Solutions to Selected Exercises

Chapter 2

2.1. For x ∈ (2, 3), let δx = min(x − 2, 3 − x).

2.3. R − {x} = (−∞, x) ∪ (x,∞). If y ∈ (−∞, x), then let δy = x − y so thatδy > 0 and (y− δy, y + δy) ⊂ (−∞, x). Hence (−∞, x) is open, and so is (x,∞)similarly. Hence R − {x} is open by Proposition 2.4, so {x} is closed.

2.5. f−1[−2, 2] = [−2, 2], f−1(2, 18) = (2, 3), f−1[2, 18) = {−1} ∪ [2, 3),f−1[0, 2] = [−

√3, 0] ∪ [

√3, 2].

Chapter 3

3.1. (1) 4. (2) At least 19.

3.2. Yes. No. Yes. No.

3.4. If f is continuous and C ⊂ T is closed, then T −C is open so f−1(T −C) isopen, and f−1(T −C) = S−f−1(C) is open, so f−1(C) is closed. The converseis proved similarly.

3.8. It is not continuous, since {0} is an open subset of Z but f−1(0) = [0, 1)which is not an open subset of R.

3.10. If w2 + x2 + y2 + z2 = 1, then w2 ≤ 1, so −1 ≤ w ≤ 1, and we canwrite w = cos θ. Then x2 + y2 + z2 = sin2 θ, so if x′ = x/ sin θ etc., then(x′)2 + (y′)2 + (z′)2 = (x2 + y2 + z2)/ sin2 θ = 1, i.e., (x′, y′, z′) ∈ S2.

M.D. Crossley, Essential Topology, Springer UndergraduateMathematics Series, DOI 10.1007/978-1-84628-194-5,c© Springer-Verlag London Limited 2010

216 Essential Topology

Chapter 4

4.2. 1) Connected, not Hausdorff. 2) Disconnected, not Hausdorff. 3) Discon-nected, Hausdorff.

4.3. Let f : R → Q be continuous, and let S ⊂ Q be the image of f . If f

is not constant, then S contains at least two points s1, s2. Hence S is discon-nected since between any two rationals there is an irrational number. Takingthe subspace topology on S allows us to restrict f to a surjection R → S from aconnected to a disconnected space, and this cannot happen by Proposition 4.11.

4.6. det is a continuous surjection GL(3,R) → R − {0} and R − {0} is notcompact (since, for example, there is an unbounded function R − {0} → R,namely x �→ 1/x). Hence GL(3,R) is not compact. Both O(3) and SO(3) arecompact; this can be proved using the Heine–Borel theorem.

4.9. It would be connected, compact, but not Hausdorff.

Chapter 5

5.3. Let (a, b) be an interval in R. Since tan is increasing on the range(−π/2, π/2), if tan−1(a) < x < tan−1(b) then a < tan(x) < b. Hencef−1(a, b) = (2 tan−1(a)/π, 2 tan−1(b)/π), which is open. Thus f is continuous.

5.7. We can define homeomorphisms f : C → A by f(x, y, z) = ((1 + z)x, (1 +z)y) and g : A → C by g(x, y) = ( x√

x2+y2, y√

x2+y2,√

x2 + y2 − 1).

5.9. Suppose that S and T are Hausdorff, and (s, t), (s′, t′) are two distinctpoints in S × T . If s �= s′, then there are two non-overlapping open subsetsQ,Q′ of S such that s ∈ Q and s′ ∈ Q′. The sets Q × T and Q′ × T are thennon-overlapping open subsets of S×T , and (s, t) ∈ Q×T and (s′, t′) ∈ Q′×T .If s = s′, then t �= t′ and a similar argument can be used.

Suppose S×T is compact. To show S is compact, let U be any open coveringof S. For each set U ∈ U , the set U × T in S × T is open as both U and T

are open. And every point in S × T will lie in one set U × T ; hence we havea cover of S × T . As this space is compact, we can refine this cover, and thecorresponding U spaces give a refinement of the original cover of S. Similarly,T is compact.

If S × T is connected, then S and T must be connected as the projectionsS × T → S and S × T → T are continuous surjections.

5.11. T 2 and G2.

Solutions to Selected Exercises 217

Chapter 6

6.2. There is only one homotopy class of maps (0, 1) → (0, 1).

6.4. If f is not surjective, then its image lies in Sn −{s} for some point s ∈ Sn.By stereographic projection, Sn − {s} is homeomorphic with Rn, so f can bethought of as a map X → Rn and, hence, homotopic to a constant map as Rn

is convex.

