| SES # | TOPICS | INSTRUCTIONS |
|---|---|---|
| 1 | Organizational Meeting | |
| 2 | n-manifolds and Orientability, Compact, Connected 2-manifolds | Lecture 1 Define the notion of an n-manifold, and give examples. Discuss orientability Some ideas: how exactly is S^n seen to satisfy the definition of an n-manifold? You could give other examples - consulting other references perhaps - but Lecture 2 is devoted to talking about the examples which are compact surfaces, so try to not dwell on these. Non-examples could be discussed. Is the closed unit disk a manifold? Why or why not? What does orientability mean intuitively? Rigorously? How does the determinant condition fit into the Mobius band example? Is S^n orientable for all n? Lecture 2 Give examples of compact connected 2-manifolds. Important examples: 2-sphere (which will probably appear in Lecture 1), the torus, the projective plane, the Klein bottle Some things to perhaps be addressed: What do these look like? (draw pictures) How do we give precise mathematical descriptions (gluing edges: the quotient topology, equations) - how can we use these precise mathematical descriptions to verify that these examples satisfy the definition of a manifold? Which ones are orientable, which ones are not? How can we visualize the projective plane? Building other examples: discuss the connect sum |
| 3 | Classification Theorem for Compact Surfaces, Triangulation | Lecture 3 State the classification theorem of compact surfaces - our short-term goal is to prove it Ideas: explain the canonical forms for our surfaces. Why do these give use the surfaces we think they do? How is the connect sum showing up in the polygons representation? How do we recognize orientability in these. Do many examples - what does a three holed torus connect sum a projective space look like in polygonal representation? Why are these things compact surfaces? Lecture 4 Explain the notion of a triangulation. Then embark in the first step of the proof of the classification of surfaces: every compact surface can be viewed as the quotient of a disk with edges appropriately identified Ideas: Give examples of triangulations, and non-examples. Try to conceptualize the elaborate argument in step 1, and give an example of it to get a feel for how it works |
| 4 | Classification Theorem for Compact Surfaces (cont.), Euler Characteristic | Lecture 5 Finish the proof of the Theorem 7.2 - every compact surface fits into the classification. (Then we just need to convince ourselves that these surfaces are all different) Ideas: These are some elaborate arguments - see if you can pick out the main points and summarize them for us. You may not have time to talk about lemma 7.1, but we can read about it if that's the case. An example would be great - probably from step 1 Lecture 6 Define the Euler characteristic. Compute the euler characteristics of all of the compact surfaces. Deduce that the surfaces occurring in the classification are all non-homeomorphic. Ideas: why is the notion of Euler characteristic as presented in this section unsatisfactory? Consider looking at exercise 8.1. Do each of the steps 2-5 of the proof of Theorem 7.2 preserve the Euler characteristic? |
| 5 | Review of Group Theory, Homotopy and the Fundamental Group | Lecture 7 Review of group theory This material does not appear in Massey. This is supposed to be a review of the most basic notions of group theory. It would be great if all of the following concepts could be briefly described - Define what a group is - Examples - (Z,+), (R,+), (R\0,*) - Multiplicative versus additive notation for Z - The cyclic groups of finite order - Subgroups - Group homomorphisms - Normal subgroups and quotients Lecture 8 Define homotopy equivalence of paths. Why is it an equivalence relation? Define the product of paths. Show the product is associative. Define the inverse path, and show it is an inverse with respect to the path product. Define the fundamental group. Do not discuss the dependence of pi_1 on the basepoint - that will be done in Lecture 9 |
| 6 | The Fundamental Group (cont.), Homotopy Equivalence and Homotopy Type | Lecture 9 Explain the dependence of the fundamental group on the basepoint. Describe the effect of a continuous mapping on the fundamental group. Show that it is a group homomorphism. Define the notion of homotopic maps, and relative homotopy. Explain theorem 4.1. Do not discuss the notion of a retract or deformation retract (that is for Lecture 12) Ideas: Explain these important notions intuitively, draw schematic pictures to accompany your abstract definitions. Discuss exercise 4.1. Give examples of homotopic maps, relatively homotopic maps, and homotopic maps which are not relatively homotopic Lecture 10 Prove Lemma 8.1 and Theorem 8.2. Define the notions of homotopy equivalence and homotopy type. Deduce Theorem 8.3 Ideas: give examples of homotopy equivalent spaces, and spaces which "appear" not homotopy equivalent. Is homotopy equivalence an equivalence relation. Why is it hard to show two spaces are not homotopy equivalent? Is the 1 point space equivalent to the (discrete) 2 point space? Look at exercise 8.2 |
