Fundamental 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.

We can regard \(\pi _1(X,x_0)\) as the set of basepoint-preserving homotopy classes of maps \((S^1,s_0) \to \) \((X,x_0)\). Let \([S^1,X]\) be the set of homotopy classes of maps \(S^1 \to X\) with no condition on basepoints. Thus there is a natural map \(\psi : \pi _1(X,x_0) \to [S^1,X]\) obtained by ignoring basepoints. Show that \(\psi \) is onto if \(X\) is path connected.

Prove or give a (justified) counterexample. If the first homology group \(H_1(X)\) of a connected CW-complex \(X\) is trivial then \(X\) is contractible.

Let \(a\) and \(b\) be the two generators of \(\pi _1(S^1 \vee S^1)\) corresponding to the two \(S^1\) summands. Draw a picture of the covering space of \(S^1 \vee S^1\) corresponding to the normal subgroup generated by \(a^2\), \(b^2\), and \((ab)^4\), and prove this covering space is indeed the correct one.

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the torus \(T = S^1 \times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to \(S^1 \times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).

2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\) coefficients.

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.

Let \(M\) be a closed, orientable 5-manifold with \(H_1(M) = \mathbb {Z} \oplus \mathbb {Z}_5\) and \(H_2(M) = \mathbb {Z}_2\). Find \(H_i(M)\) and \(H^i(M;\mathbb {Z})\) for all \(i \ge 0\).

Prove that if \(X\) and \(Y\) are path connected spaces then

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.

Draw a picture of a covering space of \(S^1\vee S^1\) whose fundamental group is the normal subgroup of \(\pi _1(S^1\vee S^1,*)\) generated by \(a^2\), \(b^2\), and \((ab)^2\), and prove your covering space is the correct one.

Let \(M\) be closed, connected, 5-manifold such that \(\pi _1(M)\cong \mathbb {Z}/7\) and \(H_2(M)=\mathbb {Z}\). Compute all homology and cohomology groups of \(M\) with \(\mathbb {Z}\)-coefficients.

Let \(W\) be the union of the three coordinate axes in \(\mathbb {R}^3\) and set \(X:=\mathbb {R}^2\setminus W\); thus \(X\) consists of points \((x,y,z)\) where none of the coordinates are zero. Compute \(\pi _1(X)\).

Let \(X\) be a CW complex consisting of one vertex p, two edges a and b attached to \(p\) along their boundaries, and two faces \(f_1\) and \(f_2\), attached along \(ab^2\) and \(ba^2\), respectively.

1. Find \(\pi _1(X)\). Is it a finite group?
2. Compute the homology groups \(H_i(X;\mathbb {Z})\) of \(X\) for all \(i\).

Consider the group \(G\) given by the presentation

where \(p\), \(q\) are given integers.

1. Construct a (connected) CW-complex \(X\) with a basepoint \(x\) such that \(\pi _1(X,x)\cong G\).
2. Compute \(H_1(X)\) and give conditions on \(p\), \(q\) that are equivalent to \(H_1(X)\) being torsion-free.

Let \(M\) be a closed connected 5-manifold such that \(H_1(M)\cong \mathbb {Z}/3\) and \(H_2(M)\cong \mathbb {Z}\). Compute \(H_i(M)\) for all \(i\).

Let \(x_0\in \mathbb {R}P^2\) be a basepoint and view the wedge \(\mathbb {R}P^2\vee \mathbb {R}P^2\) as the subspace of the product \(\mathbb {R}P^2\times \mathbb {R}P^2\) where at least one coordinate is \(x_0\).

1. Use the Seifert-van Kampen theorem to compute \(\pi _1(\mathbb {R}P^2\vee \mathbb {R}P^2)\).
2. Prove that \(\mathbb {R}P^2\vee \mathbb {R}P^2\) is not a retract of \(\mathbb {R}P^2\times \mathbb {R}P^2\).

Let \(X\) be a connected CW complex and \(A\subset X\) a connected subcomplex. Suppose that inclusion \(A\subset X\) is \(\pi _1\)-injective (for some, hence all, basepoints in \(A\)). Let \(p:\widetilde {X}\to X\) be the universal cover of \(X\). Prove that each component of \(p^{-1}(A)\) is simply connected.

