The primary objective in algebraic topology is to study the shape of geometric spaces by associating algebraic objects to them that remain unchanged under continuous deformations and homeomorphisms. The fundamental group, the higher homotopy groups, and homology groups of a topological space provide the principal examples of such invariants.
In the past century algebraic topology has evolved into an indispensable tool in topology and geometry, and it bears deeply on various other areas of mathematics, including global analysis, group theory, homological algebra, and number theory. Algebraic topology also affords novel and striking applications to physics, computer science, economics, and biology as a tool for uncovering hidden structures and for identifying obstructions to the equivalence of two objects.
MTH 635, continuation of MTH 634 offered in the fall, will be taught as an introductory course in algebraic topology with focus on cohomology groups and their applications. Emphasis will be placed on developing computational techniques required in the analysis of these key invariants. We will use the book Algebraic Topology by Allen Hatcher as the course text. A pdf-version of the book along with corrigenda can be downloaded gratis at http://www.math.cornell.edu/~hatcher/AT/ATpage.html.
Instructor: Juha PohjanpeltoPrerequisites: Singluar and simplicial homology theory as covered in the fall term MTH 634.
| Text: | Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001. |
| Topics: | |
| • Background material | |
| • Cohomology groups | |
| • Universal coefficient theorem | |
| • Cup product and cohomology ring | |
| • Poincare duality | |
| Learning objectives: Upon successfully completing this course a student | |
| • appreciates the general notions of homology and cohomology. | |
| • comprehends the concepts of comology ring of a topological space. | |
| • recognizes the essential computational techniques for determining cohomology rings and capably applies these in standard examples from topology. | |
| • is acquainted with the key applications of the cohomology to topology and geometry. | |
| The Geometry Center web site, University of Minnesota Science and Technology Center. | |
| R. Bott, L. Tu, Differential Forms in Algebraic Topology,Springer-Verlag, 1982. | |
| M.J. Greenberg, J.R. Harper, Algebraic Topology, a first course, Benjamin/Cummings, 1981. | |
| J.J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, 1986. |
Grading: Your course grade will be based on five homework assignments and a take-home final exam. The homework assignments count 60% towards the course grade, and the final exam 40%.
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