Exercises

Exercises#

1. Show that the free product \(G * H\) of nontrivial groups \(G\) and \(H\) has trivial center, and that the only elements of \(G *H\) of finite order are the conjugates of finite-order elements of \(G\) and \(H\).

2. Let \(X \subset \mathbb{R}^m\) be the union of convex open sets \(X_1 \cdots X_n\) such that \(X_i \cap X_j \cap X_k \neq \emptyset\) for all \(i,j,k\). Show that \(X\) is simply-connected.

3. Show that the complement of a finite set of points in \(\mathbb{R}^n\) is simply-connected if \(n \geq 3\).

4. Let \(X \subset \mathbb{R}^3\) be the union of \(n\) lines through the origin. Compute \(\pi_1(\mathbb{R}^3-X)\).

5. Let \(X \subset \mathbb{R}^2\) be a connected graph that is the union of a finite number of straight line segments. Show that \(\pi_1(X)\) is free with a basis consisting of loops formed by suitably chosen paths in \(X\). [Assume the Jordan curve theorem for polygonal simple closed curves, which is equivalent to the case that \(X\) is homeomorphic to \(S^1\).]

6. Use Proposition 1.26 to show that the complement of a closed discrete subspace of \(\mathbb{R}^n\) is simply-connected if \(n \geq 3\).

7. Let \(X\) be the quotient space of \(S^2\) obtained by identifying the north and south poles to a single point. Put a cell complex structure on \(X\) and use this to compute \(\pi_1(X)\).

8. 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.

../../_images/ex-1-2-9.png

9. In the surface \(M_g\) of genus \(g\), let \(C\) be a circle that separates \(M_g\) into two compact subsurfaces \(M_h'\) and \(M_k'\) obtained from the closed surfaces \(M_h\) and \(M_k\) by deleting an open disk from each. Show that \(M_h'\) does not retract onto its boundary circle \(C\), and hence \(M_g\) does not retarct onto \(C\). [Hint: abelianize \(\pi_1\).] But show that \(M_g\) does retract onto the nonseparating circle \(C'\) in the figure.

../../_images/ex-1-2-10.png

10. Consider two arcs \(\alpha\) and \(\beta\) embedded \(D^2 \times I\) as shown in the figure. The loop \(\gamma\) is obviously nullhomotopic in \(D^2 \times I\), but show that here there is no nullhomotopy of \(\gamma\) in the complement of \(\alpha \cup \beta\).

11. The mapping torus \(T_f\) of a map \(f:X \rightarrow X\) is the quotient of \(X \times I\) obtained by identifying each point \((x,0)\) with \((f(x),1)\). In the case \(X =S^1 \vee S^1\) with \(f\) basepoint-preserving, compute a presentation for \(\pi_1(T_f)\) in terms of the induced map \(f_*:\pi_1(X) \rightarrow \pi_1(X)\). Do the same when \(X=S^1 \times S^1\). [One way to do this is to regard \(T_f\) as built from \(X \vee S^1\) by attaching cells.]

../../_images/ex-1-2-12.png

12. The Klein bottle is usually pictured as a subspace of \(\mathbb{R}^3\) like the subspace \(X \subset \mathbb{R}^3\) shown in the first figure at the right. If one wanted a model that could actually function as a bottle, one would delete the open disk bounded by the circle of self-intersection of \(X\), producing a subspace \(Y \subset X\). Show that \(\pi_1(X) \approx \mathbb{Z} * \mathbb{Z}\) and that \(\pi_1(Y)\) has the presentation \(\langle a,b,c | aba^{-1}b^{-1}cb^\epsilon c^{-1} \rangle\) for \(\epsilon = \pm 1\). (Changing the sign of \(\epsilon\) gives an isomorphic group, as it happens.) Show also that \(\pi_1(y)\) is isomorphic to \(\pi_1(\mathbb{R}^3 - Z)\) for \(Z\) the graph shown in the figure. The groups \(\pi_1(X)\) and \(\pi_1(y)\) are not isomorphic, but this is not easy to prove; see the discussion in Example 1B.13.

13. The space \(Y\) in the preceding exercise can be obtained from a disk with two holes by identifying its three boundary circles. There are only two essentially different ways of identifying the three boundary circles. Show that the other way yields a space \(Z\) with \(\pi_1(Z)\) not isomorphic to \(\pi_1(Y)\). [Abelianize the fundamental groups to show they are not isomorphic.]

14. Consider the quotient space of a cube \(I^3\) obtained by identifying each square face with the opposite square face via the right-handed screw motion consisting of a translation by one unit in the direction perpendicular to the face combined with a one-quarter twist of the face about its center point. Show this quotient space \(X\) is a cell complex with two \(0\)-cells, four \(1\)-cells, three \(2\)-cells, and one \(3\)-cell. Using this structure, show that \(\pi_1(X)\) is the quaternion group \(\{ \pm1, \pm i, \pm j, \pm k\}\), of order eight.