Let \(m,n\) be positive integers and view \(S^1\) as the unit circle in the complex plane. Let \(X\) be the space obtained from the cylinder \(S^1\times [0,1]\) by identifying points \((z,0)\) and \((w,0)\) if \(z^n=w^n\), and identifying points \((z,1)\) and \((w,1)\) if \(z^m=w^m\).

1. Use the Mayer-Vietoris sequence to compute \(H_i(X)\) for all \(i\).
2. Use the Seifert-van Kampen theorem to find a presentation for \(\pi _1(X)\).

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 \(i\).

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

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to \(S^1\times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).
2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\) and \(\mathbb {Z}_m\)-coefficients.

Let \(n_1\), \(n_2\) and \(n_3\) be integers. Given an example of a topological space \(X\) with \[ \pi _1(X,x_0)=\langle a,b\mid a^{n_1}b^{n_2}a^{n_3}b\rangle . \] What is \(H_1(X)\)? Under what conditions is \(H_1(X)\) torsion free?

Find all connected Lie subgroups of \(\mathrm {SL}_2(\mathbb {R})\) that contains the subgroup \[ H= \left \{ \begin {pmatrix} 1 & x\\ 0 & 1 \end {pmatrix} : x\in \mathbb {R} \right \}. \]

Let \(X=S^1\vee S^1\), with basepoint \(*\) at the point where the two circles are glued, and with \(a,b:S^1\to X\) the generators of \(\pi _1(X,*)\) corresponding to the identity maps onto each circle in the wedge sum. Construct a covering space \(p:(E,e)\to (X,*)\) such that \(p_*\pi _1(E,e)\) is the subgroup of \(\pi _1(X,*)\) generated by \(a^2\), \(ab^{-1}\), \(ab\). Justify that your answer works in two ways:

1. using covering space theory; and
2. using the van Kampen theorem.

Let \(G\) be any group. Construct a connected topological space whose fundamental group is isomorphic to \(G\). Justify your answer.

Let \(X\) be a path-connected topological space.

1. Let \(x_0,x_1\) be points in \(X\), and let \(\alpha :I\to X\) be any path from \(x_0\) to \(x_1\). Explain how \(\alpha \) induces an isomorphism of groups \(\alpha _*:\pi _1(X,x_0)\xrightarrow {\sim }\pi _1(X,x_1)\).

2. Give an example where \(x_1=x_0\) (so \(\alpha \) is a loop at \(x_0\)), but the map \(\alpha _*:\pi _1(X,x_0)\to \pi _1(X,x_0)\) is not the identity map.
3. Give an example of a space \(X\) with \(H_1(X;\mathbb {Z})=0\) but \(\pi _1(X,x_0)\ne 0\). State carefully the results you are using.

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 \(\pi _1(S^n)=1\) if \(n\ge 2\).

Let \(p:\widetilde {X}\to X\) be a cover of connected spaces. Prove \(p_*:\pi _1\widetilde {X}\to \pi _1X\) is injective.

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

Prove \(H_1(\Sigma _2)\cong \mathbb {Z}^4\).

Prove that if \(X\) and \(Y\) are path connected spaces, then \(\pi _1(X\times Y)=\pi _1(X)\times \pi _1(Y)\).

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

Show that \(\pi _1(\Sigma _2)\) contains \(\pi _1(\Sigma _3)\) as a normal subgroup.

Suppose \(f:S^k\to S^k\) is continuous and not surjective. Prove that \(f_*:H_k(S^k;\mathbb {Z})\to H_k(S^k;\mathbb {Z})\) is the zero homomorphism.

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 \(n\).

Find all connected subgroups of \(\mathrm {SL}_2(\mathbb {R})\) that contain the group \[ V=\{g\in \mathrm {SL}_2(\mathbb {R})\mid g_{1,2}=0\text { and }g_{1,1}=g_{2,2}=1\}. \]

Let \(\Phi :\mathfrak {sl}_2(\mathbb {R})\to \mathbb {R}\) be a Lie algebra homomorphism, where \(\mathfrak {sl}_2(\mathbb {R})\) is the Lie algebra of \(\mathrm {SL}_2(\mathbb {R})\) and \(\mathbb {R}\) is the Lie algebra of the Lie group \(\mathbb {R}\). Prove that \(\Phi =0\).

