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    Hatcher, Algebraic Topology

    SRC-TEXT-HAT02

    Area
    Topology
    Status
    draft
    Source
    Contents

    437 problems.

    1.1

    1. Cancellation property for composition of paths
    2. Basepoint-change homomorphism depends only on homotopy class
    3. \(\pi_1(X)\) abelian iff basepoint-change homomorphisms depend only on endpoints
    4. Locally star-shaped paths are piecewise linear up to homotopy
    5. Three equivalent conditions for simply-connected spaces
    6. Natural map from \(\pi_1\) to unbased homotopy classes of maps \(S^1 \to X\)
    7. Homotopy of Dehn twist on \(S^1 \times I\) stationary on one boundary only
    8. Borsuk–Ulam theorem for the torus
    9. Borsuk–Ulam bisects three compact sets in \(\mathbb{R}^3\)
    10. Explicit homotopy for commuting loops in product spaces
    11. Inclusion of path-component induces isomorphism on \(\pi_1\)
    12. Every homomorphism \(\pi_1(S^1) \to \pi_1(S^1)\) is induced by a map
    13. Surjectivity of inclusion-induced map on \(\pi_1\) iff paths in \(X\) with endpoints in \(A\) are homotopic to paths in \(A\)
    14. Isomorphism \(\pi_1(X \times Y) \approx \pi_1(X) \times \pi_1(Y)\) given by projections
    15. Naturality square for basepoint-change homomorphisms
    16. No retractions in six specific cases
    17. Infinitely many nonhomotopic retractions \(S^1 \lor S^1 \to S^1\)
    18. Attaching cells of dimension \(\geq 2\) gives surjection on \(\pi_1\)
    19. Loops in 1-dimensional CW complexes are homotopic to monotone edge sequences
    20. Loops extending to loop of maps \(X \to X\) lie in the center of \(\pi_1(X)\)

    1.2

    1. Free product of nontrivial groups has trivial center and only conjugates of finite-order elements have finite order
    2. Union of convex open sets with triple intersections nonempty is simply-connected
    3. Complement of a finite set of points in \(\mathbb{R}^n\) is simply-connected for \(n \geq 3\)
    4. Fundamental group of complement of \(n\) lines through origin in \(\mathbb{R}^3\)
    5. Fundamental group of planar graph is free on boundary loops of complementary regions
    6. Complement of closed discrete subspace of \(\mathbb{R}^n\) simply-connected for \(n \geq 3\)
    7. Fundamental group of \(S^2\) with north and south poles identified
    8. Fundamental group of two tori identified along a circle
    9. Surface of genus \(g\) retracts onto nonseparating circle but not separating circle
    10. Loop nullhomotopic in \(D^2 \times I\) but not in complement of two arcs
    11. Fundamental group of mapping torus of map on wedge of circles
    12. Fundamental groups of Klein bottle and related spaces
    13. Two ways to identify boundary circles of disk with two holes give non-isomorphic fundamental groups
    14. Cube with face identifications via screw motions has quaternion group as fundamental group
    15. Canonical map from loop complex \(L(X)\) to \(X\) induces isomorphism on \(\pi_1\)
    16. Fundamental group of surface of infinite genus is free on infinitely many generators
    17. \(\pi_1(\mathbb{R}^2 - \mathbb{Q}^2)\) is uncountable
    18. Suspension vs reduced suspension of convergent sequence; reduced suspension of contractible space need not be contractible
    19. Union of spheres of radius \(1/n\) centered at \((1/n, 0, 0)\) is simply-connected
    20. Fundamental group of union of tangent circles equals that of infinite wedge, and spaces are homotopy equivalent but not homeomorphic
    21. Join of path-connected space with any nonempty space is simply-connected
    22. Wirtinger presentation for fundamental group of knot complement

