18.904 | Spring 2011 | Undergraduate

# Seminar in Topology

## Lecture Summaries

The class meets for one hour three times a week. After an introductory lecture on the main topics by the instructor, student will prepare lectures to present to the rest of the class related to that topic.

In each class session, two students will give lectures. Each lecture should be about 25 minutes long. Individual lectures will not be graded, but lectures make up a good portion of the final grade. Members of the audience will write up observations about each lecture they observe. These comments will be forwarded to the lecturing student anonymously.

Readings references are to the two required texts:

Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2001. ISBN: 9780521795401. [Preview with Google Books]

Massey, William S. A Basic Course in Algebraic Topology. Springer-Verlag, 1991. ISBN: 9783540974307.

SES # TOPICS SUMMARY AND REFERENCES
1 Organizational meeting The first meeting of class.
2 Introduction to the fundamental group Give the idea of the fundamental group and some basic examples. Also give the idea of homotopy equivalences and some basic examples.
3 Paths and homotopies

Define what a path in a space is, and what a homotopy of paths is. Showthat “homotopic”; is an equivalence relation on paths. Give an example of a homotopy between two paths. Give an example of two paths with the same endpoints which are not homotopic (proof not required). Define the composition of two paths, and show that it is well-defined on homotopy classes of paths.

References: Hatcher §1.1, Massey §II.2.

The fundamental group

Define the fundamental group of a topological space and prove that it is a group. Give an example of a space whose fundamental group is trivial and an example of a space whose fundamental group is non-trivial (without proofs). Show that a path between two points induces an isomorphism between the fundamental groups based at those points.

References: Hatcher §1.1, Massey §II.3.

4 The fundamental group of the circle Define a homomorphism Z→ π1(S1) and show that it is an isomorphism.
Applications of previous lecture

Using the isomorphism π1(S1) = Z from the previous lecture, prove the following theorems: the fundamental theorem of algebra (every non-constant polynomial with complex coefficients has a complex root), Brouwer’s fixed point theorem for the 2-disc (every continuous map from the 2-disc to itself has a fixed point) and the Borsuk-Ulam theorem for the 2-sphere (every continuous map from the 2-sphere to R2 admits two antipodal points at which it is equal).

References: Hatcher §1.1, Massey §II.6.

5 Contractible and simply connected spaces Define what it means for a space to be contractible and simply connected. Prove that a contractible space is simply connected. Show that Rn is contractible. Show that a sphere of dimension at least 2 is simply connected. (Spheres are not contractible though!)
The fundamental group of a product

Show that the fundamental group of the cartesian product of two spaces is the direct product of the fundamental groups of the two spaces. Compute the fundamental groups of tori. (A torus is just a product of circles.)

References: Hatcher §1.1, Massey §II.7.

6 Functoriality of the fundamental group Show that a continuous map f : X → Y of topological spaces induces a homomorphism of groups f* : π1(X) → π1(Y). Define what it means for two maps f, g : X → Y to be homotopic (generalizing the definition for paths). Show that if f and g are homotopic then f*=g*.
Homotopy equivalences

Define what a homotopy equivalence between two spaces is. Show that if two spaces are homotopy equivalent then their fundamental groups are isomorphic. Give several examples of spaces which are and are not homotopy equivalent.

References: Hatcher §1.1, Massey §II.8.

7 The fundamental group of S1 ∨ S1 Define the free group on n letters Fn. Define the wedge sum X ∨ Y of two topological spaces. Construct a map F2 → S1 ∨ S1 and show that it is an isomorphism. State a similar result (without proof) for an n-fold wedge sum of circles.
Amalgamated free products Define the amalgamated free product of groups. Prove the universal property of this product. Give some examples.
8 van Kampen’s theorem State van Kampen’s theorem. Prove the surjectivity part of the statement.
van Kampen’s theorem (cont.) Prove the injectivity part of van Kampen’s theorem.
9 Introduction to covering spaces Define what a covering space is. Give some ideas about the universal cover and the Galois correspondence for covers. Do some examples.
10 The universal cover

Construct the universal cover.

References: Hatcher pp. 63–5.

The universal cover (cont.)

Verify properties of the construction of the universal cover from the previous lecture.

References: Hatcher pp. 63–5.

11 Lifting properties

Prove the homotopy lifting property. Deduce the path lifting property as a corollary. Show that p* is injective on fundamental groups if p is a covering map.