Prove that the fundamental group of a finite connected graph is a free group.

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 \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

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.

Prove that the homogeneous space \(\mathrm {SL}_2(\mathbb {R})/\mathrm {SL}_2(\mathbb {Z})\) is not compact.

Let \(X=S^1\vee S^1\) and \(x_0\in X\) be the attaching point.

1. Carefully state van Kampen’s Theorem and use it to compute \(\pi _1(X,x_0)\).
2. Find two covering spaces of \(X\) with three sheets; one of which is regular (determining a normal subgroup of \(\pi _1(X,x_0)\)) and the other of which is irregular.

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 \(X\) be a path connected space that is locally simply connected. Give the construction of \(p:\widetilde {X}\to X\), the universal cover of \(X\). You don’t have to define what a universal cover is, or show that your construction is a universal cover. Just state the description of the universal cover for \(X\) used in the standard proof that the universal cover for \(X\) exists.

Let \(p,q\ge 2\) be integers and denote by \(X_{p,q}\) the join of discrete sets \(\{v_1,\ldots ,v_p\}\) and \(\{w_1,\ldots ,w_q\}\). Thus \(X_{p,q}\) is a graph with \(p+q\) vertices and \(pq\) edges.

1. What is the fundamental group of \(X_{p,q}\)?
2. Denote by \(Y_{p,q}\) the 2-complex obtained from \(X_{p,q}\) by attaching 2-cells with attaching maps \(S^1\to X_{p,q}\) homeomorphisms to all possible cycles of length 4. (Thus there are \(\binom {p}{2}\binom {q}{2}\) 2-cells.) Show that \(Y_{p,q}\) is simply connected.

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.

Prove that the fundamental group of any Lie group is abelian.

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.

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells, one with degree 3 attaching map and the other with degree 5 attaching map.

1. Using van Kampen’s theorem show that \(X\) is simply connected.
2. Show that \(X\) is homotopy equivalent to the 2-sphere. Hint: Use repeateadly the fact that the spaces \(Y\cup _f e^2\) and \(Y\cup _g e^2\) obtained from \(Y\) by attaching a single 2-cell with homotopic attaching maps \(f,g\) are homotopy equivalent.

Let \(X\) be the space obtained from the disjoint union of a circle \(S^1\) and a cylinder \(S^1\times [0,1]\) by attaching the cylinder along its boundary to the circle with the attaching map on \(S^1\times \{0\}\) having degree 2 and on \(S^1\times \{1\}\) degree 3. Find a presentation of the fundamental group of \(X\).

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.

Let \(M\) be a connected compact 3-manifold (without boundary) and assume that \(H_1(M)=\mathbb {Z}/3\mathbb {Z}\). Find \(H_2(M)\). Carefully justify the answer, and state all theorems you are using.

Let \(S^3=\{(z,w)\in \mathbb {C}^2\mid |z|^2+|w|^2=1\}\) and \(S^1\subset S^3\) is given by \(w=0\). Let \(X\) be the quotient space of \(S^3\) obtained by collapsing \(S^1\) to a point. Use any method to compute \(H_k(X;\mathbb {Z})\) for all \(k\in \mathbb {Z}\).

Let \(X\) be a path connected, locally path connected, semi-locally simply connected Hausdorff space. One of the main theorems of covering space theory asserts that there is a simply connected space \(\widetilde {X}\) and a covering map \(p:\widetilde {X}\to X\). Outline the construction of \(\widetilde {X}\) and \(p\) as follows:

1. Describe \(\widetilde {X}\) as a set.
2. Describe the topology on \(\widetilde {X}\).
3. Describe \(p\). You do not need to prove that \(p\) is a covering map.
4. Outline the proof that \(\widetilde {X}\) is simply connected.

Compute the fundamental group of the space obtained from the disjoint union \(A\sqcup B\) of two 2-tori by identifying them along 3 points, i.e. \(a_i\sim b_i\) for \(i=1,2,3\) where \(\{a_1,a_2,a_3\}\subset A\) and \(\{b_1,b_2,b_3\}\subset B\).

