Math 524: Lecture Notes and Videos on Algebraic Topology
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Copyright: I (Bala Krishnamoorthy) hold the copyright
for all lecture scribes/notes, documents, and other
materials including videos posted on these course web
pages. These materials might not be used for commercial
purposes without my consent.
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Scribes from
all lectures so far (as a single big file)
| Lec | Date | Topic(s) | Scribe | Video |
| 1 |
Aug 19 |
syllabus,
continuous functions, neighborhood of a point, topological
space, homeomorphism, showing \(\mathbb{R}^1 \not\approx
\mathbb{R}^2\)
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Scb1
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Vid1
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| 2 |
Aug 21 |
open sets, geometric independence (GI), \(n\)-simplex,
barycentric coordinates, dimension, boundary,
properties of simplices
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Scb2
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Vid2
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| 3 |
Aug 26 |
simplicial complex \(K\), subcomplex, \(p\)-skeleton
\(K^{(p)}\), underlying space \(|K|\), two topologies
for \(|K|\), properties of \(|K|\)
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Scb3
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Vid3
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| 4 |
Aug 28 |
star, closed star, link, properties, simplicial maps,
abstract simplicial complex (ASC), vertex scheme,
geometric realization
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Scb4
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Vid4
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| 5 |
Sep 2 |
ASC examples: cylinder, Möbius strip, torus,
Klein bottle; algebra review: groups, homomorphism,
finitely generated
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Scb5
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Vid5
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| 6 |
Sep 4 |
internal direct sum, results on finitely generated
abelian groups, betti number, torsion coefficients,
orientation of simplex
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Scb6
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Vid6
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| 7 |
Sep 9 |
\(p\)-chain, elementary \(p\)-chain, group of
\(p\)-chains \(C_p(K)\), boundary homo'm, fundamental
lemma of homology: \(\partial_p\partial_{p+1} = 0\)
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Scb7
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Vid7
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| 8 |
Sep 11 |
cycles, boundaries, homology group \(H_p(K) =
Z_p(K)/B_p(K)\), pushing chain off an edge, chain
carried by subcomplex
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Scb8
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Vid8
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| 9 |
Sep 16 |
homology groups of torus \((\mathbb{T}^2)\) and Klein
bottle (\(\mathbb{K}^2\)): \(H_1(\mathbb{T}^2) \simeq
\mathbb{Z}\oplus \mathbb{Z}, H_2(\mathbb{T}^2) \simeq
\mathbb{Z}\); \(H_1(\mathbb{K}^2) \simeq
\mathbb{Z}\oplus \mathbb{Z}_2\,, H_2(\mathbb{K}^2) =
0\)
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Scb9
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Vid9
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Last modified: Tue Sep 16 17:54:27 PDT 2025
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