    1.3

    1. Restriction of a covering space to a subspace
    2. Product of covering spaces
    3. Compactness lifts through finite-fiber covering spaces
    4. Simply-connected covering space of a sphere with diameter
    5. The harmonic broom has no simply-connected covering space
    6. Composition of covering spaces need not be a covering space
    7. Local path-connectedness is necessary for the lifting criterion
    8. Simply-connected covers of homotopy equivalent spaces
    9. Maps to \(S^1\) from spaces with finite fundamental group
    10. Connected covering spaces of \(S^1 \vee S^1\)
    11. Common covering space without a common cover
    12. Covering space of \(S^1 \vee S^1\) for a normal subgroup
    13. 27-sheeted covering space from the subgroup of cubes
    14. Connected covering spaces of \(\mathbb{RP}^2 \vee \mathbb{RP}^2\)
    15. Universal cover restricted to a subspace
    16. Lifting covering spaces through composites
    17. Normal covering spaces from group extensions
    18. Universal abelian covering space
    19. Abelian covering spaces of closed orientable surfaces
    20. Nonnormal covering spaces of the Klein bottle
    21. Torus with Möbius band attached
    22. Product of covering space actions
    23. Free properly discontinuous actions are covering space actions
    24. Covering spaces from subgroups of deck transformation groups
    25. Non-Hausdorff orbit space from a covering space action
    26. Covering spaces and fundamental groups
    27. Two actions of the fundamental group on the fiber
    28. Fundamental group of an orbit space
    29. Conjugate subgroups give homeomorphic orbit spaces
    30. Cayley graph of \(\mathbb{Z} * \mathbb{Z}_2\)
    31. Normal covering spaces of wedge sums are Cayley graphs
    32. Covering spaces of CW complexes from 1-skeleton coverings
    33. Quotient graphs and freeness of subgroups

    1.A

    1. Weak topology on a graph with finite vertex degree is a metric topology
    2. Connected graph retracts onto any connected subgraph
    3. Euler characteristic and rank of fundamental group of a finite graph
    4. Loop \(Y\) in finite graph is element of a basis for \(\pi_1\)
    5. Maps \(fg = \mathrm{id}\) without inducing isomorphisms on \(\pi_1\)
    6. Free generators for commutator subgroup of free group via covering space
    7. Nontrivial normal subgroup of infinite index in finitely generated free group is not finitely generated
    8. Finitely generated group has finitely many subgroups of given finite index
    9. Index \(n\) subgroup has at most \(n\) conjugates; existence of normal subgroup of finite index contained in \(H\)
    10. Finite connected subgraph of covering space extends to finite covering space
    11. Free groups are residually finite
    12. Separating an element from a subgroup by a finite index overgroup in a free group
    13. Element of free group lies in a basis for some finite-index subgroup
    14. Existence of maximal trees is equivalent to the Axiom of Choice

    1.B

    1. Free simplicial actions are covering space actions
    2. Maps to \(K(G,1)\) from spaces with trivial \(\pi_1\)-to-\(G\) homomorphisms
    3. Graph products of trivial groups are free
    4. Amalgamated products via van Kampen
    5. Baumslag–Solitar group as a graph of groups
    6. Graph of groups with universal cover \(T \times \mathbb{R}\)
    7. Bipartite realization of graph products
    8. Finite graph products of finitely generated groups
    9. Graph products of finite groups have free subgroups of finite index