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 \(i\).

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 \(\pi _1(X)\).

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells, one with degree 2 attaching map, and the other with degree 4 attaching map. What is the cardinality of the fundamental group of \(X\)? If you use van Kampen’s theorem, state it.

Let \(X\) be a topological space and \(x\in X\). Assume that for some topological space \(L\) the point \(x\) has a neighborhood \(U\) in \(X\) which is homeomorphic to the open cone \(cL=L\times [0,\infty )/L\times \{0\}\) by a homeomorphism that takes \(x\) to \(L\times \{0\}\). Prove that \(H_k(X,X-\{x\})\cong \widetilde {H}_{k-1}(L)\).

Let \(G\) be a Lie group and \(1\in G\) the identity element. Prove that \(\pi _1(G,1)\) is an abelian group.

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 \(X\) be a topological space and \(x_0\in X\) a basepoint.

1. Define \(\pi _1(X,x_0)\) (describe it as a set and define the group operation; you don’t have to prove that it is well defined or that it is a group).
2. If \(X\) is path-connected and \(x_1\in X\) prove that \(\pi _1(X,x_0)\cong \pi _1(X,x_1)\) (write down an explicit isomorphism and check that it works).

1. Define a covering space \(p:\widetilde {X}\to X\).
2. Let \(p:\widetilde {X}\to X\) be a covering space, \(\widetilde {x}_0\in \widetilde {X}\), \(x_0\in X\), and \(p(\widetilde {x}_0)=x_0\). Show that \(p_*:\pi _1(\widetilde {X},\widetilde {x}_0)\to \pi _1(X,x_0)\) is injective (carefully state the lifting property you are using).

Let \(X\) be the cell complex obtained from the circle \(S^1\) by attaching two 2-cells \(e^2_2\) and \(e^2_3\) with attaching maps of degrees 2 and 3 respectively. Compute the fundamental group of \(X\). Carefully state any theorems you are using.

1. State the universal coefficient theorem for cohomology.
2. Suppose \(X\) is a space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=\mathbb {Z}/2\mathbb {Z}\) and \(H_i(X)=0\) for \(i>1\). Compute \(H^i(X;\mathbb {Z}/2\mathbb {Z})\) for all \(i\).

Let \(X\) be a path connected, locally path connected, semi-locally simply connected Hausdorff space. One of the main theorems of covering space theory asserts that there is a simply connected space \(\widetilde {X}\) and a covering map \(p:\widetilde {X}\to X\). Outline the construction of \(\widetilde {X}\) and \(p\) as follows:

1. Describe \(\widetilde {X}\) as a set.
2. Describe the topology on \(\widetilde {X}\).
3. Describe \(p\). You do not need to prove that \(p\) is a covering map.
4. Outline the proof that \(\widetilde {X}\) is simply connected.

Let \(S^3=\{(z,w)\in \mathbb {C}^2\mid |z|^2+|w|^2=1\}\) and \(S^1\subset S^3\) is given by \(w=0\). Let \(X\) be the quotient space of \(S^3\) obtained by collapsing \(S^1\) to a point. Use any method to compute \(H_k(X;\mathbb {Z})\) for all \(k\in \mathbb {Z}\).

Give an example of a connected cell complex \(X\) such that \(H_k(X;\mathbb {Z})=0\) for all \(k>0\) but \(\pi _1(X)\ne \{1\}\). You do not have to verify the second claim, but you should argue the first.

1. Let \(G\) be the Lie subgroup of \(\mathrm {GL}_3(\mathbb {R})\) that consists of the following matrices: \[ \begin {pmatrix} a & b & x\\ c & d & y\\ 0 & 0 & 1 \end {pmatrix} \] where \(ad-bc=1\). Find a basis for the Lie algebra of \(G\), considered as a Lie subalgebra of \(M_{3\times 3}(\mathbb {R})\).
2. Let \(\mathfrak {gl}_2(\mathbb {R})\) be the Lie algebra of \(\mathrm {GL}_2(\mathbb {R})\). For \[ \begin {pmatrix}2&1\\3&-2\end {pmatrix}, \begin {pmatrix}1&3\\0&1\end {pmatrix} \in \mathfrak {gl}_2(\mathbb {R}) \] find \[ \left [ \begin {pmatrix}2&1\\3&-2\end {pmatrix}, \begin {pmatrix}1&3\\0&1\end {pmatrix} \right ]. \]

1. State Sard’s Theorem.
2. Show that any smooth map \(f:\mathrm {SL}_2(\mathbb {R})\to S^4\) is homotopic to a constant map.

