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Browse materialsMIT 18.905 · Fall 2016

Algebraic Topology I

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Haynes Miller
Term: Fall 2016
Level: Graduate
What's inside: 6 problem sets · 2 lecture notes · 38 other files · 3 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 18.905 Algebraic Topology I, Fall 2016, Prof. Haynes Miller — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

This course’s cited books, in our reading lists: Algebraic Topology

How this course was organized

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Grouped by section, as MIT organized them. Not sequenced or ordered.

Lecture Notes

38 files

Assignments

Other materials from this course

Algebraic Topology I: Lecture NotesPDF Download

MIT OpenCourseWare, Prof. Haynes Miller, 18.905 Algebraic Topology I, Fall 2016.

License: CC BY-NC-SA 4.0 · Edition: Fall 2016 · Open the original course on OCW ↗

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Algebraic Topology I

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Other materials from this course

Lecture 1: Introduction: Singular Simplices and Chains

Lecture 2: Homology

Lecture 3: Categories, Functors, Natural Transformations

Lecture 4: Categorical Language

Lecture 5: Homotopy, Star-shaped Regions

Lecture 6: Homotopy Invariance of Homology

Lecture 7: Homology Cross Product

Lecture 8: Relative Homology

Lecture 9: The Homology Long Exact Sequence

Lecture 10: Excision and Applications

Lecture 11: The Eilenberg Steenrod Axioms and the Locality Principle

Lecture 12: Subdivision

Lecture 13: Proof of the Locality Principle

Lecture 14: CW-Complexes

Lecture 15: CW-Complexes II

Lecture 16: Homology of CW-Complexes

Lecture 17: Real Projective Space

Lecture 18: Euler Characteristic and Homology Approximation

Lecture 19: Coefficients

Lecture 20: Tensor Product

Lecture 21: Tensor and Tor

Lecture 22: The Fundamental Theorem of Homological Algebra

Lecture 23: Hom and Lim

Lecture 24: Universal Coefficient Theorem

Lecture 25: Künneth and Eilenberg-Zilber

Lecture 26: Coproducts, Cohomology

Lecture 27: Ext and UCT

Lecture 28: Products in Cohomology

Lecture 29: Cup Product (cont.)

Lecture 30: Surfaces and Nondegenerate Symmetric Bilinear Forms

Lecture 31: Local Coefficients and Orientations

Lecture 32: Proof of the Orientation Theorem

Lecture 33: A Plethora of Products

Lecture 34: Cap Product and “Cech” Cohomology

Lecture 35: Cech Cohomology as a Cohomology Theory

Lecture 36: The Fully Relative Cap Product

Lecture 37: Poincaré Duality

Lecture 38: Applications