Date  Topic 
Review of and Crash Course in General Topology 
Aug 28  Topological Vocabulary:
Ch. I, Sections 23 
Aug 30  Mostly connectedness:
Ch. I, Section 4; plus definition of Hausdorff: Ch. I, Section 5 
Sep 1  Compactness, proper mappings, product topology: Ch. I, Sections 78 
Sep 6  Quotient topology: Ch. I, Section 13 
Sep 8  Homotopy: Ch. I, Section 14 
Sep 11  More Homotopy: Ch. I, Section 14 
A brief tour of differentiable manifolds:
This material provides motivation and examples for what we will be studying.
For those interested, an undergraduate level treatment can be found in Guillemin and Pollack, Differential Topology, which is inspired by the wellknown introduction by Milnor, Topology from the Differentiable Viewpoint. 
Sep 13  Definitions of (smooth) manifold and examples: Ch. II, Sections 24 
Sep 15  Poincare Conjecture, Exotic Spheres, the graph of the absolute value function in
R^{2} as a manifold (Exercise II.2.4: compare with the discussion of the cone
following Definition II.4.5), manifold with boundary (Definition II.2.7),
brief discussion without proof of classification of one and two dimensional
compact connected manifolds, allowing boundary in the 1dimensional case,
tangent vectors & differentials (Ch. II, Sec. 5),
Tangent Bundles (Ch. II, Sec. 9), and the Whitney Embedding Theorem
without proof (Ch. II, Sec. 10) 
Homotopy & Fundamental Group 
Sep 18 & 20  Definition of fundamental group and
the computation of the fundamental group for
R^{2}\{(0,0)} using "net change in argument" (or winding
numbers). This is a variation on Ch. III, Sec. 12. 
Sep 22  Spheres are simply connected (Th. III.2.2) and
fundamental group of a product space (Th. III.2.6) 
Sep 25  Covering Spaces: Ch. III, Sec. 3 
Sep 27  The Lifting Theorem: Ch. III, Secs. 4
& 5 
Sep 29  Deck Transformations: Ch. III, Sec. 6 
Oct 2  SeifertVan Kampen: Ch. III, Sec. 9 
Oct 4  Properly discontinuous actions (Ch. III,
Sec. 7), Lens Spaces, Classification & existence of coverings
(Ch. III, Sec. 8),
SO(3,R) (Ch. III, Sec. 10) 
Oct 6  TEST: The test will emphasize
definitions, vocabulary, statements of theorems, and "easy" consequences,
for example calculating the fundamental group of RP^{2}.
It will cover
Ch. I, Sec. 14 (homotopy), and Ch. III, Secs. 16
(fundamental group & covering spaces).
Click here for the test solutions. 
Singular Homology 
Oct 9  Singular simplices & definition of homology:
Ch. IV, Sec. 1 
Oct 11  Augmentation map, zero homology group, and reduced
homology: Ch. IV, Sec. 2 
Oct 13  H_{1} is isomorphic to the Abelianized
fundamental group: Ch. IV, Sec. 3 
Oct 16  A less than successful introduction to homological
algebra and diagram chasing: Ch. IV, Sec. 4 
Oct 18  A more successful day diagram chasing:
Ch. IV, Sec. 4 
Oct 20  EilenbergSteenrodMilnor Axioms: Ch. IV,
Sec. 6 
Oct 23  Homology of Spheres & Brower Fixed Point
Theorem: Ch. IV,
Sec. 6 
Oct 25  Degree of the antipodal map & consequences
for even dimensional spheres: Ch. IV, Sec. 6

Oct 27  Local Homology H_{i}(X,X\{x}); Invariance of dimension (see the discussion following Corollary IV.19.10); Homology of RP^{2}: Ch. IV, Sec. 7 
Oct 30  CW Complexes (definition) and examples:
Ch. IV, Sec. 8 
Nov 1  Start toward cellular homology: Lemma IV.10.1 
Nov 3  TEST: Chapter IV, Sections 16 (up
to and including 6.14), plus Section 8.
Click here for the test solutions. 
Nov 6  Cellular homology: page 202 and a sketch
of Theorem 10.3. Homology of real projective space: Ch. IV,
Sec. 14 
Nov 8  Chain homotopy and singular homology of
contractible spaces: Ch. IV, Sec. 15 
Nov 10  Cross product and the proof that singular homology
satisfies the homotopy axiom: Ch. IV, Sec. 16 
Nov 13  Subdivision, Excision, and MayerVietoris:
Ch. IV, Sec. 1718 
Nov 15  Embeddings of disks and spheres into spheres:
Ch. IV, Sec 19; consequences of the BorsukUlam Theorem:
Ch. IV, Sec 20 
Nov 17  BorsukUlam Theorem: Ch. IV, Sec 20 
Cohomology 
Nov 20  Differential forms & de Rham cohomology:
Ch. V 
Nov 22  Differential forms & de Rham cohomology:
Ch. V 
Nov 27  
Nov 29  
Dec 1  TEST:
Know the definitions of the following concepts and statements of the
following theorems: The cross product theorem (Th. IV.16.1),
MayerVietoris (Th. IV.18.1), Generalized
Jordan Curve Theorem (Th. IV.19.4), Invariance of Domain
(Cor. IV.19.9), BorsukUlam Theorem and its consequences (Ch. IV,
Sec. 20), differential form, exterior derivative, de Rham cohomology,
singular cohomology, de Rham's Theorem (Th. V.9.1) 
Dec 5  
Dec 7  
Dec 11  FINAL: 