Let \(M\) be a closed connected 5-manifold such that \(\pi _1(M)=\mathbb {Z}/7\mathbb {Z}\) and \(H_2(M;\mathbb {Z})=\mathbb {Z}^2\). Compute \(H_k(M;\mathbb {Z})\), \(H^k(M;\mathbb {Z})\) and \(H_k(M;\mathbb {Z}/7\mathbb {Z})\) for all \(k\).

Describe the procedure for computing (i.e. writing down a presentation of) the fundamental group of a cell complex with one 0-cell. Using this, show that \(\mathbb {C}P^n\) is simply connected and \(\pi _1(\mathbb {R}P^n)=\mathbb {Z}/2\mathbb {Z}\) for \(n\ge 2\).

Compute the fundamental group of the space \(X_{2,3}\) obtained from the cylinder \(S^1\times [-1,1]\) via identifications: \((z,-1)\sim (-z,-1)\) and \[ (z,1)\sim \left (ze^{\frac {2\pi i}{3}},1\right ) \sim \left (ze^{\frac {4\pi i}{3}},1\right ) \] for all \(z\in S^1=\{z\in \mathbb {C}\mid |z|=1\}\). (That is, in the circle \(S^1\times \{-1\}\) antipodal points are identified, and in \(S^1\times \{1\}\) two points are identified if they differ by an order 3 rotation.) Show that this group is infinite.

Let \(X\) be the union of circles in \(\mathbb {R}^2\) with centers at \((0,\frac {1}{2n})\) and radius \(1/n\). Show \(X\) doesn’t have a universal cover (show that every connected covering space of \(X\), including \(X\) itself, has non-trivial \(\pi _1\)).

Let \(X\subset \mathbb {R}^3\) be the union of \(n\)-lines through the origin. Calculate \(\pi _1(\mathbb {R}\backslash X)\).

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 \(X\) be a compact topological space and \(\widetilde {X}\) its universal cover. Show that \(\pi _1(X)\) is finite if and only if \(\widetilde {X}\) is compact.

Let \(X_n\) be the space obtained by attaching a disk \(D\) to the torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to the circle \(S^1\times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).
2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\)-coefficients and with \(\mathbb {Z}_m\)-coefficients where \(\mathbb {Z}_m\) is the cyclic group of order \(m\).

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 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 \(\mathbb {Z}\).

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).

Give an example of a space \(X\) such that \(H_1(X;\mathbb {Z})\cong \mathbb {Z}/2\) and \(H_2(X;\mathbb {Z})\cong \mathbb {Z}\).

Suppose \(X\) is a connected 2-dimensional CW complex such that \(H_1(X;\mathbb {Z})\cong \mathbb {Z}/2\) and \(H_2(X;\mathbb {Z})\cong \mathbb {Z}\). Compute \(H_i(X;\mathbb {Z}_2)\), \(H^i(X;\mathbb {Z})\) and \(H^i(X;\mathbb {Z}_2)\) for all \(i\in \mathbb {Z}\).

Compute \(\pi _2(X)\) where \(X\) is the wedge sum of \(S^1\) and \(S^2\).

The set \(SL_2(\mathbb {R})\) of \(2\times 2\) matrices with determinant 1 can be viewed as a subset of \(\mathbb {R}^4\) by choosing an ordering of the matrix entries. Show that this set is a submanifold of \(\mathbb {R}^4\) and compute the tangent space at the identity matrix.

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 \(\phi _1(x)=y\).

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.

Let \(M\) be a closed connected 5-manifold such that \(\pi _1(M)\cong \mathbb {Z}/7\). If \(H_2(M;\mathbb {Z})\cong \mathbb {Z}\), compute all other homology and cohomology groups of \(M\) with integral coefficients.

