Math 121a, Fall 2006

Contact information: The easiest way to reach me is via email: ruberman @brandeis.edu. My office is 310 Goldsmith, and my phone is x63074.

Homework assignments

 Due date Assigment Tuesday 9/19 Hatcher, 1.1/5, 3, 14, and problem A (below) Friday 9/29 Hatcher 1.2/4, 8, 10, 22 Tuesday10/17 Hatcher 1.3/8, 12, 20; 2.1/1,4, and (optional) problem B Tuesday 11/ 6 Hatcher 2.1/11, 14, 17(b), and problem C Tuesday 11/ 20 Hatcher 2.1/20, 27b (we did (a) in class), 29, 31, and problem D
1. Let X = A ∪ B, where A and B are open, path connected and simply-connected sets. Suppose also that A ∩ B is path-connected. If x ∈ A ∪ B, show that π1(X,x) is trivial. (The idea is similar to the idea in 1.14, and is a special case of van Kampen's theorem.)
2. Recall the definition of the open cone on a space Y: c(Y) = Y * [0,1)/(y,0) ∼ (y',0) ∀ y,y'. For example, you can check that c(Sn-1) ≈ int(Dn). Show that (for n ≥ 3) if c(Y) is an n-dimensional manifold, then Y must be simply-connected. Suggestion: try to show that the "cone point" (represented by (y,0) for all y) has no Euclidean neighborhood. (It is an amazing fact that there are (in all dimensions greater than 3) spaces Y that are not homeomorphic to a sphere but whose cones are manifolds.)
3. A short exact sequence 0 → A → G → C → 0 of abelian groups (where the maps are i: A → G and p: G → C) is said to split if either of the two following conditions hold:
1. There is a homomorphism s:C → G with ps=idC.
2. There is a homomorphism r:G → A with ri=idA.
Show that these conditions are equivalent, and that either one implies that G ≅ A ⊕ C with i the inclusion of A as the first factor and p the projection of G onto the second factor.
4. Let M be an n-dimensional manifold with boundary. Remember that a point y in M has a neighborhood homeomorphic to an open set in Rn+ (the set of points with xn &ge 0). A point x is defined to be a boundary point if such a homeomorphism takes y to a point in Rn-1, and an interior point otherwise. By computing the local homology of (Rn+, Rn-1) at points x with xn > 0 and xn = 0, show that this notion is independent of the choice of homeomorphism.

Math 121a Syllabus

1. Covering Spaces and Fundamental Group.
1. Basic Definitions (homotopy, fundamental group).
2. Existence and classification of covering spaces.
3. Correspondence between subgroups and covering spaces.
4. van Kampen's theorem.
2. Homology Theory.
1. Basic definitions of singular and simplicial homology.
2. Long exact sequence, excision, Mayer-Vietoris.
3. Homology of cell complexes and/or CW complexes.
4. Homology of basic spaces: sphere, projective spaces
3. Applications of homology.
1. Maps between spheres; degree of map.
2. Vector fields, fixed point theorems.
3. Separation theorems (Jordan Curve theorem).
4. Cohomology theory (as time permits).
1. Basic properties.
2. Cup and cap products.

Main text for the course will be Algebraic Topology, by Allen Hatcher. The book is available on-line on Allen Hatcher's web page but I strongly recommend against printing it. The paperbound edition of the book is inexpensive--considerably less than the cost of printing it yourself!

References: (other books that have been used in recent years).
1. Algebraic Topology: A First Course, by Greenberg and Harper
2. Elements of Algebraic Topology, by James Munkres

Last updated: August 21, 2006