Math 8306: Class Outlines, Fall 2016

12/09/16: The last class meeting this term. Homework is due at the beginning of the class. Homework discussion: Problem 8. Simplicial sets and simplicial complexes. Simplicial sets and topological spaces: the geometric realization of the singular set of a space. [Class notes. Text: Sections 2.3 (160-165), 3.1 (197-202), Section 3.2 (Example 3.7), 2.1 (128-130) and 3.B (268-273, done using cellular homology), Matt Ando's lectures online (Lecture Notes 5), Section 3.B (273-275); May's Concise Course: Sections 16.2 and 16.4]

12/07/16: The cohomology ring of the torus. Simplicial sets: geometric realization. [Class notes. Text: Section 3.2 (Example 3.7); May's Concise Course: Section 16.2]

12/05/16: Naturality of the cup product. Simplicial sets. The singular complex. [Class notes. May's Concise Course: Sections 16.1 and 4]

12/02/16: The Alexander-Whitney map: proving it is a homotopy equivalence. The cup product and the external (cross) product. [Class notes. Matt Ando's lectures online (Lecture Notes 5, Sections 1, 2, 3, and 6)]

11/30/16: The Alexander-Whitney map. [Class notes. Matt Ando's lectures online (Lecture Notes 5, Section 3)]

11/28/16: Extended Thanksgiving break: no class.

11/25/16: Thanksgiving Break: University closed.

11/23/16: Extended Thanksgiving break: no class.

11/21/16: The proof of the algebraic Künneth theorem. [Class notes. Text: Section 3.B (273-275)]

11/18/16: The proof of the Eilenberg-Zilber theorem. [Class notes. See, for example, Matt Ando's lectures online (Lecture Notes 5, Section 6)]

11/16/16: The acyclic model theorem. [Class notes. See, for example, Matt Ando's lectures online (Lecture Notes 5, Section 5)]

11/9/16: Singular cohomology. The Eilenberg-Steenrod axioms for (co)homology. [Class notes. Text: Sections 2.3 (160-165), and 3.1 (197-202)]

11/7/16: The Mayer-Vietoris sequence for singular homology. Computations. [Class notes. Text: Sections 2.1 (124-126, 149-153)]

11/4/16: Excision (proof finished). [Class notes. Text: Section 2.1 (122-124)]

11/2/16: Excision (proof). [Class notes. Text: Section 2.1 (119-122)]

10/31/16: Singular homology: (limited) computations. Excision: preliminaries. [Class notes. Text: Section 2.1 (109-110, 119)]

10/28/16: Homework is due at the beginning of the class. Discussion of Problems 2 and 5. Homotopy invariance of singular homology. [Class notes. Text: 2.2 (138-141, 144, 146-147, 153), 2.C (179-181), and 3.3 (239-242, there done differently, via cap product)); May's Concise Course: Sections 21.1 and 21.3; Text: Section 2.1 (110-113)]

10/26/16: Application of the Lefschetz fixed-point theorem to vector fields on the even-dimensional sphere and maps of even-dimensional real projective spaces. Singular homology: definition. [Class notes. Text: Section 2.1 (108-110)]

10/24/16: The Lefschetz fixed-point theorem. [Class notes. Text: Section 2.C (179-181)]

10/21/16: The Euler characteristic of a closed manifold: general properties and computations. Relation to vector fields and cobordisms. [Class notes. May's Concise Course: Sections 21.1 and 21.3]

10/19/16: Consequences of Poincaré duality. The Euler characteristic (definition). [Class notes. Text: Section 2.2 (146-147)]

10/17/16: Poincaré duality: proofs. The intersection number. [Class notes. Text: Sections 2.2 (pages 137-141, 144, 153), 3.1 (190-196, 198), 3.3 (236-238, 241, 256), 3.A (261-266); Section 3.3 (239-242, done differently, via cap product)]