Let \(X\) be the quotient space of \(S^2\) obtained by identifying the north and south poles to a single point. Carefully define a cell complex structure on \(X\) (including a description of attaching maps) and use this to compute \(\pi _1(X)\).

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.

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 \(T^2=S^1\times S^1\) 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 \(\Sigma \) be a simply connected compact 3-manifold. Prove that \(H_k(\Sigma )\cong H_k(S^3)\) for any integer \(k\).

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.

Find a non-regular cover of \(S^1\vee S^1\), the wedge of 2 circles. You should both explicitly describe the cover and the corresponding subgroup of \(\pi _1(S^1\vee S^1)\).

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.

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 \(P\) be a discrete set of points in \(\mathbb {R}^n\). Show that \(\mathbb {R}^n\setminus P\) is simply connected if \(n\ge 3\).

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 \(X\) be a connected CW-complex obtained by attaching an \((n+2)\)-dimensional cell to an \(n\)-dimensional CW-complex \(A\). What is the smallest possible dimension \(k\) where \(\pi _k(X)\) is not isomorphic to \(\pi _k(A)\)?

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 \(M\) be a Möbius strip and \(D\) a disk. Let \(f:\partial D\to \partial M\) have degree \(n\). Set \(X_n=M\coprod D/{\sim }\) where \(x\sim y\) if \(f(x)=y\). Calculate \(\pi _1(X_k)\) and construct all of its connected covering spaces.

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.

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.

Suppose \(M\) is a closed connected 5-manifold such that \(\pi _1(M)\cong \mathbb {Z}/3\mathbb {Z}\). If \(H_2(M;\mathbb {Z})\cong \mathbb {Z}\), compute all other homology and cohomology groups of \(M\) with integral coefficients.

Compute \(\pi _2(S^1\vee S^2)\) using the Hurewicz theorem.

True or false (provide justification): Let \(M\subset \mathbb {R}^n\) be a smooth submanifold of dimension \(m\). Then there are \(n-m\) smooth functions \(f_1,\ldots ,f_m:\mathbb {R}^n\to \mathbb {R}\) such that:

1. \(M=\{x\in \mathbb {R}^n:f_1(x)=\cdots =f_{n-m}(x)=0\}\).
2. For each \(x\in M\) the gradients \(\nabla (f_1)(x),\ldots ,\nabla (f_{n-m})(x)\) are linearly independent.

Express the fundamental group \(\pi _1(M\#N)\) in terms of \(\pi _1(M)\) and \(\pi _1(N)\) if \(n>1\).

Let \(\Sigma \) be a simply-connected closed 3-manifold. Prove that \(H_k(\Sigma )=H_k(S^3)\) for any integer \(k\).

Let \(X\) be a simply-connected closed 4-manifold. Prove that \(H_k(X)\) is torsion-free for any integer \(k\).

Let \(X\) be a topological space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=0\), \(H_2(X)=\mathbb {Z}_2\), \(H_3(X)=\mathbb {Z}_3\), \(H_4(X)=\mathbb {Z}\), \(H_k(X)=0\), \(k>4\). Compute \(H_*(X;\mathbb {Z}_2)\) and \(H^*(X;\mathbb {Z})\).

Let \(M\) be a closed connected 7-manifold such that \(H_1(M;\mathbb {Z})=0\), \(H_2(M;\mathbb {Z})=\mathbb {Z}\), \(H_3(M;\mathbb {Z})=\mathbb {Z}/2\). Compute \(H_i(M;\mathbb {Z})\) and \(H^i(M;\mathbb {Z})\) for all \(i\).

Let \(X\subset \mathbb {R}^3\) be the union of \(n\) lines through the origin. Compute \(\pi _1(\mathbb {R}^3\setminus X)\).

Let \(X\) be the space obtained from the torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle in the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus. Compute \(\pi _1(X)\) and describe the universal cover of \(X\).

Show that \(SL_2(\mathbb {R})\), viewed as the subset \[ \{(a,b,c,d)\mid ad-bc=1\} \] of \(\mathbb {R}^4\), is a manifold.

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 \(U(n)\) denote the set of unitary matrices (the complex \(n\times n\) matrices \(A\) which satisfy \(A\overline {A}^{T}=I\)). Show that \(U(n-1)\to U(n)\to S^{2n-1}\) is a fiber bundle. (Hint 1: Find explicit local trivializations. Hint 2: State a general theorem whose conclusion is that a smooth map between closed manifolds is a submersion and verify the hypotheses.)

