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    UNL topology workshop, May 2020

    SRC-TOP-WORKSHOP-2020

    Area
    Topology
    Status
    draft
    Source
    Contents

    42 problems.

    Worksheet 1

    1. Closure of a product equals the product of closures
    2. Closed covering by two sets with connected intersection forces both closed sets to be connected
    3. Compactness of topologies on \(\mathbb R\) generated by the Euclidean and excluded point topologies
    4. Fixed point property is preserved under retraction
    5. Diagonal of the torus is a retract but not a deformation retract
    6. For residually finite \(\pi_1\), every nontrivial loop fails to lift to a loop in some finite-sheeted cover
    7. Homeomorphic spaces have homeomorphic simply connected covering spaces
    8. Simplicial homology of the connected sum of two projective planes via a \(\Delta\)-complex structure

    Worksheet 2

    1. The definition of a topology on a set
    2. Topologies generated by bases
    3. Topologies generated by subbases
    4. Six distinct topologies on \(\mathbb R\)

    Worksheet 2A

    1. The separation axioms \(T_0\) through \(T_4\)
    2. A \(T_n\) space that is not \(T_{n+1}\)
    3. Compact subsets of Hausdorff spaces are closed; closed subsets of compact spaces are compact
    4. Points joined by a path lie in the same connected component
    5. The topologist’s sine curve is connected but not path connected
    6. An open connected subset of Euclidean space is path connected

    Worksheet 2B

    1. Retractions and deformation retractions of intervals and the circle
    2. Radial deformation retraction \(\mathbb R^n\setminus\{0\}\to S^{n-1}\)
    3. The mapping cylinder of a constant map \(D^2\to\{y\}\)
    4. Homotopy is compatible with composition, but cancellation can fail

    Worksheet 3A

    1. CW structures on \(S^2\) and \(T^2\)
    2. Seifert–van Kampen theorem
    3. Classification of compact surfaces without boundary
    4. The loop traced by the basepoint under a self-homotopy of the identity is central in \(\pi_1\)

    Worksheet 3B

    1. Universal covers of \(S^1\) and \(S^1\vee S^1\)
    2. Deck transformations of the universal cover of \(S^1\)
    3. Galois correspondence for covering spaces
    4. Maps between base spaces lift to maps between simply connected covering spaces

    Worksheet 4A

    1. A Δ-complex structure and chain complex for the standard 2-simplex
    2. A \(\Delta\)-complex structure and homology of \(S^1\)
    3. Reduced versus non-reduced homology
    4. Mayer–Vietoris homology sequence for an open cover

    Worksheet 5

    1. Projections in the product topology are continuous and open; product is Hausdorff iff factors are
    2. Compact subspace of a Hausdorff space is closed, and compact Hausdorff implies normal
    3. Quotient of \(\mathbb R^2\) by concentric circles is homeomorphic to a closed ray
    4. Local connectedness is equivalent to open components, and is preserved by quotient maps
    5. Sphere with antipodal poles identified is homotopy equivalent to \(S^1\vee S^2\)
    6. Fundamental group of a wedge sum, and the presentation complex of \(\langle a,b,c\mid a^2\rangle\)
    7. Simplicial homology of the \(\Delta\)-complex obtained from \(\Delta^3\) by identifying all \(k\)-faces
    8. Cell structure on \(S^2\) and long exact sequence computation of \(H_i(S^n)\)

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