}\). Show that this torus cannot be isometrically immersed into \(\mathbb {R}^3\).

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?)