| 7 | The Fundamental Group of a Circle, Retracts, Brower Fixed-Point Theorem | Lecture 11 Prove the Very Important Theorem: pi_1(S1) = Z Lecture 12 Define the notion of retract. Deduce that retracts give rise to monomorphisms and epimorphisms of fundamental groups. Define the notions of deformation retract, contractible, and simply connected. State and prove the Brower fixed point theorem Ideas: Give examples and non-examples of retracts, deformation retracts, simply connected spaces, contractible spaces |
| 8 | Weak Product of Groups, The Fundamental Group of a Torus, Free Abelian Groups | Lecture 13 Define the product and weak product of groups. Discuss its "universal property" of the weak product (direct sum) of Abelian groups. At least state Theorem 7.1 of chapter 2, which says that the fundamental group of a product is the product of fundamental groups - and if there is time, prove it. Explain how this allows us to compute the fundamental group of the torus Lecture 14 Explain the notion of a free abelian group, and how arbitrary abelian groups are quotients of free abelian groups by "relations". Explain what Theorem 3.6 is saying. Give examples |
| 9 | Free Products, Free Groups, Presentations of Groups | Lecture 15 Explain the free product, and what a free group is. Do not cover the commutator subgroup discussion at the end of section 5 - That will be covered in Lecture 16 Lecture 16 Discuss the notion of a presentation of a group - give examples. You can talk about the commutator subgroup (end of section 5) in connection with this |
| 10 | Siefert-Van Kampen Theorem and its Generalization | Lecture 17 State the Siefert-Van Kampen Theorem 2.1, and its generalization Theorem 2.2. Prove Lemma 2.3. This theorem has a rather abstract statement - what, intuitively, is it saying? (Consult Hatcher for a more palatable formulation, Theorem 1.20 of Chapter 1) Lecture 18 Summarize the proof of Theorem 2.2. There are a lot of details here - try to pick out the main points |
| 11 | Applications of the Siefert-Van Kampen Theorem, Structure of the Fundamental Group of a Compact Surface | Lecture 19 Prove Theorem 3.1. Compute the fundamental group of the "rose with n pedals". Deduce the fundamental group of the n-punctured plane. Give an intuitive discussion of Lemma 3.2 Lecture 20 Prove Theorem 4.1. Use it to give an alternative computation of the fundamental group of the torus (5.1). Compute the fundamental group of the projective plane (5.2) |
| 12 | Fundamental Groups on Closed Surfaces, Application to Knot Theory | Lecture 21 Use the computations of the fundamental group of the torus and projective plane to compute the fundamental groups of the rest of the closed surfaces. This is Prop 5.1. If you have time, dwell upon what the group for a 2-holed torus "looks like" Lecture 22 Explain what a knot is, and what it means for two knots to be equivalent. Explain exercise 6.1 and deduce Proposition 6.1. Briefly describe the torus knots and outline the proof of Prop 6.2. Do not worry about showing that these fundamental groups are non-isomorphic |
| 13 | Covering Spaces, Path Lifting Lemma, Homotopy Lifting Lemma | Lecture 23 Define the notion of a covering space. Select examples from the text that best illustrate the subtleties of this definition Lecture 24 Prove the very important path lifting and homotopy lifting lemmas (lemmas 3.1, 3.2, 3.3). If you have time, discuss lemma 3.4 |
| 14 | Fundamental Group of a Covering Space, Lifting of Arbitrary Maps to a Covering Space | Lecture 25 Deduce theorem 4.1 and 4.2. Select examples from section 2 to work out instances of these theorems Lecture 26 Prove Theorem 5.1. How does this specialize to prove lemmas 3.1 and 3.3? This might be a short lecture, but I don't want to tack on another section |
| 15 | Homomorphisms and Isomorphisms of Covering Spaces, Action of the Fundamental Group on Fibers of Covering Spaces | Lecture 27 Discuss homomorphisms and isomorphisms of covering spaces - consider discussing exercise 6.2 Lecture 28 Explain the action of the fundamental group on the fibers of a covering space, and identify the "isotropy" of this action. Prove theorem 7.2, which identifies the group of "Deck transformations" |
