Cornell Math - MATH 661, Fall 2000

MATH 661 — Fall 2000
Geometric Topology

Prerequisites: Undergraduate topology and algebra (as in Math 453, 434).

Course Description: This course will discuss a number of the classic theorems and examples in geometric topology, centering on topological manifolds (with no smooth or piecewise linear structure assumed), embeddings and quotients of spheres, and cartesian products.

Here is a list of possible topics. (There is more listed than can actually be done in one semester.)

A. Topological manifolds

Definition, homogeneity, the boundary is collared, uses of Invariance of Domain Theorem (which will be proved in 651). The "Alexander trick" (any homeomorphism of the boundary of an n -ball extends to a homeomorphism of the ball).

B. Embeddings of S ^n-1  in S ⁿ.

a) The Alexander horned sphere (a very bad 2-sphere in the 3-sphere)

b) Brown's Schonfliess Theorem (locally flat (n –1)-spheres are good: they always bound n-balls)

c) A closed n -manifold which is the union of two open n -balls is homeomorphic to S ⁿ. Indication of how this was used to prove the famous Poincaré conjecture in dimensions 5 and greater.

C. Cartesian products and manifolds

a) Elementary examples: X ₁ x Y  homeomorphic to X ₂ x Y  while X ₁ is not homeomorphic to X ₂.

b) Bing's dogbone space: a quotient space D  of R ³ which is not a manifold, while D  x R ¹ is homeomorphic to R ⁴.

c) J. West's theorem: If Q  is the Hilbert cube and K  is any contractible complex, the Q  x K  is homeomorphic to Q .

D. Triangulations and Quotients of the three sphere S ³

Lens spaces, the Poincaré homology sphere, the Double Suspension Theorem (the double suspension of the Poincaré homology sphere ia a triangulation of the 5-sphere which is a counterexample to the famous Hauptvermutung (conjecture on equivalence of triangulations).

A brief introduction to the fundamental group — a tool we will need, which is studied in math 651 and not assumed as a prerequisite here — will be given, as will material on triangulations at the end of the course, if required.