6.6. deg(f ◦ g) = deg(f) ∗ deg(g). Hence deg(f ◦ g) = deg(g ◦ f) and so, byTheorem 6.33, f ◦ g and g ◦ f are homotopic.

6.7. Yes. No. Yes. No.

Chapter 7

7.3. (1) 1. (2) 0. (3) 0. (4) −2. (5) 0.

7.5. Any negative integer can occur as the Euler number of a connected one-dimensional complex, by taking a triangulation of S1 and attaching V shapes to1-simplices, i.e., adding in an extra 0-simplex and two 1-simplices. This reducesthe Euler number by 1 so, by iteration, any negative integer can be achieved.However, in a connected complex, 1 is the largest Euler number possible. Ina disconnected complex, any positive integer is possible by taking a disjointcollection of 0-simplices.

Chapter 8

8.2. If Y is not path connected, then there are two points y0, y1 in Y whichcannot be joined by a path. If f : X → Y is a surjection, then there are pointsx0, x1 ∈ X such that f(xi) = yi. Since X is path connected, there is a pathp : [0, 1] → X with p(0) = x0, p(1) = x1. Then f ◦ p will be a path from y0

to y1 if f is continuous. This cannot happen, by assumption, so f cannot be acontinuous surjection X → Y .

8.4. It is multiplication by n.

Chapter 9

9.2. H0(square;Z/2) = Z/2, Hi(square;Z/2) = 0 for i > 0. H0(annulus;Z/2) =Z/2, H1(annulus;Z/2) = Z/2 and Hi(annulus;Z/2) = 0 for i > 1.

218 Essential Topology

Chapter 10

10.1. H0(R2 − {0}) = Z, H1(R2 − {0}) = Z, Hi(R2 − {0}) = 0 otherwise.H0(R3 − {0}) = Z, H2(R3 − {0}) = Z, Hi(R3 − {0}) = 0 otherwise.

10.2. H0(R2 − S0) = Z, H1(R2 − S0) = Z ⊕ Z, Hi(R2 − S0) = 0 otherwise.

10.5. The sequence looks like

· · · 0 → H2(X) → Z → Z ⊕ Z → H1(X) → Z → Z ⊕ Z → Z = H0(X).

Working from the right-hand end, we see that the function H1(X) → Z mustbe 0, so we have 0 → H2(X) → Z → Z ⊕ Z → H1(X) → 0. Hence H2(X) is asubgroup of Z, i.e., either 0 or Z. If H2(X) = 0, then H1(X) = Z⊕Z/m for someinteger m, or H1(X) = Z/m⊕Z/n for some integers m,n. In fact, however, itcan be shown geometrically that the map H1(U ∩ V ) → H1(U) ⊕ H1(V ) is 0,so H1(X) = Z ⊕ Z and H2(X) = Z.

10.7. Hi(Q) = 0 for i > 0 and H0(Q) = C0(Q) is a free Abelian group withone generator for each rational number.

Bibliography

[1] M.F. Atiyah, K-Theory, Benjamin, 1966.

[2] Glen E. Bredon, Topology and Geometry, Springer-Verlag, 1993.

[3] S. Eilenberg, N.E. Steenrod, Foundations of Algebraic Topology, PrincetonUniv. Press, 1954.

[4] William Fulton, Algebraic Topology: A First Course, Springer-Verlag, 1995.

[5] Allen Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002.

[6] Dale Husemoller, Fibre Bundles, third ed., Springer-Verlag, 1994.

[7] William S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag,1991.

Index

Δn, see standard n-simplexΩ, 205Σ, 203a, 150[f ], 130(a, b), 8, 56, 61, 98[0, 1], 24, 39, 50, 61–63, 79–81, 86, 93,

97, 102, 200, 207[S, T ], 95[a, b], 8CX, 201C×, 23, 99DX, 201D2, 147Dn, 60, 110, 113, 114, 138, 172, 191e, 32, 42, 64, 207GL(3,R), 26, 29, 30, 42G2, 25I, 24, 93O(3), 26, 30, 42, 69Q, 22, 53, 71, 144R, 16– R − {0}, 69– R ∪ {∞}, 59– and homeomorphisms, 56, 59, 61–63– and homotopy, 93, 97, 102– basis for, 31– properties of, 37, 38, 47RPn, 27– and bundles, 206, 207, 210– and homeomorphisms, 58, 82, 84, 210– properties of, 42, 84R2, 19, 59, 71, 93R2 − {0}, 23, 74, 139

