Introductory Topology II

Introductory Topology II (Math 752), Spring 2011
MWF 9:55 AM - 10:45 AM in Van Vleck B131

with Autumn Kent.

Office: Van Vleck 615

Office hours: TBA

Text: Algebraic topology, by Allen Hatcher.

Homework, exams, and grades:

There will be two midterm exams at times to be determined. These will be like miniature qualifying exams.

There will be regular homework assignments (due every couple of weeks or so) that will be added to the bottom of this page.

Assignments with positive numbers will be to turn in, and those with negative numbers will not be graded, but will be discussed.

Your grade will be based on the following weights:
Midterm 1: 30%, Midterm 2: 40%, Homework: 30%.

Exam dates: TBD

Homework:

HW 1 (due 2/9/11):

1. We say a simplicial complex is finite if it has finitely many simplices. Recall that a subcomplex of a simplicial complex X is a subspace that is a union of simplices of X. Show that if X is a finite simplicial complex there is an N such that X is a subcomplex of the standard N-simplex.

2. A triangulation T of a space X is a simplicial complex T and a homeomorphism T --> X. Two simplicial complexes are isomorphic if there are homeomorphic via a map that takes simplices to simplices via linear homeomorphisms. Two triangulations f: S --> X and g: T --> X of a space X are said to be equivalent if S and T have isomorphic subdivisions.
Show that any two triangulations of the torus are equivalent.
Update: As an easier exercise, show that any two triangulations of the disk are equivalent.

3. Let T be a finite dimensional simplicial complex (so that there is maximum to the dimension of its simplices). Let K be a subcomplex of T. Show that K has a neighborhood N such that K is a deformation retract of N.

4. (Problem #2 in Hatcher's Appendix.) Let X be a cell complex and x any point in X. Show that there is a cell structure on X such that x is a 0-cell and each cell in the old structure is a union of cells in the new one.

HW −1 (never due):

1. Show that the long line is not contractible.

HW 2 (due 3/2/11):

1. (Car crash lemma) Suppose that the 2-sphere is equipped with a cell structure such that each closed 2-cell is embedded. On the boundary of each 2-cell, there is a taxi driving counterclockwise (from the perspective of the 2-cell), and it will travel around the boundary infinitely many times. Show that there is a crash.

2. A manifold is closed if it is compact and without boundary. Let X be a closed surface. Let Z be the integers. Let c be an element of H¹(X;Z). Show that there is a continuous map f: X ---> S¹ such that
c = f _* : H₁(X) ---> H₁(S¹).

Problems from Hatcher:
3.1: 2, 9, 11

HW −2 (never due):

1. (Schur-Zassenhaus theorem for abelian kernels). Let m and n be relatively prime. Let G be a finite group with a normal abelian subgroup A of order m. Show that G is isomorphic to a semidirect product of A and G/A.

HW 3 (due 3/11/11):

Problems from Hatcher: 3.2: 1, 3, 4, 7, 8, 18

HW 4 (due 4/8/11):

1. Let L be the long line. Show that the cohomology of L is zero in nonzero dimensions with any ring of coefficients.

Problems from Hatcher: 3.3: 7, 11, 12, 14, 25

HW 5 (not due):

1. Let M be a closed oriented smooth manifold and let α be a class in H¹(M; Z). Suppose that α is represented by a smooth map f: M --> S¹ (meaning f_* = α : H₁(M;Z) --> Z) of which r is a regular value. Show that the homology class of f⁻¹(r) is the Poincaré dual Dα of α.

2. (3-manifolds are null cobordant) Use Wall's theorem that every smooth closed 3-manifold embeds in the 5-sphere to show that every orientable 3-manifold is homeomorphic to the boundary of some 4-manifold.

3. (Milnor's counterexample to the Hauptvermutung is not a manifold.) Let p > 1 and q be relatively prime and consider the lens space L(p,q). Let Dⁿ be a closed n-ball. Let X_pqn be the space obtained from L(p,q) × Dⁿ by attaching the cone on ∂(L(p,q) × Dⁿ) along ∂(L(p,q) × Dⁿ). Show that X_pqn is not a manifold.

4. (Čech cohomology is well defined.)
Recall that the nerve NU of an open cover U of a space X is the simplicial complex whose k-simplices are the subsets S of U with k+1 elements such that ∩S is nonempty.
If V is a refinement of U, meaning that each element of V is contained in an element of U, we may define a simplicial map NV --> NU by sending each vertex (an element of V) to an element of U containing it (a vertex of NU) and extending linearly.
This requires the axiom of choice, but any two of these maps NV --> NU are homotopic. Prove this.

5. Find an example of spaces X and Y and maps f, g : X --> Y such that f and g induce the same maps on all homotopy groups, and yet f and g are not homotopic.

Problems from Hatcher:

4.1: 2, 8, 10, 11, 12, 16, 20

4.2: 1, 8, 9, 12, 14, 15, 19