10/14/16: Homework is due at the beginning of the class. Discussion of homework. Poincaré duality: construction of a dual CW complex for a triangulated compact manifold. [Class notes. Text: Sections 2.2 (pages 137-141, 144, 153), 3.1 (190-196, 198), 3.3 (236-238, 241), 3.A (261-266); Section 3.3 (239-242, done differently, via cap product)]

10/12/16: Cellular homology. Poincaré duality. [Class notes. Text: Sections 2.2 (138-141, 144, 153) and 3.3 (241)]

10/10/16: CW complexes. [Class notes. Text: Chapter 0 (p. 5), Appendix (519-521), Section 2.2 (137-138)]

10/7/16: Fundamental classes. CW complexes. [Class notes. Text: Sections 3.3 (236-238), 2.2 (137-138), and Appendix]

10/5/16: The universal coefficient theorems. [Class notes. Text: Sections 3.1 (195, 198), 3.A (264)]

10/3/16: The long exact sequences for Tor's and Ext's. Computations of Tor's and Ext's. [Class notes. Text: Section 3.1 (195-196), Section 3.A (263-264, 265-266)]

9/30/16: Homework is due at the beginning of the class. Discussion of homework: Problem 6. [Class notes. Text: Section 2.2 (pages 110, 158 (Exercise 32), 147-150), Section 3.1 (185-193 through the Exercise, 197-200, 195), Section 3.A (263)]

9/28/16: The groups Tor and Ext. [Class notes. Text: Section 3.1 (190-195, 198), Section 3.A (261-264)]

9/26/16: Split short exact sequences. Cohomology of a pair. The long exact sequence of a pair. Computations of Hom's between abelian groups. [Class notes. Text: Section 2.2 (147-148), Section 3.1 (199-200)]

9/23/16: Simplicial cohomology: (cumbersome) duality between cohomology and homology with coefficients in a field. [Class notes. Read a brushed-up duality argument here. Text: Section 3.1 (191-193 through the Exercise and 198). Beware that in the text these are done for singular homology, whereas we will be doing all that for simplicial homology.]

9/21/16: The suspension isomorphism. Simplicial cohomology. General discussion of duality between homology and cohomology. [Class notes; Text: Section 2.2 (page 158 (Exercise 32)), Section 3.1 (185-191, 197-198). Beware that in the text these are done for singular homology, whereas we will be doing all that for simplicial homology.]

9/19/16: Reduced homology. The Mayer-Vietoris sequence. [Class notes. Text: Section 2.2 (pages 110, 149-150). Beware that in the text these are done for singular homology, whereas we will be doing all that for simplicial homology.]

9/16/16: Homework is due at the beginning of the class. Discussion of homework. Some applications of homology. More homological algebra and applications to relative homology. [Class notes. Text: Section 2.1 (pages 104-107, 110-113, 115-117). Beware that in the text these are done for singular homology, whereas we will be doing all that for simplicial homology.]

9/14/16: The functoriality and invariance of simplicial homology. Relative homology. The long exact sequence of a pair. [Class notes. Text: Section 2.1 (pages 110-113, 115-117). Beware that in the text these are done for singular homology, whereas we are doing all that for simplicial homology.]

9/12/16: Simplicial homology: first computations. [Class notes. Text: Section 2.1 (pages 106, 109-110). Beware that in the text these are done for Δ-complexes and their simplicial homology or for singular homology, whereas we are doing that for simplicial complexes and their simplicial homology.]

9/9/16: Abstract simplicial complexes. Relation to simplicial complexes. Geometric realization. Simplicial homology (definition). [Class notes. Text: Section 2.1 (pages 104-106, 107).]

9/7/16: Introduction. Syllabus handed out. Simplices in Rⁿ and related notions. The notion of a simplicial complex in Rⁿ. [Syllabus. Class notes. Text: Chapter 2 (pages 97-103, 107); note the difference between the notion of a simplicial complex and a Δ-complex. We will not be using Δ-complexes.]

Last modified: (2016-12-09 16:27:20 CST)