Rn, 21, 40, 48, 50Rn − {0}, 23, 40SO(3), 26, 27, 30, 43, 69, 84, 210S0, 37, 61, 62, 67, 69, 99, 207S1, 22, 147, 202– and fibre bundles, 207, 209– and homeomorphisms, 27, 58, 59,

62–64, 73, 74, 77, 79, 82– and homotopy, 93, 99, 101, 139– and vector fields on D2, 113– as simplicial complex, 117, 119–123– homology of, 154, 163, 190– homotopy groups of, 133, 209– maps D2 → S1, 111, 138– maps S1 → S1, 102–110– properties of, 42, 46S2, 22, 202– and bundles, 135, 211, 212– and homeomorphisms, 59, 77, 79, 123– and vector fields, 113, 115– as simplicial complex, 122– homology of, 156, 163, 191Sn, 23, 211– and SO(3), 30– and bundles, 207– and connectivity, 42, 85– and homeomorphisms, 79, 82– and homotopy groups, 129, 135, 146Sn, 129T 2, 24– and homeomorphisms, 74, 81, 84– as simplicial complex, 121, 123– homology groups of, 155, 158, 164, 193– homotopy groups of, 136

222 Index

Z, 21, 43, 61, 144#, see addition of maps S1 → S1

�, see disjoint union∼=, see homeomorphic�, see homotopic⊗, 165×, see Cartesian product∨, see wedge product

Abelian group, classification of, 165acyclic, 155, 171, 173acyclic models, method of, 185addition, 19– of maps S1 → S1, 128– of maps Sn → X, 130adjoint functors, 205annulus, 74, 98, 117, 119, 121, 163arc, open, 33

ball, open, 19barycentre, 180barycentric coordinates, 118barycentric subdivision, 183barycentric vertex, 180base point, 128base space, 207basis, 31, 32, 72bottle, Klein, 81, 158, 164, 165boundary– in simplicial chain complex, 153, 162– in singular chain complex, 170– of a simplex, 118– of simplex, 150– of singular simplex, 168boundary operator– simplicial, mod 2, 151– simplicial, integral, 161– singular, 169bounded– function, 44– subset, 48breathing space, 9Brouwer’s fixed-point theorem, 110, 191bundle– fibre, see fibre bundle– Hopf, 135, 211– normal, 212– tangent, 212– vector, 211

Cartesian product, 71chain, 161– simplicial, mod 2, 151– simplicial, integral, 161

– singular, 168chain complex, 153– for simplicial homology, mod 2, 153– for simplicial homology, integral, 162– for singular homology, 170chain homotopy, 175chain map, 174chain splitting, 185characteristic, Euler, see Euler numbercircle, see S1

classification of surfaces, 125classifying space, 134closed interval, 8closed set, 11, 16– and continuity, 66closed surface, 125coffee, stirring, 112commutative diagram, 83compact-open topology, 205compactness, 45– and disjoint unions, 70– and homeomorphisms, 62, 65– and homotopy equivalence, 101– and product spaces, 75– and quotient spaces, 84– and simplicial complexes, 119– Heine–Borel Theorem, 48complement, 11, 16component– path, 144composition, 18, 57, 173cone, 201connected, 38, 41, 68– n-connected, 135, 204– components, 71– path connected, 141connecting homomorphism, 190connectivity– and disjoint unions, 70– and homeomorphisms, 62– and homotopy equivalence, 100– and product spaces, 75– and quotient spaces, 85continuity, 8– ε-δ definition, 8– at a point, 8– by closed sets, 66– by open sets, 13– in a topological space, 17– of polynomials, 20– using bases, 32continuous, 8, 17contractible, 97, 98

Index 223

contravariant, 214convex, 93, 118, 172– and path connectivity, 173– homotopy groups of convex set, 133– singular homology of convex set, 173cube, 60cycle– simplicial, 153, 162– singular, 170cylinder, 25, 73, 80, 125, 136, 206, 212

degree, 105, 154determinant, 29, 42diagonal map, 20diagram– commutative, 83diameter, 106dimension, 154disc, 60, 77disconnected, 38, 41, 71discrete space, 22, 43discrete topology, 16, 18disjoint union, 67– and compactness, 70– and connectivity, 70– and Hausdorff property, 70– and singular homology, 172– continuous maps from, 68– continuous maps to, 68division, 24domain splitting, 106, 146, 180, 186, 208double cone, 201double point, 52, 63doughnut, see tea cup, 61

