Math 7520: Fall 2010: Algebraic Topology: Cohomology & Duality

Math 7520: Algebraic Topology: Cohomology & Duality

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    Math 7520:      Algebraic Topology: Cohomology & Duality
    Time/Place:     10:40 am - 11:30 am --- 130 Lockett Hall.
    Instructor:     Neal Stoltzfus
    Office:         Lockett 258:   578.1656
    Office Hours:   MW 1:30pm or by appointment
    Web Page:	URL:
                    This site will contain this document on course information, 
                    and links to additional web resources.
    Prerequisite:   Point-set topology, fundamental groups and modules over rings, Topology II: homology theory.
    Textbook:   Allen Hatcher:  Algebraic Topology  (Available on Cornell Math website)
    	    James Munkres:  Algebraic Topology

    Project: The course project will develops a topic related to cohomology theory. An oral presentation of your project will be made after mid-terms. A written report will be due on the Final Exam date.

    Additional Project Information .

    Grades will be based on your Project.

    Description: This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group in MATH 7510, the homology groups in MATH 7512) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., isomorphic groups). Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent. The focus of this course will be on cohomology theory, dual to the homology theory developed in MATH 7512. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications.

    In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, we may pursue some of these connections, such as cohomology of groups or the De Rham theorem.

    Last update: 22 August, 2010