References: Hatcher Prop. 1.30 and Prop. 1.31.

Lifting properties (cont.)

Prove the criterion for lifting and uniqueness of lifts.

References: Hatcher Prop. 1.33 and Prop. 1.34.

12 Existence of covers

Show that for each subgroup of the fundamental group, there exists a corresponding cover. Show that this cover is Galois if and only if the subgroup of the fundamental group is normal.

References: Hatcher Prop. 1.36 and Prop. 1.39.

The Galois correspondence

Finish off the proof of the Galois correspondence.

References: Hatcher Prop. 1.37 and Thm. 1.38.

13 Category theory Define what a category is and what a functor is. Give several examples.
The Galois correspondence in categorical form

State and discuss the proof of how to state the Galois correspondence as an equivalence of categories involving permutation representations of the fundamental group.

References: Hatcher pp. 68–70.

14 Covering spaces of S1 ∨ S1

Describe covers of the space S1 ∨ S1 in terms of 2 oriented graphs. In particular, explicitly describe the universal cover. Also, give some examples of Galois and non-Galois covers.

References: Hatcher p. 57.

Quotients by finite groups

Discuss the theory of quotients by discontinuous group actions, in particular, how the fundamental group changes. Compute the fundamental group of real projective space.

References: Hatcher Prop. 1.40.

15 Overview of homology An overview of what we’ll cover on homology.
16 Chains and the boundary operator

Define what a singular simplex is. Define the group of singular chains and the boundary operator on it. Prove that the square of the boundary operator is 0.

References: Hatcher p. 103 and p. 108.

Definition of homology and first calculations

Define the group of cycles and boundaries. Define singular homology. Compute H0 and the homology of a point.

References: Hatcher p. 108, Prop. 2.7 and Prop. 2.8.

17 Chain complexes

Define what a chain complex is. Define what a map of chain complexes is. Define what a homotopy of maps of chain complexes is. Show that a map of chain complexes induces a map on homology, and that homotopic maps induce the same map on homology. This talk is pure algebra, and does not involve topology in any way.

References: Some discussion in Hatcher pp. 110–3.

Functoriality of homology

Show that homology is naturally a functor on the category of topological spaces. Show that homotopic maps induce the same map on homology.

References: Hatcher pp. 110–3, esp. Thm. 2.10.

18 The long exact sequence

Show that there is a natural long exact sequence in homology associated to a short exact sequence of chain complexes. This talk is pure algebra, and does not involve topology in any way.

References: Some discussion in Hatcher pp. 115–8.

Relative homology

Define the relative homology groups. Show that there is a long exact sequence relating the homology of a space, a subspace and the relative homology groups. Do the example of a disc relative to its boundary.

References: Hatcher pp. 115–8, esp. Example 2.17.

19 Excision

State the theorem of excision. State Proposition 2.21 of Hatcher, and prove the excision theorem assuming this proposition.

References: Hatcher p. 119 (esp. Thm. 2.20) and p. 124.

Homology of a quotient

Define reduced homology groups. Relate the reduced homology groups of a quotient space to relative homology groups using excision. Compute the homology of a sphere.

References: Hatcher p. 110, Cor. 2.14 and Prop. 2.22.

20 Proof of Prop. 2.21, part 1

Begin the proof of Proposition 2.21 of Hatcher. This is the main step in the proof of excision.

References: Hatcher pp. 119–24.

Proof of Prop. 2.21, part 2

Finish the proof of Proposition 2.21 of Hatcher. This is the main step in the proof of excision.

References: Hatcher pp. 119–24.

21 Review Pause and take stock of where we are. Review what we have done so far and where we’re going.
Naturality of connecting homomorphisms

Prove the naturality of connecting homomorphisms in long exact sequences.

References: Hatcher pp. 127–8.

22 Axioms for homology

State the Eilenberg–Steenrod axioms and the theorem that they uniquely characterize singular homology.

References: Hatcher pp. 160–2.

The Mayer–Vietoris sequence

Establish the Mayer–Vietoris sequence using the axioms of homology. Do some examples.

References: Hatcher pp. 153–4 and pp. 161–2.

23 Homology with coefficients

Define homology with coefficients in an arbitrary abelian group.

References: Hatcher pp. 153–4.

The universal coefficient theorem

Review tensor products and the Tor functor. State the universal coefficient theorem, relating homology with coefficients to usual homology (i.e., homology with integer coefficients.) Do not prove the theorem!

