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.
   
   
3. Take-home midterm: Due Monday May 6, available Apr 30.
   
   
4. Problem Set 3: No due date.
   
   
5. Take-home final: Due Tuesday June 11, available June 7.
   
   

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.
   
   
9. Friday Apr 26: Cellular homology.
   
   Cellular chains and differential. Equivalence with singular homology.
   Computations of homologies of complex projective spaces and spheres.
   
   Textbook Reading (Apr 26): Chapter 2.2, page 137.
   
   
10. Monday Apr 29: Cellular boundary formula.
    
    Cellular boundary formula: statement and proof. Examples.
    Computations of homologies of surfaces and real projective spaces.
    
    Textbook Reading (Apr 29): Chapter 2.2, page 140.
    
    
11. Wednesday May 1: Mayer-Vietoris Sequence.
    
    Short Mayer-Vietoris Sequence and its Long Exact Sequence.
    Examples: spheres and complex projective spaces.
    
    Textbook Reading (May 1): Chapter 2.1, page 149.
    
    
12. Friday May 3: Midterm Exam.
    
    
13. Monday May 6: Coefficients and Axiomatic Approach.
    
    Homology with coefficients. Example for real projective spaces.
    Discussion on meaning of coefficients and generalizations.
    Eilenberg–Steenrod axioms. Dimensional axiom and other homology theories.
    
    Textbook Reading (May 6): Chapter 2.2 and 2.3 pages 153 and 160.
    
    
14. Wednesday May 8: Cech cohomology I: Integral coefficients.
    
    Cochain complexes. Examples: dual of chain complexes.
    Open covers and Cech cochain complex. Example of the circle.
    
    Textbook Reading (May 8): Bott-Tu's Differential Forms in Algebraic Topology, page 110.
    
    
15. Friday May 10: Cech cohomology II.
    
    Directed sets and directed systems. Colimits. Examples.
    Construction of directed system from open covers. Cech cohomology.
    
    Textbook Reading (May 10): Bott-Tu's Differential Forms in Algebraic Topology, page 110.
    
    
16. Monday May 13: Sheaf cohomology.
    
    Presheaves and sheaves: definitions and examples.
    Sheaf cohomology. Examples: smooth functions on the circle.
    
    Textbook Reading (May 13): Bott-Tu's Differential Forms in Algebraic Topology, Section 10.
    
    
17. Wendesday May 15: Sheaf cohomology II.
    
    Examples of sheaves. Example of pushfoward from universal cover.
    Correspondence between locally constant sheaves and covers.
    Description as representations of fundamental group. Examples.
    Local systems on a circle: geometric meaning of cohomology in terms of monodromy.
    
    Textbook Reading (May 15): Bott-Tu's Differential Forms in Algebraic Topology, Section 10.
    
    
18. Friday May 17: Sheaf cohomology III.
    
    Review: monodromy of local systems and computation of cohomology.
    Derived functors: definition of sheaf cohomology via resolutions.
    An application of sheaf cohomology: Leray's spectral sequence.
    Comparison between different cohomology theories.
    
    Textbook Reading (May 17): Bott-Tu's Differential Forms in Algebraic Topology, page 110.
    
    
19. Monday May 20: Ext and Tor groups.
    
    Examples of failure of commutativity of homology and algebraic operations.
    Duality in the category R-mod. Right exactness of Hom functor and failure to left exactness.
    Statement of universal coefficients. Construction of Ext groups via projective resolutions.
    
    Textbook Reading (May 20): Chapter 3.1.
    
    
20. Wednesday May 22: Ext groups.
    
    Properties of Ext groups. Computation in the case of Abelian groups.
    Proof of universal coefficients. Connection to extensions of groups.
    
    Textbook Reading (May 22): Chapter 3.1.
    
    
21. Friday May 24: Ext groups and loop spaces.
    
    Review of universal coefficient theorem. Computation for real projective spaces.
    Loop spaces. Serre fibrationa and homotopy groups of loops spaces. Remarks on loop spaces.
    Adams' Theorem computing the homology of loop spaces in terms of Ext groups.
    
    Textbook Reading (May 24): Chapter 3.1, Morse Theory (Chapter I and III.1) and Adams' paper.
    
    
22. Wednesday May 29: Kunneth Formula and Tor groups.
    
    Properties of Ext groups. Computation in the case of Abelian groups.
    Proof of universal coefficients. Connection to extensions of groups.
    
    Textbook Reading (May 29): Chapter 3, Parts of Appendices 3.A and 3B.
    
    
23. Friday May 31: Serre's intersection formula.
    
    Review of derived functors. An exercise in the intersection of algebraic varieties.
    Failure of (underived) tensor product to give intersection multiplicity. (Example 1.1.1 in Appendix A.1).
    Serre's Intersection formula. Computation of Tor groups in example and correct multiplicity.
    
    Textbook Reading (May 31): Appendix A.1 of Hartshorne's Algebraic Geometry.
    
    
24. Monday June 3: h-cobordism Theorem I.
    
    Introduction to the problem. Statement of h-cobordism theorem.
    Existence of exotic spheres: Milnor's construction and Brieskorn spheres.
    Sketch of the proof of h-cobordism theorem.
    
    Textbook Reading (June 3): Milnor's Lectures on the h-cobordism theorem.
    
    
25. Wedday June 5: h-cobordism Theorem II.
    
    Corollaries of h-cobordism theorem: characterization of disks.
    Generalized Poincare conjecture. Proofs of corollaries.
    The geometric and algebraic cancellation lemmas.
    
    Textbook Reading (June 5): Milnor's Lectures on the h-cobordism theorem.