Math 290B-C Material

Eric:

1. Derive long exact sequence of a pair (X, A). Give examples of spaces (X, A) where the maps in the long exact sequence are neither one to one nor onto.

2. Prove 5 lemma, conditions on when A is a retract of X.

Chris:

3. Compute homology of a suspension of a space (15.24).

4. All the sphere problems from Section 16 except 16.9. Note we prove that the nth homotopy group of real projective space Pⁿ is nontrivial.

Oded:

5. In Section 17, computation of the homology of a torus, and double torus, homology of wedge product XÚY is an algebraic retract of homology of X ×Y, but as spaces this may not be the case. 17.17, 17.18, 17.14 for k = 2, 17.15.

6. Construct a one to one continuous map f such that the induced homology map is not one to one. Construct an onto continuous map g such that the induced homology map is not onto.

Andy:

7. The adjunction space construction in Section 19. Know how the real, complex and quaternionic projective spaces are cell complexes. In dimension 1, the various projective spaces are spheres. The attaching map to obtain real, complex or quaternionic projective spaces in degree 2 become homotopy classes in the homotopy groups of spheres. Later, using Poincaré duality, these elements will be nontrivial elements of the homotopy groups of spheres.

Nate:

8. 19.12, the lens space, homology of a cell complex with finitely many cells attached has finitely generated homology. Be able to compute the homology of various compact surfaces that are obtained from a polygon where various sides have been identified.

Manda:

9. For a collared pair, the reduced homology of the pair (X, A) is the same as the homology of X/A. 19.42, 19.43, 19.44. Given any finitely generated abelian group G, construct a connected space X with nth homology group isomorphic to G and all other homology groups of X are zero except in degree 0. Be able to construct two spaces with isomorphic homology groups but different homotopy types.

Ross:

10. Computation of the Euler characteristic for various rings. Relation of relative Euler characteristic to absolute. Attach a cell, what is the effect on the Euler characteristic.

11. Section 21. The covering space of a cell complex is a cell complex. The mapping cone of a map. The induced long exact sequence relating mapping cones and the induced homology map. 21.14-17. A map f from X to Y is null homotopic if and only if Y is a retract of Cf.

Evan

12. Orientation, its definition, creation of the covering space and the orientation sheaf. 22.20, 22.22, Z₃-orientable implies orientable, a topological group that is a manifold is orientable. A simply connected manifold is orientable. Any compact connected space with no reduced homology cannot have the homotopy type of a manifold.

Paul:

13. Cohomology, the universal coefficient theorem. How to compute Ext for finitely generated abelian groups. Construction of Ext. The most recently assigned exercises about homology with Z_n coefficients and the universal coefficient theorem.

Ananda:

14. Definition of cup and cap products. Computation of the orientation class for the homology of the torus. Computation of the cup and cap product structure of the torus.

Evan:

15. Definition of fundamental group. For a path connected space X, the fundamental group at different basepoints is isomorphic. Given a simply connected topological group G and H a discrete subgroup, the p₁( G/H, 1) is isomorphic to H. p₁( G, e) is commutative for any topological group G. The one point union of two circles cannot have the homotopy type of a topological group. Sketch of 4.12, 4.13, 4.14 (van Kampen theorem). Examples of covering spaces of one point union of two circles. Given a finitely generated possibly nonabelian group G, construct a space X whose fundamental group is G.

Oded:

16. Covering spaces. Assume throughout that all such spaces are path connected, locally path connected and semi-locally simply connected. Given E a covering space over X, the induced map from p₁(E,e₀) to p₁(X, x₀) is one to one. The cokernel of this map is in one to one correspondence with the fibre p^-1 ( x₀). If E is simply connected, then p₁(X,x₀) is isomorphic as groups to the group of covering transformations of E. 5.10, 5.11, 5.12. Notice that the lens space is covered by the 3-sphere is an example of 5.10.

Chris:

17. Universal covering space is a simply connected space E that covers X. Show E is unique up to homeomorphism. Show all covering spaces of X are obtained by taking a subgroup H of p₁(X,x₀) and acting on the universal cover by H. This produces a covering space of X. Use 6.2 and 6.9. Classify all covering spaces of Pⁿ, all covering spaces of the lens space L(p, q) where p is a prime, all covering spaces of the torus, what other spaces can you think of?

Nate:

18. The covering space of a topological group is again a topological group. The covering space of a cell complex is again a cell complex. The covering space of a manifold is again a manifold. Higher homotopy groups. The higher homotopy groups of the circle and real projective spaces. We now know that certain higher homotopy groups are nontrivial such as p_n(Pⁿ), p_2n(CPⁿ), p₄(S³), p₇(S⁴), p₁₅(S ⁸). p₁( X, x₀) abelianized is isomorphic to H₁(X; Z).

Ross:

19. Cohomology with compact supports and limits. How to define the Poincaré duality map and its compatibility with continuous maps. Limits commute with exact sequences, limits over different index sets, the relations between applying cap product and limits that give us a commutative diagram involving inclusions. Limits over a final subset.

Manda and Ananda:

20. Computing the cohomology rings of the torus, double torus, projective spaces, real, complex and quaternionic, Euler characteristics of compact orientable manifolds. 26.18-26.26. Z_k orientability for k > 2 implies orientability.

Paul:

21. Products Künneth theorem and universal coefficient theorem for homology. As a corollary, products of orientable manifolds are orientable. Euler characteristic of products of two spaces is the product of the Euler characteristics. Cohomology ring structure of X ×Y in terms of cohomology of X and cohomology of Y. Relative cup products. 29.38-29.42

Andy:

22. Natural transformations. Acyclic models. Sign commutativity of cup product. Trivial cup products for suspensions. Coalgebras and Hopf algebras. The even sphere is not a topological group. What other spaces are not topological groups? Which Pⁿ can be?