References: Hatcher Appendix 3.A.

24 CW complexes, part 1

Define what a CW complex is. Mention that these spaces are all locally contractible.

References: Hatcher, Appendix.

CW complexes, part 2

Give examples of CW complexes, especially the two “standard” CW structures on the n-sphere.

References: Hatcher, Appendix.

25 CW homology, part 1

Introduce the cellular chain complex associated to a CW complex. State Theorem 2.35, that the homology of this complex is naturally isomorphic to singular homology. Also, mention the cellular boundary formula.

References: Hatcher pp. 137–141.

CW homology, part 2

Do some examples of cellular homology, for instance, recompute the homology of the n-sphere, compute the homology of complex projective space.

References: Hatcher pp. 141–6.

26 Definition of cohomology

Discuss the functor Hom(-, G). Define the cohomology of a space, with coefficients in G.

References: Hatcher pp. 197–8.

Overview of formal properties

Mention the formal properties of cohomology, which parallel those of homology. Be sure to point out the places where there are small differences (e.g., when the arrows go the other way). Also state the universal coefficient theorem, which let’s one compute cohomology from homology.

References: Hatcher pp. 199–204.

27 The cup product, part 1

Define the cup product, say that it makes cohomology into a ring. Show that the map on cohomology induced from a map on spaces is a ring homomorphism. Finally, state and try to prove Theorem 3.14, which says that the cup product is graded commutative.

References: Hatcher pp. 206–7, Prop. 3.10, Thm. 3.14.

The cup product, part 2

Compute the cohomology ring of the wedge sum S2 ∨ S4 and of complex projective space CP2. Observe that the underlying groups are isomorphic, but since the ring structure is different the two spaces are not homotopy equivalent. This is something that cannot be seen purely in terms of homology, and relies on the additional structure in cohomology.

References: Hatcher Example 3.13.

28 The Kunneth formula, part 1

Define the cross product in cohomology. State the Kunneth formula in the case where cohomology is finitely generated and free, and in the general case. Do an example, e.g., compute the cohomology of a torus.

References: Hatcher pp. 218–9, Appendix 3.B.

The Kunneth formula, part 2

Sketch a proof of the Kunneth formula in the simple case where the cohomology is finitely generated and free.

References: Hatcher pp. 220–1.

29 Overview of Poincare duality

Review the definition of a manifold. State a primitive version of Poincare duality. Explain what this means in the case of a torus.

References: Hatcher p. 231.

Orientations

Define what a local orientation is and what a (global) orientation is. Discuss the two sheeted cover which determines if there is an orientation. Define what a section is, in this context.

References: Hatcher pp. 233–5.

30 The fundamental class, part 1

State Theorem 3.26 and define the term fundamental class. State and prove Lemma 3.27.

References: Hatcher p. 236.

The fundamental class, part 2

Prove Theorem 3.26.

References: Hatcher p. 327.

31 Direct limits

Define and discuss direct limits of abelian groups. State and prove Proposition 3.33.

References: Hatcher pp. 243–4.

Cohomology with compact support

Define cohomology with compact support, and give the characterization of it as a direct limit. Compute the cohomology of Rn with compact supports.

References: Hatcher pp. 242–5.

32 The cap product

Define the cap product.

References: Hatcher pp. 239–41.

Statement of Poincare duality

State Poincare duality precisely, in both the compact and non-compact cases.

References: Hatcher Thms 3.30 and 3.35.

33 Proof of Lemma 3.36

State and prove Lemma 3.36.

References: Hatcher Lemma 3.36.

Proof of Poincare duality

Complete the proof of Poincare duality.

References: Hatcher pp. 247–8.

34 Closing lecture I will briefly discuss some things we didn’t get to and discuss some other places in math where cohomology comes up.
35 Sheaf cohomology  Student Presentations of Final Papers
Morse theory  Student Presentations of Final Papers
36 Hopf fibrations  Student Presentations of Final Papers
The Gauss—Bonnet theorem  Student Presentations of Final Papers
37 K-theory  Student Presentations of Final Papers
De Rham cohomology  Student Presentations of Final Papers
38 The Hurewicz isomorphism  Student Presentations of Final Papers
Orbifold fundamental groups  Student Presentations of Final Papers
39 Simplicial sets  Student Presentations of Final Papers

## Course Info

Spring 2011
##### Learning Resource Types
Problem Sets with Solutions
Projects with Examples
Instructor Insights
Online Textbook