ERRATA FOR. A First Course in Geometric Topology and Differential Geometry Ethan D. Bloch Birkhäuser, Last Updated August 20, 2005

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1 ERRATA FOR A First Course in Geometric Topology and Differential Geometry Ethan D. Bloch Birkhäuser, 997 Last Updated August, 5 Below is an updated list of errata. The fault for all the errors in the book is my own, and I offer my sincere apologies for any inconvenience caused by the errors in the book. This list was compiled with the generous assistance of: Bill Bloch, Jonathan Dent, John Engbers, Kyle Glashower, Tsz Ho Ip, Gerard Venema, Yumi Watanabe, Peter Maria Wirtz, and Bard students Matthew Brophy, Vasilica Crecea, Tim Goldberg, Jiaming Mao, Kenneth Ober. If you find any additional errors in the book, or any errors in this list of errors, I would very much appreciate it if you would let me know by or regular mail at the following address: Ethan D. Bloch Bard College Annandale-on-Huon, NY 54 bloch@bard.edu Page Line/Item Text Comment/Should be Exercise.. ( a single point in R n Should be a single point in R Exercise..6 containing p Should be such that p V U 3 Exercise..7 be a closed set Should be be a non-empty closed set 8 l. 3 f i = f π i Should be f i = π i f 4 l. 6 f is surjective Should be q is surjective 7 l. 7 i,...,i r {,...,p} Should be i,...,i r {,,...,p} 7 Exercise.4.9 We need to assume that X i and Y i are both open or both closed in X i Y i for i =,. 8 l. -3 B [b,b ] Should be B [b,b ] 39 l. -6 [a, y Should be [a, y]

2 53 l. 7 Remove containing B 63 l. 3 q : D Q Should be q : D X 75 l. 3 A. Should be A. 8 Exercise.6. K # P, K # K and T # T Should be K # P and K # K 89 l. - k times in two places Should be d times in both places 9 l. 6 k times in two places Should be d times in both places 93 l. h( xy = xy Should be h( xy = h(xh(y 98 Figure A..7(i D 4 Should be D n 98 Figure A..7(i α 4 Should be α n 99 l Should be l Should be Exercise A... The number of the Exercise should be A.. 7 Exercise A... The number of the Exercise should be A.. l. -5 p to q Should be v to w 5 l. -5 {a,...,a i } Should be {a,...,a k } 5 l. -4 A face of σ that is a k-simplex is called a k-face Should be A face of σ that is an i-simplex is called an i-face 6 l. 6 S k = {x R k x < } Should be S k = {x R k x =} 8 l. 9 affine linear take Should be affine linear map takes 4 l. 7 Exercise Should be Lemma l. 4 Exercise Should be Lemma 3.3.4

3 3 Exercise η Int σ in two places Should be Int η Int σ in both places 3 l. -4 that is simplicial complex Should be that is a simplicial complex 34 l. There should not be a line break in the middle of star(w, K {w} 4 Exercise If P is a -dimensional cell complex Should be If P is a -dimensional cell complex such that P is a topological surface, 54 l. 4 Should be ηv η v 55 l. 6 ηv Should be η v 55 l. 7 ηv Should be η v 55 Corollary Let K be a -complex Should be Let K be a simplicial surface 6 l. - f Should be r 68 l. is subsequent Should be in subsequent 74 l. -7 Remove one definition 75 l. - h (s =/q (s Should be h (s =/q (h(s 78 Exercise 4.3. (ii t ln t t Should be t ln t 79 Exercise smooth curve Should be regular curve 79 l. 4 Remove the second would 79 l. - chose Should be choose 8 Exercise 4.3. Hint: Use Exercise 4..; let G be as in Exercise 4.., and then note that (G c is smooth, and that c c =(G c G c, and this latter expression is smooth 3

4 8 l. -6 c (t = 8 l. -6 c (t c (t = Should be c (t = Should be c (t c (t = 9 l. g (t Should be g (t, 94 l. -7 the claim Should be Exercise l. - the claim Should be Exercise l. 4 loose Should be lose 97 l. 6 counterclockwise Should be clockwise 97 l. 9 is clockwise Should be is counterclockwise Exercise 4.7. This exercise nee the formula computed in Exercise Exercise 4.7. (ii R 3 Should be R Exercise Hint for Exercise 4.7.3: Start as on p. 9, and obtain a formula for c (tintermsof T and N. Then take the inner product with (c 6 l. - rank, and hence D(x φ has rank. Should be rank. Because Dφ is a matrix, it follows that D(x φ hasrank. 7 Figure 5.. A x Should be A xy 7 Figure 5.. A y Should be A yx 8 l. A x Should be A x y 8 l. 3 A y Should be A y x 8 l. 5 A x Should be A x y 4

5 8 Proposition 5..5 Remove and let p M beapoint from the statement of the proposition l. in M Should be in N 3 l. -3 smooth surfaces Should be smooth surfaces in R 3,where we think of U as sitting in the x-y plane in R 3 6 l. -3 missing the north and south poles Should be missing half a great circle from the north to the south pole l. - F (a Should be F (a 4 l. - a curve Should be a smooth curve ( t 6 Equation 5.4. x (t, θ Should be x ( θ 7 l. 4 Remove such that p x(u 7 l. 8 n is a Should be n as a 8 Exercise Remove this exercise 9 l. -7 the basis B Should be the given basis 33 Exercise t +t Should be t t 34 Exercise s τ (s Should be + t τ (s 34 Exercise R cos t cos θ R cos t sin θ R sin t Should be 35 l. - (a, b Should be ( ɛ, ɛ cos t cos θ cos t sin θ sin t 35 l. -8 x c(t Should be x c (t andx c (t respectively 39 Lemma (iii ( v fz(p+f(p( v Z Should be ( v fz(p+f(p( v Z ( ( 4 Exercise 5.6. p = S Should be p = S R 5

