MAT 540 Advanced Topology, Geometry I, Fall 2020.

  • Instructor: Oleg Viro, office 5-110 Math Tower, e-mail:
  • Office hours: Monday and Wednesday 11:30am-1:00pm in 5-110 Math Tower or by appointment.
  • Grader: Aleksandar Milivojevic Math Tower 3-104
  • Textbook : Algebraic Topology by Allen Hatcher
  • Class meetings: Monday and Wednesday, 9:45-11:05pm, Mathematics P127.

    Everyone participating in this class must wear a mask or face covering at all times or have the appropriate documentation for medical exemption. Please contact Student Accessibility Support Center (SASC) at if you need special accommodations. Any student not in compliance with this policy will be asked to leave the class.

  • Homework: will be assigned weekly through Blackboard (Assignments). Your solutions should be submitted to Blackboard. Each submission should contain a single pdf-file. You may use any app which consolidate your pictures in a single pdf-file (for example, CamScan). Submission in a wrong format (multiple files, jpg-format, links to the cloud, etc.) will be accepted but with reduced score. Late submission will be accepted but with reduced score.

    Homework 1 due by 9/9.

    Homework 2 due by 9/9.

    Homework 3 due by 9/16.

    Homework 4 due by 9/23.

  • Quizzes will be given weekly in class.
  • Exams: two midterms (in class) and final exam.
    The Final Exam will be on Wednesday, December 16, 2:15PM-5:00PM online.
  • Grading system: your grade for the course will be based on: homework 15%, quizzes 15%, two midterms 25% each, final exam 20%.

  • Content:
    1. Homotopy
      • Homotopy as a map, as a family of maps and as a relation.
      • Relative homotopy.
      • Homotopy equivalences and deformation retractions.
    2. Natural group structures
      • Multiplication of paths and their homotopy classes.
      • Homotopy properties of path multiplication.
      • Natural group structures on sets of homotopy classes.
    3. Homotopy groups
      • Fundamental group and high homotopy groups
      • Homomorphisms induced by a continuous map.
      • Homotopy groups of a product.
      • $n$-connectedness.
      • Extension of homotopy.
      • Dependence of homotopy groups on the base point.
      • $n$-simple spaces.
      • Relative homotopy groups.
      • Homotopy sequences of a pair and of a triple.
    4. Bundles
      • The language of bundles.
      • Fibre bundles.
      • Coverings.
      • Lifting homotopy property.
      • Serre bundles.
      • Homotopy sequence of a Serre bundle.
      • Calculation of fundamental groups using covering spaces.
      • Classification of coverings.
      • Automorphisms of a covering.
    5. CW-complexes
      • Cellular maps
      • Cellular approximation theorem.
      • Calculation of the fundamental group of a CW-complex.
      • Van-Kampen theorem.
      • Homotopy classification of 1-dimensional CW-complexes.
      • The Euler characteristic.
      • Whitehead theorem.
      • Homotopy excision.
      • Suspension theorem.
      • Stable homotopy groups.
      • Simplest calculations of homotopy groups.
      • Applications.
    6. Manifolds
      • Topological manifolds
      • Interior and boundary of a manifold.
      • Product of manifolds.
      • Dimension.
      • The boundary of Euclidean half-space.
      • One-dimensional manifolds.
      • Topological classification of one-dimensional manifolds.
      • Triangulated two-dimensional manifolds and their topological classification.

    Student Accessibility Support Center (SASC) statement: If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact SASC (631) 632-6748 or They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and SASC. For procedures and information go to the following website:

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