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    Review sheet for the qualifying exam — Hatcher selections and practice problems

    SRC-QUAL-REVIEW-HATCHER

    Area
    Topology
    Status
    draft
    Source
    Contents

    122 problems.

    Hatcher §2.1 selections

    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. Retract induces injective map on homology
    11. Chain homotopy is an equivalence relation
    12. Short exact sequences of cyclic groups
    13. Relative homology of surfaces with point and circle subsets
    14. Reduced homology of suspension is shift of reduced homology
    15. Homology of finite-dimensional CW complexes by induction on dimension
    16. Map that is homotopy equivalence on \(X\) and \(A\) induces isomorphism on relative homology
    17. Local homology groups of cone on 1-skeleton of \(\Delta^3\)
    18. \(S^1 \times S^1\) and \(S^1 \lor S^1 \lor S^2\) have isomorphic homology but different universal covers
    19. Five-lemma example with zero outer maps and nonzero middle map

    Hatcher §2.2 selections

    1. Map of \(S^{2n}\) has fixed point or antipodal point; \(\mathbb{RP}^{2n}\) maps have fixed points
    2. Degree zero map has fixed and antipodal points; Hairy Ball Theorem
    3. Surjective map \(S^n \to S^n\) of degree zero
    4. Any two reflections of \(S^n\) are homotopic through reflections
    5. Every map \(S^n \to S^n\) can be homotoped to have a fixed point
    6. Induced map on local homology of invertible linear transformation is \(\pm 1\) according to sign of determinant
    7. Homology of four specific 2-complexes
    8. Homology of \(S^2\) with equatorial antipodal identification, and \(S^3\) with equatorial \(S^2\) antipodal identification
    9. Homology of cube with opposite faces identified via one-quarter twist (quaternion group)
    10. 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
    11. Even maps \(S^n \to S^n\) must have even degree; degree zero when \(n\) even
    12. Ranks of simplicial chain groups of \(\Delta^n\) and homology of skeleta
    13. \(H_i(\mathbb{RP}^n/\mathbb{RP}^m)\) by cellular homology
    14. If \(M_g\) covers \(M_h\) then \(g = n(h-1)+1\)
    15. Splitting of chain short exact sequences does not give splitting of homology
    16. Mayer–Vietoris computations for torus with Möbius band and \(\mathbb{RP}^2\) with Möbius band
    17. Homology of double of genus-\(g\) surface via Mayer–Vietoris
    18. Mayer–Vietoris gives \(\tilde{H}_n(X \vee Y) \approx \tilde{H}_n(X) \oplus \tilde{H}_n(Y)\) under neighborhood condition
    19. Mayer–Vietoris gives suspension isomorphism \(\tilde{H}_n(SX) \approx \tilde{H}_{n-1}(X)\)
    20. Nerve-type vanishing: \(\tilde{H}_i(X) = 0\) for \(i \geq n-1\) when intersections are acyclic
    21. Künneth formula \(H_i(X \times S^n) \approx H_i(X) \oplus H_{i-n}(X)\)
    22. Universal coefficient short exact sequence for homology with \(\mathbb{Z}_n\) coefficients
    23. Splitting of chain complexes and universal coefficient formula for \(H_n(X; G)\)

    Hatcher §2.B selections

    1. Real division algebras and the Borsuk–Ulam theorem

    Hatcher §1.1 selections

    1. Cancellation property for composition of paths
    2. Basepoint-change homomorphism depends only on homotopy class
    3. Three equivalent conditions for simply-connected spaces
    4. Natural map from \(\pi_1\) to unbased homotopy classes of maps \(S^1 \to X\)
    5. Borsuk–Ulam theorem for the torus
    6. Explicit homotopy for commuting loops in product spaces
    7. Surjectivity of inclusion-induced map on \(\pi_1\) iff paths in \(X\) with endpoints in \(A\) are homotopic to paths in \(A\)
    8. No retractions in six specific cases

    Hatcher §1.2 selections

    1. Fundamental group of \(S^2\) with north and south poles identified
    2. Fundamental group of two tori identified along a circle
    3. Loop nullhomotopic in \(D^2 \times I\) but not in complement of two arcs
    4. Fundamental group of mapping torus of map on wedge of circles
    5. Cube with face identifications via screw motions has quaternion group as fundamental group
    6. Canonical map from loop complex \(L(X)\) to \(X\) induces isomorphism on \(\pi_1\)
    7. Fundamental group of surface of infinite genus is free on infinitely many generators
    8. Join of path-connected space with any nonempty space is simply-connected