Eilenberg–MacLane space, 134– uniqueness of, 134Eilenberg–Steenrod axioms for homol-

ogy, 196Eilenberg–Steenrod theorem, 196equivalence relation, 80equivalence, homotopy, 96equivalent, 212Euclidean plane, see R2, 19Euclidean space, see Rn

Euler characteristic, see Euler numberEuler number, 120, 123– and homology, 158– and homotopy equivalence, 123, 196– classification of surfaces, 125– independence of triangulation, 123,

196– of S1, 120– of S2, 122

– of T 2, 121– of annulus, 121– of square, 121exact sequence, 188, 209exponential map, 32, 64, 102

face, 118fibre, 207fibre bundle, 207– base space, 207– exact sequence of, 209– total space, 207– trivial, 207figure of eight, 141, 156, 192, 200finite refinement, 45fixed-point set, 50fixed-point theorem– Brouwer’s, 110– for [0, 1], 39– for n-disc, 191– for disc, 110Freudenthal suspension theorem, 204function, see mapfunctor, adjoint, 205fundamental group, 140– of a union, 145

general linear group, see GL(3,R)general position, 118genus, 25gluing, 79gluing lemma, 87, 104, 127, 130, 132, 145group– fundamental, see fundamental group– general linear, see GL(3,R)– homology, see homology groups– homotopy, see homotopy groups– orthogonal, see O(3)– special orthogonal, see SO(3)– topological, 30

hairy ball theorem, 113half-open interval, 8, 64hat ( ) notation, 150Hausdorff property, 50, 65– and disjoint unions, 70– and homeomorphisms, 62, 65– and product spaces, 75– and quotient spaces, 85Heine–Borel Theorem, 48, 186hole counting, 149homeomorphic, 56, 97homeomorphism, 56, 65– and compactness, 62

224 Index

– and connectivity, 62– and Hausdorff property, 62– and homotopy groups, 140homologous, 154homology– Eilenberg–Steenrod axioms, 196– relation between simplicial and

singular, 195– simplicial, mod 2, 162– simplicial, mod 2, 154– simplicial, integral, 162– singular, 170homology groups– and Euler number, 158– of loop space, 205– of suspension, 202, 204– of wedge product, 201– relation between simplicial and

singular, 195– simplicial– – 0th, 156, 164– – mod 2, 162– – mod 2, 154– – and path components, 156, 164– – integral, 162– – of S1, 155, 163– – of S2, 156, 163– – of T 2, 155, 164– – of annulus, 163– – of figure of eight, 156– – of Klein bottle, 164, 165– – of square, 155, 163– – relation between mod 2 and

integral, 165– singular, 170– – 0th, 171– – and disjoint unions, 172– – and homotopies, 175– – and homotopy groups, 194– – and path components, 171– – induced homomorphism on, 173– – of S1, 191– – of S2, 191– – of T 2, 194– – of convex set, 173– – of disc, 172– – of figure of eight, 193homotopic, 92, 96, 108homotopy, 92– and singular homology, 175– chain, 175– pointed, 129homotopy classes, 95

– of maps S1 → S1, 102–110homotopy equivalence, 96– and compactness, 101– and connectivity, 100– and homotopy groups, 139, 147– and the Euler number, 123, 196homotopy exact sequence of a fibre

bundle, 209homotopy groups, 133– 0th, 141– and fibre bundles, 208– and homeomorphisms, 140– and homotopy equivalence, 139, 147– and path components, 144– and singular homology, 194– induced homomorphism on, 137– of Q, 144– of RPn, 210– of Z, 144– of S1, 133, 209– of S2, 135, 211– of Sn, 135, 146– of T 2, 136– of convex set, 133– of cylinder, 136– of loop space, 205– of suspension, 204– stable, 204homotopy lifting, 106, 208Hopf bundle, 135, 211Hopf map, 135, 211Hurewicz homomorphism, 195Hurewicz theorem, 195

indiscrete topology, 16, 18induced homomorphism– from a chain map, 174– on homotopy groups, 137– on singular homology, 173induced topology, see subspace topologyintegers, set of, 21, 43, 144integral homology group, 162interior, 118intermediate value theorem, 39intersection of open sets, 10interval– closed, 8, 50– connectivity, 39– half-open, 8, 64– open, 8, 47invariant, topological, 89inversion of matrices, 29invertible matrices, set of, 26

Index 225

K-theory, 213kernel, 209Klein bottle, 81, 158, 164, 165

Lebesgue lemma, 107lifting– homotopies, 106– in a bundle, 208– paths, 102long exact sequence of a fibre bundle,

