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