    Hatcher §4.1 selections

    1. Alternative sum of maps and abelian \(\pi_n\)
    2. Functoriality of \(\pi_n\) under homotopy equivalence

    Hatcher §4.2 selections

    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. Split sequences from nullhomotopic fiber inclusions
    4. Fiber bundles \(S^k \to S^m \to S^n\)
    5. Homotopy groups if all Hopf-like bundles existed

    Hatcher §1.3 selections

    1. Restriction of a covering space to a subspace
    2. Product of covering spaces
    3. Simply-connected covering space of a sphere with diameter
    4. The harmonic broom has no simply-connected covering space
    5. Maps to \(S^1\) from spaces with finite fundamental group
    6. Connected covering spaces of \(S^1 \vee S^1\)
    7. Covering space of \(S^1 \vee S^1\) for a normal subgroup
    8. 27-sheeted covering space from the subgroup of cubes
    9. Connected covering spaces of \(\mathbb{RP}^2 \vee \mathbb{RP}^2\)
    10. Lifting covering spaces through composites
    11. Universal abelian covering space
    12. Nonnormal covering spaces of the Klein bottle
    13. Torus with Möbius band attached
    14. Covering spaces and fundamental groups
    15. Two actions of the fundamental group on the fiber

    Hatcher §3.1 selections

    1. Functoriality of \(\operatorname{Ext}(H,G)\) in both variables
    2. Multiplication by \(n\) induces multiplication by \(n\) on \(\operatorname{Ext}\)
    3. Simplicial cohomology of the torus, \(\mathbb{RP}^2\), and the Klein bottle
    4. \(\operatorname{Hom}(H_n(X),\mathbb Z)\) is not a cohomology theory
    5. Cohomology of spheres, good pairs, and retracts
    6. Degree \(d\) maps act on \(H^n(S^n;G)\) by multiplication by \(d\)

    Hatcher §3.2 selections

    1. Cup product structure on closed orientable surfaces
    2. Maps \(\mathbb{RP}^n\to\mathbb{RP}^m\) for \(n>m\) and the Borsuk–Ulam theorem
    3. The \(d\)-th power map on \(\mathbb{CP}^n\) in cohomology
    4. \(\mathbb{RP}^3\) is not homotopy equivalent to \(\mathbb{RP}^2\vee S^3\)
    5. Spaces with isomorphic integral but distinct \(\mathbb Z_p\) cohomology rings
    6. Maps \(S^{k+\ell} \to S^k \times S^\ell\) are trivial on top homology
    7. Torsion in products of CW complexes
    8. Cup product nondegenerality on surfaces

    Hatcher §3.A selections

    1. Rational and mod-\(p\) cohomology determine integral homology

    Hatcher §3.B selections

    1. Homology and cohomology of \(\mathbb{RP}^m \times \mathbb{RP}^n\)
    2. Chain homotopy as a chain map on the cone

    Hatcher §3.3 selections

    1. Deleting a point does not affect orientability
    2. Orientability of products
    3. Degree 1 maps to \(S^n\)
    4. Associativity of cap product
    5. Compactly supported \(H^0\) of noncompact spaces
    6. Compactly supported cohomology and one-point compactification
    7. Torsionfree \(H_{k-1}\) implies torsionfree \(H_k\)
    8. Cup product on connected sums of products

    More practice problems — source-faithful positions

    1. Kronecker pairing descends to homology and cohomology
    2. Total Hom complexes preserve chain homotopies
    3. Cup-length obstruction from a contractible cover
    4. Degree of the coordinate-squaring map of the sphere
    5. Homology and cohomology of twice-punctured real projective space
    6. Classify a closed surface drawn as a circular chain of tori
    7. Classify a circular chain of tori with two links replaced by projective planes
    8. Covering-space existence among spheres and the torus
    9. A unital multiplication on a positive-dimensional sphere forces odd dimension
    10. Cohomological matrices of homeomorphisms of a product of even-dimensional spheres
    11. Realize the Klein bottle as a quotient of the plane
    12. The dimension of a manifold is intrinsic
    13. The figure-eight is not a retract of the torus
    14. Complex projective space does not split off a sphere wedge
    15. A product decomposition of complex projective space has a cohomologically trivial factor
    16. Degree of a normalized polynomial counts roots in the unit disk

    Browse these problems in source order