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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| 10 |
Sep 18 |
\(\mathbb{R}P^2, H_1(\mathbb{R}P^2) \simeq
\mathbb{Z}_2\), \(k\)-fold dunce hat: \(H_1(D_k)
\simeq \mathbb{Z}_k\), connected sum, \(\mathbb{R}P^2
\# \mathbb{R}P^2 \approx \mathbb{K}^2\),
\(0\)-dimensional homology
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Scb10
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Vid10
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| 11 |
Sep 23 |
proof: \(0\)-dim homology, reduced homology,
\(\tilde{H_i}(\sigma) = 0 \, \forall i\) for
\(p\)-simplex \(\sigma\), relative chains, boundary,
and homology groups
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Scb11
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Vid11
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| 12 |
Sep 25 |
examples of relative homology, intuition in 3D,
torsion in \(H_1(K,K_0)\) for Möbius strip for
\(K_0=\) edge, excision theorem
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Scb12
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Vid12
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| 13 |
Sep 30 |
homomorphisms induced by simplicial maps, \(\partial
f_{\#} = f_{\#} \partial \), functoriality, maps from
1-cycle \(\mathbf{z}\) to torus, \(g_{\#}(\mathbf{z})
\sim h_{\#}(\mathbf{z})\)
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Scb13
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Vid13
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| 14 |
Oct 2 |
chain homotopy, \( \partial D + D \partial =
g_{\#}\)\(-\)\(f_{\#}\), contiguous maps, contiguous
maps of pairs in relative homology, star condition
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Scb14
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Vid14
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| 15 |
Oct 7 |
simplicial approximation: \(h( \mathrm{St } \ v )
\subset \mathrm{St} f(v) \,\forall v \in K^{(0)}\),
subdivision \(K'\) of \(K\) and properties, star
condition for subdivision
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Scb15
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Vid15
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| 16 |
Oct 9 |
cone of \(K\) with vertex \(\mathbf{w}\), barycentric
subdivision \(\mathrm{Sd} K\), simplices in
\(\mathrm{Sd} K\), \(\mathrm{diam}(\sigma) <
\epsilon\, \forall \sigma \in \mathrm{Sd}^rK\) for
large enough \(r\)
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Scb16
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Vid16
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| 17 |
Oct 14 |
simplicial approximation, subdivision of \(K\) holding
\(K_0\) fixed, exact sequence: \(\mathrm{im} \,
\phi_{i-1}\)\(=\)\(\mathrm{ker} \, \phi_i\), short
exact sequence (SES)
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Scb17
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Vid17
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| 18 |
Oct 16 |
chain map, SES splits, connecting homo'm
\(\partial_{\star} : H_p(K,K_0)\)\(\to\)\(
H_{p-1}(K_0)\), long exact homology sequence of pair
\((K,K_0)\)
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Scb18
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Vid18
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| 19 |
Oct 21 |
SES of chain complexes, zig-zag lemma, long exact
homology sequence, connecting homomorphism \(\partial_{\star}\),
diagram chasing
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Scb19
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Vid19
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| 20 |
Oct 23 |
proof of zig-zag lemma, commuting diagram for chain
complexes, Steenrod five lemma, Meyer-Vietoris
sequence (MVS)
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Scb20
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Vid20
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| 21 |
Oct 28 |
proof of MVS,
\(\phi(\mathbf{c})\)\(=\)\((i^{'}_{\#}(\mathbf{c}^{'}),
\)\( -i^{''}_{\#}(\mathbf{c^{''}})) \), structure of
\(H_p(K)\), examples, definition of \(\partial_{*}\),
homology of \(d\)-sphere using MVS
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Scb21
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Vid21
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| 22 |
Oct 30 |
\(d\)-sphere: \(\tilde{H}_p(\mathbb{S}^d) \simeq
\tilde{H}_{p-1}(\mathbb{S}^{d-1})\), suspension
\(S(K)\), \(\tilde{H}_p(S(K)) \simeq
\tilde{H}_{p-1}(K)\), cutting Klein bottle into two
Möbius strips
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Scb22
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Vid22
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Last modified: Thu Oct 30 17:04:27 PDT 2025
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