Week 1


March 26

 Topology review, surfaces, connected sum of surfaces.
 Read: Chapter 1, Sections 1  5, Section 8 (lightly)
 Homework for Week 1 can be found at Homework Week 1.

March 28

 Cut and paste, classification of surfaces, homotopy.

March 30

 Homotopy: retracts, deformation retracts, contractible spaces, homotopy equivalences.
 Read: Chapter 2, pp. 43  46

Week
2


April 2

 Paths in a space, the fundamental group.
 Read: Chapter 2, Sections 1  4
 Homework for Week 2 can be found at Homework Week 2.

April 4 
 Homework for Week 1 due today.
 The fundamental group: Change of base point, the fundamental group of a product, abelian fundamental groups.
 Read: Chapter 2, Sections 7 & 8

April 5

 Xhour, 1:00  1:50
 Free products of groups, the Seifertvan Kampen theorem.
 Read: Class notes; Chapter 4, Sections 1 & 2 (lightly)

April 6 
 Calculation of fundamental groups.
 Read: Chapter 4, Sections 3  5

Week
3


April 9 
 Covering spaces: definitions and examples.
 Read: Chapter 5, Sections 1 & 2
 Homework for Week 3 can be found at Homework Week 3.

April 11 
 Homework for Week 2 due today.
 Lifting properties of covering spaces.
 Read: Chapter 5, Sections 3  5

April 13 
 Applications: The degree of a map, the fundamental group of a circle, the fundamental theorem of algebra.
 Read: Class notes

Week
4


April 16

 Conjugate subgroups of the fundamental group, the Galois correspondence between subgroups and coverings.
 Read: Class notes
 Homework for Week 4 can be found at Homework Week 4.

April 18 
 Homework for Week 3 due today.
 The Galois correspondence between subgroups and coverings, the group of deck transformations.
 Read: Class notes; Chapter 5, Section 6

April 19 
 Xhour, 1:00  1:50
 Consequences of the Galois correspondence, free groups and the fundamental group of a graph.

April 20 
 Sketch of possible future topics: groups operating properly on a space, higher homotopy groups.
 For next class review freeabelian groups (Chapter 3, Section 3)

Week
5


April 23 
 We begin homology theory. We will follow the text more closely than before.
 The (cubical) singular chain complex, definition of homology groups.
 Homework for Week 4 is due today.
 Read: Chapter 7, Sections 1 & 2

April 25 
 Connectedness and zero dimensional homology, induced homomorphisms
 Read: Chapter 7, Sections 2 & 3

April 27 
 The homotopy theorem, homotopy type
 Read: Chapter 7, Section 4
 The takehome midterm is due to be handed in today.
 Homework for Week 6 can be found at Homework Week 6.

Week
6


April 30

 Chain complexes, the five lemma
 Read: Chapter 10, Sections 1 & 2 (up to top of p. 258), Chapter 7, p. 184

May 2

 Homework for Week 6 is due today.
 Homology of a pair, the exactness theorem
 Read: Chapter 7, Section 5

May 4 
 Vsmall theorem, excision theorem, EilenbergSteenrod axioms
 Read: Chapter 7, Section 6 and class notes
 Homework which covers Week 6 can be found at Homework Week 7 (revised).

Week
7


May 7 
 Homology groups of the ball and sphere, degree of a map
 Chapter 8, Sections 1 & 2

May 9 
 Applications: The Brouwer fixed point theorem, vector fields, maps of spheres
 Read: Chapter 8, Section 2
 Homework Week 7 (revised) is due today.

May 11 
 Attachment of cells
 Read Chapter 9, Sections 1 & 2
 Homework which covers Week 7 can be found at Homework Week 8.

Week
8


May 14

 CWcomplexes
 Read: Chapter 9, Section 3

May 16 
 Homework Week 8 is due today.
 Homology of CWcomplexes, isomorphism between singular and CW homology
 Read: Chapter 9, Section 4

May 17 
 Xhour, Thursday, 1:00  1:50
 Homology of CWcomplexes continued, Euler characteristic

May 18

 Homework which covers Week 8 can be found at Homework Week 9.
 Computations in CW homology

Week
9


May 21 
 The MayerVietoris sequence
 Read: Chapter 8, Section 5

May 23

 Homework Week 9 is due today.
 Applications: The JordanBrouwer theorem, invariance of domain
 Read: Chapter 8, Section 6

May 25 
 Introduction to cohomology theory
 Read: Chapter 12, Sections 1  3

Week
10


May 28 
 First day of reading period  NO CLASS

May 30

 LAST CLASS
 Cohomology theory
