Homotopy 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.

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

Prove 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.

Show that \(S^2\vee S^4\) is not homotopy equivalent to a compact manifold without boundary.

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 \(i\).

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.

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

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 \(x_0\in \mathbb {R}P^2\) 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 \(n_1\), \(n_2\) 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?

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 \(X\) 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.

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.

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

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

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 \(X=S^1\vee S^1\) 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.

Define the concepts of chain maps and chain homotopies, and prove that chain homotopy is an equivalence relation.

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

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).

Define the concepts of chain maps and chain homotopies, and prove that chain homotopy is an equivalence relation.

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.

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 \(n\ge 2\).

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 \(\pi _1(\mathbb {R}\backslash X)\).

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.

Show that \(S^2\vee S^4\) and \(\mathbb {C}P^2\) are not homotopy equivalent.

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 \(\mathbb {Z}\).

Define the higher homotopy groups, \(\pi _i\), for \(i\ge 2\) and prove that they are abelian.

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

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 \(\pi _1(X)\).

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 \(\mathbb {Z}\).

Show that \(S^2\vee S^4\) and \(\mathbb {C}P^2\) are not homotopy equivalent.

Define the higher homotopy groups, \(\pi _i\), for \(i\ge 2\) and prove that they are abelian.

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 \(p:S^3\to \mathbb {C}P^1=S^2\) be the map that sends \((z,w)\in S^3\subset \mathbb {C}^2\) to its equivalence class \([z:w]\in \mathbb {C}P^1\).

1. Prove that \(p\) is a fiber bundle by explicitly writing down local trivializations.
2. Write down the associated long exact sequence of homotopy groups.
3. Compute \(\pi _3(S^2)\). Justify all auxiliary results you are using.

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 \(\pi _1(X)\).

Show that \(S^2\vee S^4\) is not homotopy equivalent to \(\mathbb {C}P^4\).

Calculate \(\pi _3(S^2)\).

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.)

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.

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.

Let \(p:S^3\to \mathbb {C}P^1=S^2\) be the map that sends \((z,w)\in S^3\subset \mathbb {C}^2\) to its equivalence class \([z:w]\in \mathbb {C}P^1\).

1. Prove that \(p\) is a fiber bundle by explicitly writing down local trivializations.
2. Write down the associated long exact sequence of homotopy groups.
3. Compute \(\pi _3(S^2)\). Justify all auxiliary results you are using.

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

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

1. Define carefully the Hopf map \(h:S^3\to S^2\).
2. Define what it means for a map to be a fiber bundle and sketch a proof that \(h\) is a fiber bundle.
3. State the long exact sequence in homotopy groups associated to a fiber bundle and compute \(\pi _3(S^2)\).

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 \(X\).

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 \(\pi _1(X)\).

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

Explicitly describe a nontrivial homotopy class in \(\pi _3(S^2)\). Show that it is nonzero.

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 \(X\) 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.

True or false: Suppose that \(X\) is a finite CW-complex and \(A\subset X\) is a subcomplex. Then the relative homotopy groups \(\pi _j(X,A)\) are isomorphic to the homotopy groups \(\pi _j(X/A)\), where \(X/A\) is the CW-complex obtained from \(X\) by collapsing \(A\) to a point. Justify your answer.

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

II. Pick four of the following seven questions to answer

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

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

Find the homology, cohomology and homotopy groups

a. \(H_*(\mathbb {R}P^3,\mathbb {Z})\);

b. \(H_*(\mathbb {R}P^3,\mathbb {Z}/2\mathbb {Z})\);

c. \(H^*(\mathbb {R}P^3,\mathbb {Z})\);

d. \(\pi _k(\mathbb {R}P^3,x_0)\), for \(k=1,2,3\).