    2.1

    1. Quotient \(\Delta\)-complex of 2-simplex with two edges identified
    2. Edge identifications of \(\Delta^3\) producing Klein bottle, torus, \(S^2\), \(\mathbb{RP}^2\)
    3. \(\Delta\)-complex structure on \(\mathbb{RP}^n\) from \(S^n\)
    4. Simplicial homology of triangular parachute
    5. Simplicial homology of Klein bottle
    6. Simplicial homology of \(\Delta\)-complex from \(n+1\) 2-simplices with cyclic edge identifications
    7. \(\Delta\)-complex structure on \(S^3\) with a single 3-simplex
    8. Simplicial homology of lens space from cyclic tetrahedra
    9. Homology of \(\Delta\)-complex from \(\Delta^n\) with all faces of same dimension identified
    10. Quotient of disjoint 2-simplices with edge identifications is always a surface
    11. Retract induces injective map on homology
    12. Chain homotopy is an equivalence relation
    13. Homotopic maps induce equal maps on reduced homology
    14. Short exact sequences of cyclic groups
    15. Middle term of exact sequence is zero iff outer maps have expected kernel/cokernel
    16. When relative homology \(H_0(X,A)\) and \(H_1(X,A)\) vanish
    17. Relative homology of surfaces with point and circle subsets
    18. \(H_1(\mathbb{R}, \mathbb{Q})\) is free abelian
    19. Homology of subspace of \(I \times I\) with rational first coordinate
    20. Reduced homology of suspension is shift of reduced homology
    21. Explicit chain maps inducing suspension isomorphism
    22. Homology of finite-dimensional CW complexes by induction on dimension
    23. Second barycentric subdivision of \(\Delta\)-complex is simplicial
    24. \(n\)-simplices in barycentric subdivision defined by inequalities on barycentric coordinates
    25. Explicit noninductive formula for barycentric subdivision operator
    26. \(H_1(X, A)\) not isomorphic to \(\tilde{H}_1(X/A)\) for shrinking wedge of circles
    27. Map that is homotopy equivalence on \(X\) and \(A\) induces isomorphism on relative homology
    28. Local homology groups of cone on 1-skeleton of \(\Delta^3\)
    29. \(S^1 \times S^1\) and \(S^1 \lor S^1 \lor S^2\) have isomorphic homology but different universal covers
    30. Five-lemma: if all but one map in commutative diagram are isomorphisms, so is the remaining one
    31. Five-lemma example with zero outer maps and nonzero middle map

    2.2

    1. Brouwer fixed point theorem via degree theory
    2. Map of \(S^{2n}\) has fixed point or antipodal point; \(\mathbb{RP}^{2n}\) maps have fixed points
    3. Degree zero map has fixed and antipodal points; Hairy Ball Theorem
    4. Surjective map \(S^n \to S^n\) of degree zero
    5. Any two reflections of \(S^n\) are homotopic through reflections
    6. Every map \(S^n \to S^n\) can be homotoped to have a fixed point
    7. Induced map on local homology of invertible linear transformation is \(\pm 1\) according to sign of determinant
    8. Degree of one-point compactification of polynomial equals degree as polynomial; local degree at roots equals multiplicity
    9. Homology of four specific 2-complexes
    10. Homology of \(S^2\) with equatorial antipodal identification, and \(S^3\) with equatorial \(S^2\) antipodal identification
    11. Homology of cube with opposite faces identified via one-quarter twist (quaternion group)
    12. Quotient map \(S^1 \times S^1 \to S^2\) inducing isomorphism on \(H_2\); all maps \(S^2 \to S^1 \times S^1\) nullhomotopic
    13. 2-complex from \(S^1\) with two 2-cells of degrees 2 and 3
    14. Even maps \(S^n \to S^n\) must have even degree; degree zero when \(n\) even
    15. \(H_n(X^n)\) is free as kernel of cellular boundary map
    16. Ranks of simplicial chain groups of \(\Delta^n\) and homology of skeleta
    17. Cellular map induces chain map compatible with singular homology isomorphism
    18. Relative cellular chain complex for CW pair
    19. \(H_i(\mathbb{RP}^n/\mathbb{RP}^m)\) by cellular homology
    20. Euler characteristic is multiplicative for products of finite CW complexes
    21. Euler characteristic of union of subcomplexes
    22. Euler characteristic of \(n\)-sheeted covering space is \(n\) times base
    23. If \(M_g\) covers \(M_h\) then \(g = n(h-1)+1\)
    24. 1-skeleton of \(S^2\) from polygon identifications cannot be certain graphs
    25. Unique Euler characteristic function on finite CW complexes
    26. Retraction of \(X \cup CA\) onto \(X\) iff \(A\) contractible in \(X\); homology decomposition
    27. Splitting of chain short exact sequences does not give splitting of homology
    28. Mayer–Vietoris computations for torus with Möbius band and \(\mathbb{RP}^2\) with Möbius band
    29. Homology of double of genus-\(g\) surface via Mayer–Vietoris
    30. Homology of mapping tori via long exact sequence
    31. Mayer–Vietoris gives \(\tilde{H}_n(X \vee Y) \approx \tilde{H}_n(X) \oplus \tilde{H}_n(Y)\) under neighborhood condition
    32. Mayer–Vietoris gives suspension isomorphism \(\tilde{H}_n(SX) \approx \tilde{H}_{n-1}(X)\)
    33. Nerve-type vanishing: \(\tilde{H}_i(X) = 0\) for \(i \geq n-1\) when intersections are acyclic
    34. Nonorientable surface or complex with torsion in \(H_1\) cannot embed in \(\mathbb{R}^3\) with mapping cylinder neighborhood
    35. Künneth formula \(H_i(X \times S^n) \approx H_i(X) \oplus H_{i-n}(X)\)
    36. Elementary derivation of Mayer–Vietoris sequence in simplicial homology
    37. Algebraic lemma: diagram with two exact rows and isomorphisms every third vertical map yields long exact sequence
    38. Relative Mayer–Vietoris sequences for CW pairs from algebraic lemma
    39. Universal coefficient short exact sequence for homology with \(\mathbb{Z}_n\) coefficients
    40. Euler characteristic equals alternating sum of dimensions of \(H_n(X; F)\) over a field
    41. Finite group of homeomorphisms of graph injects into \(GL_n(\mathbb{Z})\) acting on \(H_1\)
    42. Splitting of chain complexes and universal coefficient formula for \(H_n(X; G)\)

