Algebraic Topology - Lecture Notes and Videos

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 \(\mathbb{R}^1 \not\approx \mathbb{R}^2\) scribe video
2 Aug 24 open set, topological space, geometrically independent (GI), \(n\)-plane, \(n\)-simplex, barycentric coordinates, \(2\)-simplex scribe video
3 Aug 29 properties of simplices, vertex, dimension, faces, boundary, simplicial complex \(K\), subcomplex, \(p\)-skeleton \(K^{(p)}\) scribe 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 scribe video
5 Sep   5 simplicial maps, abstract simplicial complex (ASC), geometric realization theorem, ASCs for cylinder and Möbius strip scribe video
6 Sep   7 ASCs for torus and Klein bottle, groups, homomorphism, finitely generated, torsion, internal direct sum, direct product scribe video
7 Sep 12 results on finitely generated abelian groups, orientation of simplex, \(p\)-chain, group of \(p\)-chains \(C_p(K)\), elementary chains scribe video
8 Sep 14 boundary homomorphism \(\partial_p(K)\), fundamental lemma of homology: \(\partial\partial = 0\), homology group \(H_p(K) = Z_p(K)/B_p(K)\) scribe 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 scribe 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 scribe 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 scribe 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: \(\tilde{H_i}(K) = 0 \, \forall i\) scribe video
13 Oct   3 relative chains, relative boundary and homology, examples, torsion in \(H_1(K,K_0)\) for Möbius strip, excision theorem scribe video
14 Oct   5 proof of excision, chain maps and homomorphisms induced by simplicial maps, \(\partial f_{\#} = f_{\#} \partial \), functoriality, \(\epsilon \circ f_\# = \epsilon\) scribe video
15 Oct 10 make-up lecture: chain homotopy, \( \partial D + D \partial = g_{\#} - f_{\#}\), contiguous maps, relative homology: contiguous maps of pairs scribe 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 scribe video
17 Oct 17 another make-up lecture: subdivision, cone of \(K\) with vertex \(\mathbf{w}\), base of cone, barycenter, barycentric subdivision \(\mathrm{Sd}^r K\) scribe video
18 Oct 19 simplices in \(\mathrm{Sd} K\), diameter, simplicial approximation, finite case, Lebesgue number, subdivision of \(K\) holding \(K_0\) fixed scribe 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 scribe 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 scribe video
21 Oct 31 SES of chain complexes, zig-zag lemma, long exact homology sequence, connecting homomorphism, "diagram chasing" scribe video
22 Nov  2 proof of zig-zag lemma, commuting diagram for chain complexes, five lemma, Meyer-Vietoris sequence (MVS), blog scribe 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 \(\partial_{*}\) scribe 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 scribe video
25 Nov 14 Meyer-Vietoris sequences of Klein bottle and torus, use of lemmas from Lecture 19, categories, \(\mathrm{hom}(X,Y)\) scribe video
26 Nov 16 categories: associativity, identity, inverse, functors, covariant and contravariant functors, examples, natural transformation scribe 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 scribe 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 scribe 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 scribe video
30 Dec   7 examples of relative cohomology, \(H^2(\mbox{Möbius,edge}) \simeq \mathbb{Z}_2\), statements of Poincaré and Alexander duality scribe video

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