209loop space, 205– homology groups of, 205– homotopy groups of, 205

map, 17– bounded, 44– composition, 18, 57– continuous, 8, 17– exponential, 32– open, 65– pointed, 129, 199– restriction, 28matrix inversion, 29matrix multiplication, 29Mayer–Vietoris sequence, 190, 202– applied to S1, 191– applied to S2, 191– applied to T 2, 193– applied to figure of eight, 193Mayer–Vietoris Theorem, 190meteorology, 115method of acyclic models, 185metric spaces, 21Mobius band, 26, 80, 124, 206, 207, 212mod 2 homology, 162multiplication, 20– of matrices, 29

neighbourhood, open, 9non-orientable, 125normal bundle, 212null-homotopic, 133

one-point union, see wedge product, 199open arc, 33open ball, 19open cover, 44, 45– refinement, 45open interval, 8open map, 65open neighbourhood, 9open set, 9, 15orientable, 125

oriented simplex, 160orthogonal group, see O(3)

pair, topological, 129path components, 144– and simplicial homology, mod 2, 156– and simplicial homology, integral, 164– and singular homology, 171path connected, 141path lifting, 102, 208permutation, sign of, 183plane– Euclidean, see R2

– projective, see RPn

pointed homotopy, 129pointed map, 129, 199pointed space, 128, 199polynomial, 20preimage, 11product, 71, 206– and compactness, 75– and connectivity, 75– and Hausdorff property, 75– as fibre bundle, 207– Cartesian, 71– continuous maps to, 73– homotopy groups of, 136product topology, 72projection, stereographic, 59, 114, 147projective space, see RPn

proper subset, 103property, topological, 89pull-back, 213

quotient group, 153quotient space, 78, 80– and compactness, 84– and connectivity, 85– and Hausdorff property, 85– continuous maps from, 83, 84

rationals, set of, 22, 53, 71, 144real line, see R, 16, 38real line with a double point, 52, 63, 85real projective space, see RPn

reciprocal map, 23refinement, of open cover, 45relation, equivalence, 80restriction, 28Riemann sphere, 60rising sun, 67rotation, 30

sequence, exact, 188, 209

226 Index

sign, of a permutation, 183simplex, 118, 150, 168– boundary, 118, 150– face, 118– interior, 118– oriented, 160– singular, 168– standard, 167– subsimplex, 118– vertices, 118simplicial boundary operator, 151, 161simplicial chain complex, 153simplicial complex, 119, 168– and compactness, 119– and homotopy equivalence, 147simplicial homology, see homology

groups, simplicialsingular n-chains, 168singular n-simplex, 168singular boundary operator, 169singular chain complex, 170singular homology, see homology groups,

singularspace, 15– Euclidean, see Rn

– pointed, 128, 199– projective, see RPn

special orthogonal group, see SO(3)sphere, see S0, S1, S2, S3, Sn

– Riemann, 60spiral, 33, 102, 124square, 60, 117, 119, 121, 155, 163stable homotopy group, 204standard n-simplex, 167stereographic projection, 59, 114, 147stirring coffee, 112subdivision, 180– barycentric, 183subsimplex, 118subspace, 21subspace topology, 21, 24, 28surface, 123– and Euler number, 123– classification of, 125– closed, 125– non-orientable, 125– orientable, 125surface of genus two, 25suspension, 203– Freudenthal’s theorem, 204– homology groups of, 202, 204– homotopy groups of, 204

systems of equations, 192

tangent bundle, 212tangential vector field, see vector fieldtea cup, see doughnut, 61tensor product, 165topological group, 30topological invariant, 89topological pair, 129topological property, 89topological space, 15topology, 15– basis, 31– discrete, 16, 18– indiscrete, 16, 18– product, 72– subspace, 21, 24, 28Tor, 165torsion, 165torus, see T 2

total space, 207train tracks, 67, 75triangulable, 121triangulation, 121– of S1, 121, 122– of S2, 122trivial fibre bundles, 207Tychonov’s theorem, 76

union, 145, 185– disjoint, 67– of open sets, 9– one-point, 199universal coefficient theorem, 165unreduced suspension, 203

Van Kampen theorem, 145, 147, 180,186, 188

vector bundle, 211vector field, 112– on S2, 113– zero of, 113vertices, 118

weak topology, see subspace topologywedge product, 195, 199– homology groups of, 201well defined, 83, 137Whitehead theorem, 147wind, 112, 115winding number, 105