    2.3

    1. Torsion subgroup and mod-torsion functor do not define homology theories
    2. Candidate homology theory \(\Pi_i \tilde{H}_i(X)/\oplus_i \tilde{H}_i(X)\) satisfies all axioms except wedge axiom
    3. Reduced homology theory vanishes on point; suspension isomorphism follows
    4. Wedge axiom follows from other axioms for finite wedge sums

    2.B

    1. Homology of complements of wedges and disjoint unions in spheres
    2. Homology of complements of finite graphs in spheres
    3. Complement of a disk pair in a sphere pair
    4. Linking numbers via Mayer–Vietoris
    5. Homology of complements of spheres in spheres
    6. Alexander horned sphere with non-simply-connected complement
    7. Modified Alexander horned sphere
    8. Real division algebras and the Borsuk–Ulam theorem
    9. Transfer sequence for trivial coverings
    10. Homology of \(\mathbb{RP}^\infty\) via transfer
    11. Homology of \(X \times \mathbb{RP}^\infty\) via transfer

    2.C

    1. Minimum number of edges for simplicial map \(S^1 \to S^1\) of degree \(n\)
    2. Lefschetz fixed point theorem: map \(S^n \to S^n\) has fixed point unless degree equals antipodal degree
    3. Conjugate-linear map on \(\mathbb{C}^{2k}\) induces fixed-point-free map on \(\mathbb{CP}^{2k-1}\)
    4. Lefschetz number of simplicial homeomorphism equals Euler characteristic of fixed point set
    5. Fixed-point-free homotopies of reflected surfaces
    6. Even-genus analog of Lefschetz example: replace central torus by sphere with antipodal map
    7. Lefschetz fixed point theorem with field coefficients
    8. At most countably many homotopy classes of maps between simplicial spaces
    9. Countably many homotopy types of finite CW complexes