Deduce that \(\pi _r(U(n))\simeq \pi _r(U(n+1))\) for \(r<2n\). What are \(\pi _1(U(3))\), \(\pi _2(U(3))\), \(\pi _3(U(3))\)?

Prove that \(\pi _2(\Sigma _2)=0\).

Suppose that \(K_1\supset K_2\supset K_3\supset \cdots \) is a nested sequence of compact connected subsets of \(S^2\) such that \(\pi _1(K_j)=\mathbb {Z}\) for all \(j\). Prove or provide a counterexample for each of the following statements:

1. \(\bigcap _{j=1}^{\infty }K_j\) is connected.
2. \(\pi _1\left (\bigcap _{j=1}^{\infty }K_j\right )\cong \mathbb {Z}\).

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.

Let \(X=S^1\vee S^2\vee T^2\) be the one point union. Compute \(\pi _1(X)\) and show that \(\pi _2(X)\) is infinitely generated.

Show that \(\pi _1(\mathbb {R}P^2\#\mathbb {R}P^2\#\mathbb {R}P^2)\) is infinite.

Let \(p(z)=a_nz^n+\cdots +a_0\) be a non-constant complex polynomial. Prove \(p:\mathbb {C}\to \mathbb {C}\) is proper, has only a finite number of singular values, that the inverse images of regular values have constant cardinality. Deduce the Fundamental Theorem of Algebra: \(p\) has a root.

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.

Consider the “tripod” \(T\) with the central point \(O\) (i.e. the union of three segments in the plane which intersect only at the origin \(O\)). Let \(P=(O,O)\) be the point on \(X=T\times T\) which projects to the point \(O\) in the both factors. Compute \(H_1(X,X-P;\mathbb {Z})\).

Diagram description: the tripod \(T\) consists of three line segments meeting at the central point \(O\): one vertical segment above \(O\), one diagonal segment down and to the left, and one diagonal segment down and to the right.

Suppose \(M\) is a connected manifold. Calculate \(H_1(M\times S^1)\) in terms of the homology groups of \(M\).

Calculate \(\pi _2(\mathbb {R}P^3)\).

II. Pick four of the following seven questions to answer

Let \(M_2(\mathbb {R})\) be the set of all \(2\times 2\) matrices with real coefficients, and identify it with \(\mathbb {R}^4\) in the usual way. Let \(SL(2,\mathbb {R})\) be the subset of \(M_2(\mathbb {R})\) consisting of those matrices with determinant equal to one.

(a) Show \(SL(2,\mathbb {R})\) is a submanifold of \(M_2(\mathbb {R})\).

(b) Exhibit the tangent space to \(SL(2,\mathbb {R})\) at the identity matrix.

Consider the domain \(D\) in \(\mathbb {R}^3\) which is the complement to the union of two unit balls with centers at the points \((0,0,2)\) and \((0,2,0)\). Calculate \(H_2(D,\mathbb {Z})\).

Let \(K\subset S^3\) be a knot (a homeomorph of \(S^1\)). Then \(\pi _1(S^3-K,x_0)=\mathbb {Z}\).

\[ \pi _k\left (\underbrace {S^1\times \cdots \times S^1}_{n\ \mathrm {times}},x_0\right ) = \begin {cases} \mathbb {Z}^n, & \text {if } k=1,\\ 0, & \text {if } k>1. \end {cases} \]

If \(X\) and \(Y\) are compact connected smooth \(n\)-manifolds then for \(0<k<n\), the singular homology satisfies \[ H_k(X\times Y,\mathbb {Z})=H_k(X,\mathbb {Z})\oplus H_k(Y,\mathbb {Z}). \]

Let \(X\) be a finite connected graph. Then the fundamental group \(\pi _1(X,x_0)\) is a free group with \(1-\chi (X)\) generators. (\(\chi \) is the Euler characteristic.)

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?

Let \(X\) be a connected, locally arcwise connected, semilocally simply connected, compact metric space. Show that \(\pi _1(X,x_0)\) is finitely generated.