Course Information

Lectures: MWF at 3:10pm in Bainer Hall 1128. Attendance not mandatory.

Textbook: Algebraic Topology by A. Hatcher.

Office Hours: The office hours of Professor Casals (casals--at--ucdavis.edu) are Fridays 2-3pm at MSB 3214. Always feel free to ask after class.

Important Dates: First Day (Apr 1), Midterm Exam (May 3) and Final Exam (June 10).

Grade: The grade will be determined by Problem Sets (40%), Midterm Exam (20%) and a Final Exam (40%).

Problem Sets: Weekly assignments due on Fridays at 9pm, submitted through Gradescope.

Problem Sets

1. Problem Set 1: Due Friday Apr 12, available Apr 4.
   
   
2. Problem Set 2: Due Friday May 3, available Apr 25.
   
   

Math Diaries

1. Monday Apr 1: Introduction to MAT-215B.
   
   Topological spaces and algebraic invariants. Whitehead's Theorem on homotopy equivalence.
   Algebraic invariants from topological spaces. Computability issues with homotopy groups.
   Instances of homology theories based on presentation of a space: singular, simplicial, cellular & Cech.
   Definition of Chain complexes. Examples: deRham and simplicial. Definition of homology.
   
   Textbook Reading (Apr 1): Chapter 2.0 (Intro to Chapter 2 until 2.1, not included).
   
   
   
2. Wednesday Apr 3: Simplicial Homology.
   
   Review of strategy. Definition of n-simplex and examples.
   Decompositions with simplicies: simplicial, Δ, CW and cell.
   Boundary of a simplex. The boundary of a boundary is zero.
   Simplicial chain complex and simplicial homology.
   Definition of Δ-complex and examples.
   
   Textbook Reading (Apr 3): Chapter 2.1 (until singular homology, not included).
   
   
3. Friday Apr 5: Computations and Examples in Simplicial Homology I.
   
   Homology of the circle with two different Δ-complexes.
   Homology of a wedge of circle (and so any open surface).
   Argument for independence of Δ-structure.
   Issues with functoriality and homotopy invariance.
   Homology for the closed orientable genus g surface.
   
   Textbook Reading (Apr 5): Chapter 2.1, until singular homology, and exercises.
   
   
4. Monday Apr 8: Computations and Examples in Simplicial Homology II.
   
   Homology under operations: wedges, connected sums and products.
   Homology under suspension. Universal covers and Eilenberg-MacLane spaces.
   Homology of the 2-sphere. Intuition for simplicial cycles.
   Homology versus homotopy of 2-torus. Torsion in homology.
   
   Textbook Reading (Apr 8): Chapter 2.1, until singular homology, and exercises.
   
   
5. Wednesday Apr 10: Singular Homology.
   
   Definition of singular chains and differencial. Singular homology.
   First properties and computations (case of a point, disjoint unions). Remarks.
   Definition of chain maps. Maps induced in homology by chain maps.
   Continuous maps induce chain maps in singular chains. Examples and non-examples.
   
   Textbook Reading (Apr 10): Chapter 2.1, until homotopy invariance.
   
   
6. Friday Apr 12: Homotopy invariance of Singular Homology.
   
   Invariance of homology under homeomorphism.
   Chain maps. Chain homotopy between chain maps.
   The prism operator associated to a homotopy of maps.
   Homotopic maps induce chain homotopic maps. Homotopy invariance.
   
   Textbook Reading (Apr 12): Chapter 2.1, until excision.
   
   
7. Monday Apr 22: Equivalence between simplicial and singular homology.
   
   Naturality of l.e.s. for maps of pairs. (Functoriality.)
   Reduction from the relative case to the absolute case.
   Five-Lemma and its proof: diagram chasing arguments.
   Proof that simplicial and singular homologies are isomorphic.
   
   Textbook Reading (Apr 22): Chapter 2.1, until page 131.
   
   
8. Wednesday Apr 24: Summary thus far and Applications.
   
   Textbook Reading (Apr 22): Chapter 2.2.