    3.1

    1. Functoriality of \(\operatorname{Ext}(H,G)\) in both variables — Exercise 3.1.1
    2. Multiplication by \(n\) induces multiplication by \(n\) on \(\operatorname{Ext}\) — Exercise 3.1.2
    3. \(\operatorname{Ext}^n_{\mathbb Z_4}(\mathbb Z_2,\mathbb Z_2)\) is nonzero for all \(n\) — Exercise 3.1.3
    4. Homology defined by \(\operatorname{Hom}(G,C_n(X))\) — Exercise 3.1.4
    5. 1-cocycles as functions on paths and \(H^1(X;G)\to\operatorname{Hom}(\pi_1(X),G)\) — Exercise 3.1.5
    6. Simplicial cohomology of the torus, \(\mathbb{RP}^2\), and the Klein bottle — Exercise 3.1.6
    7. \(\operatorname{Hom}(H_n(X),\mathbb Z)\) is not a cohomology theory — Exercise 3.1.7
    8. Cohomology of spheres, good pairs, and retracts — Exercise 3.1.8
    9. Degree \(d\) maps act on \(H^n(S^n;G)\) by multiplication by \(d\) — Exercise 3.1.9
    10. Cohomology of lens spaces with various coefficients — Exercise 3.1.10
    11. Moore spaces show the universal coefficient splitting is not natural — Exercise 3.1.11
    12. Vanishing of \(H^k(X,X^n;G)\) for \(k\le n\) — Exercise 3.1.12
    13. Maps to \(K(G,1)\) represent \(H^1(X;G)\) — Exercise 3.1.13

    3.2

    1. Cup product structure on closed orientable surfaces — Exercise 3.2.1
    2. Cup products vanish on unions of contractible open sets — Exercise 3.2.2
    3. Maps \(\mathbb{RP}^n\to\mathbb{RP}^m\) for \(n>m\) and the Borsuk–Ulam theorem — Exercise 3.2.3
    4. Fixed points of self-maps of \(\mathbb{CP}^n\) — Exercise 3.2.4
    5. Cohomology ring \(H^*(\mathbb{RP}^\infty;\mathbb Z_{2k})\) — Exercise 3.2.5
    6. The \(d\)-th power map on \(\mathbb{CP}^n\) in cohomology — Exercise 3.2.6
    7. \(\mathbb{RP}^3\) is not homotopy equivalent to \(\mathbb{RP}^2\vee S^3\) — Exercise 3.2.7
    8. Spaces with isomorphic integral but distinct \(\mathbb Z_p\) cohomology rings — Exercise 3.2.8
    9. \(H^*(X;\mathbb Z_p)\cong H^*(X;\mathbb Z)\otimes\mathbb Z_p\) as rings when homology is free — Exercise 3.2.9
    10. Failure of the Künneth cross product for infinite discrete spaces — Exercise 3.2.10
    11. Maps \(S^{k+\ell} \to S^k \times S^\ell\) are trivial on top homology
    12. Spaces with isomorphic cohomology but different homotopy type
    13. Cohomology of \(\mathbb{CP}^\infty/\mathbb{CP}^1\)
    14. Surjection from \(\mathbb{CP}^\infty\) to \(\mathbb{RP}^\infty\) on cohomology
    15. Poincaré series and products
    16. Torsion in products of CW complexes
    17. Cup product nondegenerality on surfaces

    3.3

    1. Nonorientable 1-manifolds without Hausdorff
    2. Deleting a point does not affect orientability
    3. Covering spaces of orientable manifolds
    4. Quotients by orientation-preserving actions
    5. Orientability of products
    6. Connected sums of manifolds
    7. Degree 1 maps to \(S^n\)
    8. Degree as sum of local signs
    9. Degree of covering space projections
    10. Degree 1 implies surjection on \(\pi_1\)
    11. Degree 1 maps between surfaces
    12. Commutator length in free groups
    13. No retraction onto subsurface of high genus
    14. Shrinking wedge of circles
    15. Local homology as sections of a covering
    16. Associativity of cap product
    17. Direct limits commute with homology
    18. Direct limits of torsionfree groups
    19. Countability of homology of open subsets of \(\mathbb{R}^n\)
    20. Compactly supported \(H^0\) of noncompact spaces
    21. Compactly supported cohomology and one-point compactification
    22. Compactly supported cohomology and \(\times \mathbb{R}\)
    23. Simplicial vs. singular compactly supported cohomology
    24. Homology of closed 3-manifolds
    25. Torsionfree \(H_{k-1}\) implies torsionfree \(H_k\)
    26. Cup product on connected sums of products
    27. Normal form for skew-symmetric forms over \(\mathbb{Z}\)
    28. Nondegeneracy on large subspaces
    29. Retractions of surfaces onto graphs
    30. Boundary of orientable manifold is orientable
    31. Boundary map sends fundamental class to fundamental class
    32. Compact manifolds do not retract onto boundary
    33. Boundary of contractible manifold is a homology sphere
    34. Poincaré duality diagram for manifolds with boundary
    35. Poincaré duality for noncompact manifolds with boundary

