Math 581 (Fall 2017) - Lecture Notes and Videos on Algebraic topology

**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.

Scribes from all lectures so far (as a single **big** file)

Lec # Date Topic(s) Scribe Panopto
1
Aug 22
syllabus,
continuous functions, neighborhood of a point, topological
space, homeomorphism, showing \(\mathbb{R}^1 \not\approx
\mathbb{R}^2\)
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video
2
Aug 24
open set, topological space, geometrically independent (GI),
\(n\)-plane, \(n\)-simplex, barycentric coordinates,
\(2\)-simplex
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video
3
Aug 29
properties of simplices, vertex, dimension, faces, boundary,
simplicial complex \(K\), subcomplex, \(p\)-skeleton \(K^{(p)}\)
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video
4
Aug 31
underlying space \(|K|\), infinite complex, properties of
\(|K|\), star, closed star, link \(\mathrm{Lk } \ \mathbf{v} =
\overline{\mathrm{St } \ \mathbf{v}} - \mathrm{St } \
\mathbf{v}\), locally finite
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video
5
Sep 5
simplicial maps, abstract simplicial complex (ASC), geometric
realization theorem, ASCs for cylinder and Möbius strip
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video
6
Sep 7
ASCs for torus and Klein bottle, groups, homomorphism, finitely
generated, torsion, internal direct sum, direct product
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video
7
Sep 12
results on finitely generated abelian groups, orientation of
simplex, \(p\)-chain, group of \(p\)-chains \(C_p(K)\),
elementary chains
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video
8
Sep 14
boundary homomorphism \(\partial_p(K)\), fundamental lemma of
homology: \(\partial\partial = 0\), homology group \(H_p(K) =
Z_p(K)/B_p(K)\)
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video
9
Sep 19
examples, homologous chains: \(\mathbf{c}_1 \sim \mathbf{c}_2\)
if \(\mathbf{c}_1 - \mathbf{c}_2 = \partial \mathbf{d}\),
"pushing a chain off an edge", chain carried by subcomplex
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video
10
Sep 21
homology of torus: \(H_1(\mathbb{T}^2) \simeq \mathbb{Z}\oplus
\mathbb{Z}, H_2(\mathbb{T}^2) \simeq \mathbb{Z}\), homology of
Klein bottle: \(H_1(\mathbb{K}^2) \simeq \mathbb{Z}\oplus
\mathbb{Z}_2 \), projective plane
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video
11
Sep 26
\(k\)-fold dunce hat, \(H_1(D_k) \simeq \mathbb{Z}_k\),
Möbius strip, cylinder, connected sum, \(H_1(\mathbb{R}P^2
\# \mathbb{R}P^2) \simeq \mathbb{Z} \oplus \mathbb{Z}_2 \),
\(0\)-dim homology
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video
12
Sep 28
*make-up
lecture*: reduced homology groups, \(\tilde{H_0}(K)
\oplus \mathbb{Z} \simeq H_0(K)\), homology of \(p\)-simplex:
\(\tilde{H_i}(K) = 0 \, \forall i\)
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video
13
Oct 3
relative chains, relative boundary and homology, examples,
torsion in \(H_1(K,K_0)\) for Möbius strip, excision
theorem
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video
14
Oct 5
proof of excision, chain maps and homomorphisms induced by
simplicial maps, \(\partial f_{\#} = f_{\#} \partial \),
functoriality, \(\epsilon \circ f_\# = \epsilon\)
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video
15
Oct 10
*make-up
lecture*: chain homotopy, \( \partial D + D \partial =
g_{\#} - f_{\#}\), contiguous maps, relative homology:
contiguous maps of pairs
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video
16
Oct 12
*make-up
lecture*: star condition, simplicial approximation to
\(h: |K| \to |L|\) by \(f: K \to L\) with \(h( \mathrm{St } \ v
) \subset \mathrm{St} f(v) \), example
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video
17
Oct 17
*another
make-up lecture*: subdivision, cone of \(K\) with vertex
\(\mathbf{w}\), base of cone, barycenter, barycentric
subdivision \(\mathrm{Sd}^r K\)
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video
18
Oct 19
simplices in \(\mathrm{Sd} K\), diameter, simplicial
approximation, finite case, Lebesgue number, subdivision of
\(K\) holding \(K_0\) fixed
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video
19
Oct 24
exact sequence: \(\mathrm{im} \, \phi_{i-1} = \mathrm{ker} \,
\phi_i\), short exact sequence (SES), chain complex, chain map,
homology group, SES that splits
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video
20
Oct 26
connecting homomorphism \(\partial_{\star} : H_p(K,K_0) \to
H_{p-1}(K_0)\), long exact homology sequence of a pair, use in
examples
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video
21
Oct 31
SES of chain complexes, zig-zag lemma, long exact homology
sequence, connecting homomorphism, "diagram chasing"
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video
22
Nov 2
proof of zig-zag lemma, commuting diagram for chain complexes,
five lemma, Meyer-Vietoris sequence (MVS), blog
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video
23
Nov 7
proof of Meyer-Vietoris Sequence, \( \phi(\mathbf{c}) =
(i^{'}_{\#}(\mathbf{c}^{'}), \)\( -i^{''}_{\#}(\mathbf{c^{''}}))
\), structure of \(H_p(K)\), examples, definition of
\(\partial_{*}\)
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video
24
Nov 9
homology of sphere: \(\tilde{H}_p(\mathbb{S}^d) \simeq
\tilde{H}_{p-1}(\mathbb{S}^{d-1})\), suspension:
\(\tilde{H}_p(S(K)) \simeq \tilde{H}_{p-1}(K) \), cut Klein
bottle into Möbius strips
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video
25
Nov 14
Meyer-Vietoris sequences of Klein bottle and torus, use of
lemmas from Lecture 19, categories,
\(\mathrm{hom}(X,Y)\)
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video
26
Nov 16
categories: associativity, identity, inverse, functors,
covariant and contravariant functors, examples, natural
transformation
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video
27
Nov 28
Hom functor, dual homomorphism, \( \tilde{f}(\phi) = \phi \circ
f\), simplicial cochains, coboundary \(\langle \delta \phi,
\mathbf{c} \rangle = \langle \phi, \partial \mathbf{c}
\rangle\), cohomology group
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video
28
Nov 30
elementary cochains, \(\delta \phi^p = \sum g_{\alpha} (\delta
\sigma_{\alpha}^*) \), examples of cochains and coboundaries,
cohomologous cochains on the torus
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video
29
Dec 5
examples of cohomology groups, \(H^2(\mathbb{K}^2,\mathbb{Z})
\not\simeq H_2(\mathbb{K}^2) = 0\), 0-dimensional cohomology,
relative cohomology groups
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video
30
Dec 7
examples of relative cohomology, \(H^2(\mbox{Möbius,edge})
\simeq \mathbb{Z}_2\), statements of Poincaré and
Alexander duality
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video

Last modified: Thu Dec 07 19:17:27 PST 2017