| 16 | Regular Covering Spaces and Quotient Spaces, Borsuk-Ulam Theorem for the 2-sphere | Lecture 29 The most important thing here is to show that the covering space may be regarded as a quotient of a space by a suitably "discrete" group action. There are two counterexamples at the end of the chapter exhibiting some bizarre phenomena - they are rather detailed, but you can touch on the main point of these counter examples. Lecture 30 The Borsuk-Ulam theorem is one of those wonderful theorems that we now have the technology to prove. This section is short - you could also do the exercises |
| 17 | The Existence Theorem for Covering Spaces, Induced Covering Space over a Subspace | Lecture 31 We've talked about the properties of covering spaces, but do these fabulous beasts always exist? The answer is - yes - provided the space is "semi-locally simply connected" Lecture 32 For us a covering space has always been path connected. If you restrict a covering to a subspace, it will be a covering, except for the fact that it might not be path connected. Prop 11.2 gives a nice criterion. Work through some examples |
| 18 | Graphs, Trees, Fundamental Group of a Graph | Lecture 33 These sections (Chapter 6, sections 2-3) contain many definitions and concepts concerning graphs. Explain these, and draw a lot of pictures to illustrate them Lecture 34 Define the notion of a tree, explain why trees are contractible, and explain the notion of maximal trees. Show that the fundamental group of a graph is free, and if there is time, discuss the number of generators. We already touched on some of this in our discussions, but it is worthwhile to prove this in detail |
| 19 | Euler Characteristic and Coverings of Graphs, Generators of Subgroups of Free Groups | Lecture 35 Discuss the relationship between Euler characteristic and the fundamental group of a graph. Discuss what the coverings of a graph look like. Prove the all important theorem 7.2 Lecture 36 Explain how to produce generators of subgroups of free groups. Discuss this remarkable theory in the context of many examples |
| 20 | Delta Complex, Singular Chains, Homology | Lecture 37 Define a Delta complex. Define the singular chains of a Delta complex, and define the boundary homomorphism (up to lemma 2.1). Do not prove lemma 2.1 Lecture 38 Begin with the proof of lemma 2.1. Define homology. Present some of the simple computations that follow |
| 21 | Singular Homology, The Homomorphism pi_1(X) -> H_1(X) | Lecture 39 Define singular homology. Hatcher gives a very interesting interpretation of homology: cycles are maps of delta complexes into the space X. What does it mean if a cycle is the boundary of another chain? Ideas: Examples of this philosophy could be used in the examples of the homology of the torus and the projective plane (as described in the last Lecture). What is the geometric meaning of the 2-torsion in H_1(P)? Prove 2.6, 2.7, 2.8. Do not define reduced homology Lecture 40 Prove that there is a homomorphism pi_1(X) -> H_1(X). Show that this is the abelianization. Deduce, from our fundamental group computations, that an orientable surface of genus g has an H_1 of rank 2g. What do the generators look like? |
| 22 | Degree of a Map and its Applications, Higher Homotopy Groups | Lecture 41 Last time I outlined how the Meier-Vietoris sequence computes H_n(S^n). Explain how this computation may be also deduced from simplicial homology, just in the case of n=2, where you can be explicit. (This is easiest to do with the delta-complex structure where you glue two 2-simplices together along their common boundary) Discuss some of the applications given in Hatcher (as time allows): the degree of a map between spheres, Theorem 2.28, and Proposition 2.29. Some of the degree discussion relies on the homotopy invariance of homology - just assume this, even though we have not proved it Lecture 42 Discuss the following: - Definition of higher homotopy groups - Prove the higher homotopy groups are abelian - explain Hatcher's picture based argument - Define the induced map (middle of page 342) - Prove Proposition 4.1. Use more detail than Hatcher does, and try to phrase the proof in using the lifting theorems of Massey. Use this to compute the higher homotopy groups of S^1 and T^2 Ignore the material discussing the action of pi_1 on pi_n |
| 23 | Cell Complex, Whitehead's Theorem | Lecture 43 Define Cell complex, aka CW complex. Give some of the examples on page 6. State, but do not prove, Whitehead's Theorem 4.5 on page 346. This illustrates the fundamental importance of homotopy groups for homotopy theory. This lecture might be a little short Final project draft due |
| 24 | Presentations of Final Projects | Final project due |
| 25 | Presentations of Final Projects (cont.) |