    3.A

    1. Euler characteristic is independent of coefficient field
    2. Tor with \(\mathbb{Q}/\mathbb{Z}\) and the torsion subgroup
    3. Rational and mod-\(p\) cohomology determine integral homology
    4. Tensor product and Tor commute with direct limits
    5. Tor vanishes for torsionfree modules
    6. Tor is always a torsion group

    3.B

    1. Homology and cohomology of \(\mathbb{RP}^m \times \mathbb{RP}^n\)
    2. Chain homotopy as a chain map on the cone
    3. Non-naturality of the Künneth splitting
    4. Cross product of fundamental classes
    5. Slant products

    3.C

    1. H-space structure and strict identity
    2. Retracts of H-spaces
    3. Path-components of homotopy-associative H-spaces
    4. Lifting H-space structure to universal covers
    5. Fundamental group of an H-space is abelian
    6. Spheres as H-spaces via \(J_2(S^n)\)
    7. Primitive elements of \(\mathbb{Z}_p[x]\)
    8. Tensor product of Hopf algebras
    9. Euler characteristic of finite H-spaces
    10. Product of maps in an H-space
    11. Pontryagin ring of the torus
    12. Pontryagin product in lens space homology
    13. Self-duality of Hopf algebras
    14. Coproduct from the diagonal map
    15. Pontryagin ring from polynomial cohomology
    16. Classification of Hopf algebras over \(\mathbb{Z}\)

    3.D

    1. Topological groups with finite CW structure are manifolds
    2. Fundamental group of \(SO(n)\)
    3. Pontryagin ring of \(SO(5)\)

    3.E

    1. Cohomology of \(K(\mathbb{Z}_m, 1)\) with \(\mathbb{Z}_k\) coefficients
    2. Homotopy classification of lens spaces
    3. Bockstein homomorphisms on smash products of Moore spaces
    4. \(SO(5)\) is not a product of two CW complexes

    3.F

    1. Reverse and double mapping telescopes
    2. Mittag–Leffler condition and \(\varprojlim^1\)
    3. \(\operatorname{Ext}(A, \mathbb{Q}) = 0\)
    4. Divisible groups and vanishing Ext
    5. \(\operatorname{Ext}(A, \mathbb{Z})\) as a cokernel
    6. \(\operatorname{Ext}(\mathbb{Z}_{p^\infty}, \mathbb{Z}_p)\)
    7. Moore spaces as quotients
    8. Bockstein sequence on Moore spaces
    9. Spliced exact sequences for \(p\)-primary components

    3.H

    1. Homology of \(S^1\) with twisted \(\mathbb{Z}\) coefficients
    2. Homology of nonorientable surfaces with local coefficients
    3. Classification of group bundles via \(B\operatorname{Aut}(G)\)
    4. Proper homotopy equivalence of universal covers
    5. Cohomology of graphs with group ring coefficients
    6. Locally finite homology

