18.905
Algebraic Topology I
Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.
Fall 2016
Topics
- Introduction: Singular Simplices and Chains
- Homology
- Categories, Functors, Natural Transformations
- Categorical Language
- Homotopy, Star-shaped Regions
- Homotopy Invariance of Homology
- Homology Cross Product
- Relative Homology
- The Homology Long Exact Sequence
- Excision and Applications
- The Eilenberg Steenrod Axioms and the Locality Principle
- Subdivision
- Proof of the Locality Principle
- CW-Complexes
- CW-Complexes II
- Homology of CW-Complexes
- Real Projective Space
- Euler Characteristic and Homology Approximation
- Coefficients
- Tensor Product
- Tensor and Tor
- The Fundamental Theorem of Homological Algebra
- Hom and Lim
- Universal Coefficient Theorem
- Künneth and Eilenberg-Zilber
- Coproducts, Cohomology
- Ext and UCT
- Products in Cohomology
- Cup Product (cont.)
- Surfaces and Nondegenerate Symmetric Bilinear Forms
- Local Coefficients and Orientations
- Proof of the Orientation Theorem
- A Plethora of Products
- Cap Product and “Cech” Cohomology
- Cech Cohomology as a Cohomology Theory
- The Fully Relative Cap Product
- Poincaré Duality
- Applications
Readings
- Dold, Albrecht. Lectures on Algebraic Topology (Grundlehren Der Mathematischen Wissenschaften Series) . Springer-Verlag, 1980. ISBN: 9780387103693. (Wonderful technique)
- Hatcher, Allen. Algebraic Topology . Cambridge University Press, 2001. ISBN: 9780521791601. (Chatty)
- Munkres, James R. Elements of Algebraic Topology . Addison Wesley Publishing Company, 1984. ISBN: 9780201045864. (Meticulous)
Prerequisites & course information
Prereq: 18.901 and ( 18.701 or 18.703 ) G (Spring) 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Fall 2016
- MIT OCW syllabus ↗
- MIT OCW calendar ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.