6 4 Exercise 5.6. T p S ( R in two places Should be T p S R inbothplaces xy xy 4 l. - f(( Should be Z(( z z 45 l. x i ( p, xk ( px j Should be x i ( p, xk ( px j 46 Equation ( A ij A ij ( gj u Should be i g j u i + g i u j + g i u j g ij u g ij u 47 l. - we turn tangent Should be we turn to tangent 49 l. -6 v = ( /x ( p + ( /x ( p Should be v = x ( p+ x ( p 49 l. -5 v = v = / Should be v = v = 6

7 5 l. The displayed equation is: ( v Z = + Γ k ij( p Zi ( p x k ( p k= j= i= =Γ ( p Z ( p x ( p+γ ( p Z ( p x ( p +Γ ( p Z ( p x ( p = / +( / / +( /4 / = /. /4 The displayed equation should be: v Z = k= j= ( + Γ k ij ( p Z i ( p x k ( p i= =Γ ( p Z ( p x ( p+γ ( p Z ( p x ( p +Γ ( p Z ( p x ( p = / +( / / +( / / =. / 55 l. 7 Exercise Should be Exercise l. -4 F (W Should be f(w 59 l. 7 p such that Should be p such that 59 l. 8 W = f(t M Should be W = x(t M 7

8 6 l. as x Should be as x V 6 l. -5 f x has rank Should be D(f x has rank 6 l. 7 local isometry Should be local isometry, where we think of R as the x-y plane in R 3 6 l. 4 x x Should be (f x (f x 6 Exercise R Should be R {O } 63 Exercise This exercise uses Exercise Exercise Add and that d( M p = I, where M is the identity map on M and I is the identity matrix at the end of the exercise 63 l Should be 3 3matrix 74 l. plane that contain Should be plane that contains 74 l. Π TpM : U T P M Should be Π TpM : U T p M 77 l. = R = R v Should be = R 77 l. 3 I I Should be II = R v 8 l. 5 κ(s+t κ(sτ (s Should be κ(s+t κ(sτ (s tτ (s 83 l. -9 oriented Should be ordered 84 l. 5-6 smooth curve such that c Ω (( ɛ, ɛ is an open subset of Ω M and c Ω ( = p, Should be smooth, unit speed curve such that c Ω (( ɛ, ɛ is an open subset of Ω M, that c Ω ( = p and that c Ω ( = v Ω, 84 l. -4 ˆn c Ω (,c Ω ( Should be (ˆn c Ω (,c Ω ( 85 l. 3 I I p Should be II p 85 l. 4 I I p Should be II p 8

9 85 l. 5 eigenvalue Should be eigenvector 85 l. 6 I I p Should be II p 85 l. -3 (k k sinθ cos θ Should be (k k sinθ cos θ 88 Figure 6.3. (iii K(p < Should be K(p = 9 Exercise 6..6 Should be Exercise Exercise 6..7* Should be Exercise 6.3.7* 96 l. - Theorem 6.6. Should be Theorem Equation Γr u Should be Γr u ( t 3 l. 4 θ ( t Should be θ 3 l. 7 The f is Should be The map f is 3 Example 7.. ( We want find Should be We want to find 3 l. 3 3t 3t Should be 34 l. 4 c j Should be c J 38 l. 4 Section 7.3 Should be Section 7. 3 l. c: ( ɛ, ɛ U Should be c: ( ɛ, ɛ x(u 34 l. 4 b a Should be 34 l. 6 Dc (s (s Should be Dc (s (s 34 l. 6 assume that Dc (s (s > Should be hence Dc (s (s > 34 l. 7 Drop the other case is similar 34 l. 7 Dc (s (s > Should be Dc (s (s > 9 y x 6t 6t

10 34 l. -3 (s η, s + ηc (x(u Should be (s η, s + η c (x(u 35 Equation Dc (s Should be 35 Equation Dc (s Should be 36 l. Dc (s Should be Dc (s Dc (s Dc (s 33 l. exp p (O δp (O 3,T q M Should be exp q (O δp (O 3,T q M 33 l. 3 exp p (O δp (O 3,T q M Should be exp q (O δp (O 3,T q M 335 l. J p (B Should be J p (B 344 l. 4 geodesic Should be non-constant geodesic 347 Figure 8.4. higher D x (x z Should be D x (xz 347 Figure 8.4. lower Dx (x z Should be Dx (xy 35 l. 4 Should be σv σ v 358 l. -4 find surfaces Should be find such surfaces 366 l. 3 Remove the line break 373 l. 7 vertical line Should be horizontal line 373 l. 8 vertical line Should be horizontal line 385 l. - {,...,k} Should be {,...,k} 4 l. - D Should be D 4 l. - D Should be D 4 l. -7 D Should be D 45 l Should be 4..4

11 45 l. 7 h(d, e(a, b Should be h: (d, e (a, b 45 l. 8 h(t Should be h (t 46 l. -4 a curve Should be a smooth curve 46 l. - x ( k w Should be x ( k w 48 l. A = U Should be A = x(u 48 l. (dy p (v Should be (dy p (v 48 l Should be Exercise 6.5. Should be Exercise l. asimplicial Should be a simplicial 48 Kline bottle Should be Klein bottle