    4.1

    1. Alternative sum of maps and abelian \(\pi_n\)
    2. Functoriality of \(\pi_n\) under homotopy equivalence
    3. H-space multiplication gives group structure on \(\pi_n\)
    4. Deck transformation action on \(\pi_n\) of universal cover
    5. Relative \(\pi_1\) as cosets
    6. Covering spaces and relative homotopy groups
    7. Change-of-basepoint for relative homotopy groups
    8. Exactness of \(\pi_1\) sequence for pairs
    9. Extending the long exact sequence to \(\pi_0\)
    10. Quasi-circle has trivial homotopy but is not contractible
    11. Contractibility from nullhomotopic inclusions
    12. \(n\)-connected \(n\)-dimensional CW complexes are contractible
    13. CW complexes retract onto contractible subcomplexes
    14. Homotopy equivalent skeleta
    15. Every self-map of \(S^n\) is a multiple of the identity
    16. Factoring maps through \(n\)-connective covers
    17. Connectivity of products and smash products
    18. Asymmetric weak homotopy equivalence
    19. Weak homotopy equivalence via common CW approximation
    20. Finiteness of \([X, Y]\) for finite CW complexes
    21. Postnikov towers and covering spaces commute
    22. Countable CW approximation for countable \(\pi_n\)
    23. Mapping cylinders and cones have CW type

    4.2

    1. No retraction \(\mathbb{RP}^n \to \mathbb{RP}^k\) via homotopy groups
    2. Action of \(\pi_1\) on \(\pi_n\) of \(\mathbb{RP}^n\)
    3. Homotopy groups of punctured lens spaces
    4. Homotopy of infinite wedge of spheres
    5. Short exact sequence in mapping torus
    6. Relative Hurewicz implies absolute
    7. CW complexes with prescribed homotopy groups
    8. Suspension of acyclic CW complex is contractible
    9. Contractible mapping cone implies homotopy equivalence
    10. Finiteness of \(\pi_n'\) vs. \(\pi_n\)
    11. \(\pi_2(X, X^1)\) and the kernel of \(\pi_1(X^1) \to \pi_1(X)\)
    12. Whitehead theorem via universal covers
    13. Isomorphism on \(\pi_i\) for \(i \leq n\) in \(n\)-dimensional complexes
    14. Inclusion of \(S^n\) subcomplex is injective on \(\pi_n\)
    15. Simply-connected closed 3-manifolds are \(S^3\)
    16. Surfaces with infinite \(\pi_1\) are \(K(\pi, 1)\)’s
    17. Maps to \(K(G,n)\) are determined by \(\pi_n\)
    18. Wedge equals product for coprime homology
    19. \(\pi_n(X^n)\) is free abelian for \(K(\pi,1)\)’s
    20. Trivial \(\pi_1\)-action on \(\pi_n\) of products
    21. Constructing spaces with arbitrary \(\pi_n\)
    22. \(H_{n+1}(K(G,n); \mathbb{Z}) = 0\) for \(n > 1\)
    23. Surjectivity of Hurewicz in dimension \(n+1\)
    24. Moore spaces \(M(G,1)\) and \(H_2(K(G,1))\)
    25. Hurewicz cokernel for spaces with few nonzero \(\pi_i\)
    26. Isomorphic homotopy but different homotopy type
    27. Image of \(\pi_2(X) \to \pi_2(X,A)\) is central
    28. \(\mathbb{Z}_p \times \mathbb{Z}_p\) cannot act freely on spheres
    29. Homotopy classification of lens spaces
    30. Fiber bundle projections from subspaces of \(\mathbb{R}^2\)
    31. Split sequences from nullhomotopic fiber inclusions
    32. Fiber bundles \(S^k \to S^m \to S^n\)
    33. Homotopy groups if all Hopf-like bundles existed
    34. Hopf bundle composed with torus collapse
    35. Quotient fiber bundle from quaternionic Hopf bundle
    36. Whitehead product for \(S^1\) and \(S^n\)
    37. Whitehead products vanish in H-spaces
    38. \(\pi_3(S^1 \vee S^2)\) is not finitely generated
    39. Indeterminacy of Toda brackets

