18.905 | Fall 2016 | Graduate

Algebraic Topology I

Lecture Notes

These lecture notes are based on a live LaTeX record made by Sanath Devalapurkar with images by Xianglong Ni, both of whom were students in the class at the time it was taught on campus.

I. Singular Homology
1 Introduction: Singular Simplices and Chains (PDF)
2 Homology (PDF)
3 Categories, Functors, Natural Transformations (PDF)
4 Categorical Language (PDF)
5 Homotopy, Star-shaped Regions (PDF)
6 Homotopy Invariance of Homology (PDF)
7 Homology Cross Product (PDF)
8 Relative Homology (PDF)
9 The Homology Long Exact Sequence (PDF)
10 Excision and Applications (PDF)
11 The Eilenberg Steenrod Axioms and the Locality Principle (PDF)
12 Subdivision (PDF)
13 Proof of the Locality Principle (PDF)
II. Computational Methods
14 CW-Complexes (PDF)
15 CW-Complexes II (PDF)
16 Homology of CW-Complexes (PDF)
17 Real Projective Space (PDF)
18 Euler Characteristic and Homology Approximation (PDF)
19 Coefficients (PDF)
20 Tensor Product (PDF)
21 Tensor and Tor (PDF)
22 The Fundamental Theorem of Homological Algebra (PDF)
23 Hom and Lim (PDF)
24 Universal Coefficient Theorem (PDF)
25 Künneth and Eilenberg-Zilber (PDF)
III. Cohomology and Duality
26 Coproducts, Cohomology (PDF)
27 Ext and UCT (PDF)
28 Products in Cohomology (PDF)
29 Cup Product (cont.) (PDF)
30 Surfaces and Nondegenerate Symmetric Bilinear Forms (PDF)
31 Local Coefficients and Orientations (PDF)
32 Proof of the Orientation Theorem (PDF)
33 A Plethora of Products (PDF)
34 Cap Product and “Cech” Cohomology (PDF)
35 Cech Cohomology as a Cohomology Theory (PDF)
36 The Fully Relative Cap Product (PDF)
37 Poincaré Duality (PDF)
38 Applications (PDF)

Course Info

As Taught In
Fall 2016
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Lecture Notes