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 634 will be taught as an introductory course in algebraic topology with focus on simplicial and singular homology and 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 for free at http://www.math.cornell.edu/~hatcher/AT/ATpage.html.

Prerequisites: The course will be mainly self-contained but a working knowledge of the rudiments of abstract algebra and topology will be assumed.

Text:

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001.

Topics:
	• Background material
	• Δ–complexes and simplicial homology
	• Singular homology
	• Computational techniques and applications of homology
	• Cohomology rings

Learning objectives: Upon successfully completing this course a student
	• appreciates the general notions of homology and cohomology.
	• comprehends the concepts of simplicial and singular homologies of a topological space.
	•  recognizes the essential computational techniques for determining singular cohomology and capably applies these in standard examples from topology.
	• is acquainted with the key applications of the singular homology to topology and geometry.

Grading: Your course grade will be based on three homework assignments, a midterm exam, and a take-home final exam. The homework assignments count 30% towards the course grade, and the midterm and final exams 35% each.

• The homework assignments will be due October 14, November 11, and December 2.
• The midterm exam will be held on Friday, October 28, during the class hour. The questions in the exam will be drawn from a pool of 12–15 problems which will be posted on the web about a week in advance of the test.
• The due date for the take-home final exam will be announced at a later time.

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