15. Given a space \(X\) with basepoint \(x_0 \in X\), we may construct a CW complex \(L(X)\) having a single \(0\)-cell, a \(1\)-cell \(e^1_\gamma\) for each loop \(\gamma\) in \(X\) based at \(x_0\), and a \(2\)-cell \(e^2_\tau\) for each map \(\tau\) of a standard triangle \(PQR\) into \(X\) taking the three vertices \(P,\, Q\), and \(R\) of the triangle to \(x_0\). The \(2\)-cell \(e^2_\tau\) is attached to the three \(1\)-cells that are the loops obtained by restricting \(\tau\) to the three oriented edges \(PQ,\, PR\), and \(QR\). Show that the natural map \(L(X) \rightarrow X\) induces an isomorphism \(\pi_1(L(X)) \approx \pi_1(X, x_0)\).

16. Show that the fundamental group of the surface of infinite genus shown below is free on an infinite number of generators.

../../_images/ex-1-2-16.png

17. Show that \(\pi_1(\mathbb{R}^2 - \mathbb{Q}^2)\) is uncountable.

18. In this problem we use the notions of suspension, reduced suspension, cone, and mapping cone defined in Chapter 0. Let \(X\) be the subspace of \(\mathbb{R}\) consisting of the sequence \(1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots\) together with its limit point \(0\).

  1. For the suspension \(SX\), show that \(\pi_1(SX)\) is free on a countably infinite set of generators, and deduce that \(\pi_1(SX)\) is countable. In contrast to this, the reduced suspension \(\sigma X\), obtained from \(SX\) by collapsing the segment \(\{0\} \times I\) to a point, is the shrinking wedge of circles in Example 1.25, with an uncountable fundamental group.

  2. Let \(C\) be the mapping cone of the quotient map \(SX \rightarrow \sigma X\). Show that \(\pi_1(C)\) is uncountable by constructing a homomorphism from \(\pi_1(C)\) onto \(\prod_\infty \mathbb{Z} / \bigoplus _\infty \mathbb{Z}\). Note that \(C\) is the reduced suspension of the cone \(CX\). Thus the reduced suspension of a contractible space need not be contractible, unlike the unreduced suspension.

19. Show that the subspace of \(\mathbb{R}^3\) that is the union of the spheres of radius \(\frac{1}{n}\) and center \((\frac{1}{n},0,0)\) for \(n=1,2,\cdots\) is simply-connected.

20. Let \(X\) be the subspace of \(\mathbb{R}^2\) that is the union of the circles \(C_n\) of radius \(n\) and center \((n,0)\) for \(n=1,2,\cdots\). Show that \(\pi_1(X)\) is the free group \({\Large *}_n\pi_1(C_n)\), the same as for the infinite wedge sum \(\bigvee _\infty S^1\). Show that \(X\) and \(\bigvee _\infty S^1\) are in fact homotopy equivalent, but not homeomorphic.

21. Show that the join \(X * Y\) of two nonempty spaces \(X\) and \(Y\) is simply-connected if \(X\) is path-connected.

22. In this exercise we desribe an algorithm for computing a presentation of the fundamental group of the complement of a smooth or piecewise linear knot \(K\) in \(\mathbb{R}^3\), called the Wirtinger presentation. To begin, we position the knot to lie almost flat on a table, so that \(K\) consists of finitely many disjoint arcs \(\alpha_i\) where it intersects the table top together with finitely many disjoint arcs \(\beta_l\) where \(K\) crosses over itself. The configuration at such crossing is shown in the first figure below.

../../_images/ex-1-2-22.png

We build a \(2\)-dimensional complex \(X\) that is a deformation retract of \(\mathbb{R}^3-K\) by the following three steps. First, start with the rectangle \(T\) formed by the table top. Next, just above each arc \(\alpha_i\) place a long, thin rectangular strip \(R_i\), curved to run parallel to \(\alpha_i\) along the full length of \(\alpha_i\) and arched so that the two long edges of \(R_i\) are identified with points of \(T\), as in the second figure. Any arcs \(\beta_l\) that cross over \(\alpha_i\) are positioned to lie in \(R_i\). Finally, over each arc \(\beta_l\) put a square \(S_l\), bent downward along its four edges so that these edges are identified with points of three strips \(R_i,\quad R_j\), and \(R_k\) as in the fthird figure; namely, two opposite edges of \(S_l\) are identified with short edges of \(R_j\) and \(R_k\) and the other two opposite edges of \(S_l\) are identified with two arcs crossing the interior of \(R_i\). The knot \(K\) is now a subspace of \(X\), but after we lift \(K\) up slightly into the complement of \(X\), it becomes evident that \(X\) is a deformation retract of \(\mathbb{R}^3-K\).

(a) Assuming this bit of geometry, show that \(\pi_1(\mathbb{R}^3-K)\) has a presentation with one

generator \(x_i\) for each strip \(R_i\) and one relation of the form \(x_i x_j x_i^{-1} = x_k\) for each square \(S_l\), where the indices are as in the figures above. [To get the correct signs it is helpful to use an orientation of \(K\).]

(b) Use this presentation to show that the abelianization of \(\pi_1(\mathbb{R}^3-K)\) is \(\mathbb{Z}\).