    4.3

    1. Map \(\mathbb{RP}^\infty \to K(\mathbb{Z},2)\) trivial on homology
    2. Group structure on \(\langle X, S^1 \rangle\)
    3. Circle retract from direct summand on \(H_1\)
    4. Maps between Eilenberg–MacLane spaces
    5. \([X, S^n] \approx H^n(X; \mathbb{Z})\) for \(n\)-dimensional \(X\)
    6. H-space structure on \(K(G,n)\)
    7. Addition in \(\langle X, K(G,n) \rangle\) matches cohomology
    8. Fibrations from sections of \(E^I \to E_p\)
    9. Linear projection is fibration but not fiber bundle
    10. Action of \(\pi_1(E)\) on \(\pi_n(F)\) for fibrations
    11. Fibration classes and homotopy equivalence
    12. Homotopic maps give fiber homotopy equivalent fibrations
    13. Pre-composing with homotopy equivalence
    14. Fibration classes bijection with map classes
    15. Homotopy equivalence fibration is fiber homotopy trivial
    16. Whitehead theorem via homotopy fiber
    17. \(\Omega X\) is an H-space
    18. Long exact sequence for \(\langle X, - \rangle\)
    19. Exactness improvement via \(B\)-action
    20. Postnikov towers for \(\Omega X\)
    21. H-space Postnikov towers
    22. Principal fibrations with sections split
    23. Uniqueness of Quillen plus construction
    24. Obstructions for homotopy of lifts

    4.A

    1. Homotopies of maps to topological groups are basepoint-preserving
    2. \(\pi_1\)-action on \(\langle X, Y \rangle\) and Eilenberg–MacLane spaces
    3. Automorphisms of \(K(\pi, 1)\) are outer automorphisms of \(\pi\)
    4. Automorphisms of wedge sums of spheres
    5. Eilenberg–MacLane spaces and Postnikov towers

    4.B

    1. Hopf invariant of compositions
    2. Fiber bundles \(S^k \to S^m \to S^n\) and the Hopf invariant

    4.D

    1. Cup product structure on pullbacks of \(\mathbb{CP}^3 \to S^4\)
    2. Cohomology of \(K(\mathbb{Z}_p, 1)\) via Leray–Hirsch
    3. Cohomology of Stiefel manifolds via Leray–Hirsch
    4. Cohomology of complex flag manifolds
    5. Gysin sequence and Hopf invariant of sphere bundles
    6. Characterization of \(\mathbb{CP}^n\)-like manifolds
    7. Thom class implies orientability
    8. Thom space of a product bundle
    9. Torus inclusion in \(U(n)\) is nullhomotopic
    10. Eigenvalues of quaternionic matrices

    4.F

    1. Direct limit axiom vs. wedge sum axiom
    2. Suspension sequences of CW complexes
    3. Homotopy colimit commutes with loop space

    4.G

    1. Infinite mapping cylinder deformation retracts onto telescope
    2. Projection from \(\Delta X\) is a fiber bundle
    3. Nerve of the open star cover
    4. Nerve lemma for subcomplex covers of CW complexes

    4.H

    1. Cofibrations give fibrations on function spaces
    2. Cofibrations are preserved by pushout
    3. Fiber-preserving homotopy equivalences are fiber homotopy equivalences
    4. Dual of an iterated mapping cylinder

    4.I

    1. Suspension of a retract splits as a wedge
    2. Suspension of \(K(\mathbb{Z}_m \times \mathbb{Z}_n, 1)\) for coprime \(m, n\)
    3. Skeletons of suspended lens spaces

    4.K

    1. Mayer–Vietoris for CW subcomplexes
    2. Contractible fibers imply homotopy equivalence
    3. \(SP_n(I) = \Delta^n\)
    4. \(SP_2(S^1)\) is a Möbius band
    5. Quasifibrations
    6. Projection from \(\Delta X\) is a quasifibration

    4.L

    1. Cohomology operations on \(H^1\) and \(H^2\)
    2. Cohomology operations distinguish quotients
    3. No bundle \(S^4 \to \mathbb{HP}^5 \to \mathbb{OP}^2\) exists
    4. No bundle \(S^7 \to S^{23} \to \mathbb{OP}^2\) exists
    5. Subalgebra of \(\mathcal{A}_2\) generated by \(Sq^1\) and \(Sq^2\)

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