Knots and 3-manifolds

Math 87100, Spring 2022

Thursdays, 11:45 - 1:45 pm, Room 6417

Mathematics Program, The Graduate Center, CUNY


Suggested Readings and Problems


Homework 1
  1. Explain why a line in the plane passing through the origin with irrational slope projects to a dense embedding on the torus. Is it a submanifold ? (Hint: equidistribution theorem)
  2. Explain in your own words why the 3-sphere is a union of two solid tori.
  3. Read about the Hopf fibration e.g. Wiki (excellent)   Elementary introduction to Hopf Fibration & Quaternions   Dimensions YouTube Video (many videos on YouTube on Hopf fibration)

    4. More dimensions videos: Dimensions videos


Homework 2
  1. Read a proof of the Dehn-Lickorish Theorem which says that the mapping class group of a surface of genus g is generated by Dehn twists on (3g-1) loops. Reference: (1) Rolfsen - Chpt 9I, (2) Lickorish - Chptr 12
  2. Read different descriptions of Lens Spaces .
  3. Compute fundamental group of a Lens space L(p,q) using the Dehn surgery description and the Heegard splitting description.
  4. Verify that the only 3-manifold of Heegaard genus zero is the 3-sphere. (see visualization of Alexander's trick. )
  5. Suppose the surface homeomorphism of a Heegaard splitting is a composition of two homeomorphisms f and g, verify that they can be "stacked".


Homework 3
  1. Using similar technique as used to prove Alexander's theorem, prove the smooth version of Schonflies Theorem - A smoothly embedded circle in the plane bounds an embedded disk.
  2. Prove 3-manifold quotients of irreducible 3-manifolds are irreducible. See Hatcher: pg 11
  3. Let M be a compact 3-manifold with boundary which is a submanifold of the 3-space. If the first homology of M is trivial then it is simply connected.
  4. Normal surfaces enbaled the use of computational methods in 3-manifold topology. Here are two nice surveys about this topic - Lackenby, Hews


Homework 4
  1. English translation of Seifert's original paper on SFS
  2. What manifold do we get if all the coefficients in the SFS Dehn filling coefficients are 0/1 i.e fiber-parallel Dehn filling ? For more read Martelli: 10.3.13
  3. Compute the homology of SFS. For more details see Seifert's paper or Martelli: 10.3.5.
  4. We can expect SFS to be covered by circle bundles. See Martelli: 10.3.6-10.3.8.


Homework 5
  1. Show that if S is a 1-sided connected surface in M, then pi_1(S) injects into pi_1(M) iff the boundary of the regular neigbourhood of S (i.e. the closed orientable surface which is the boundary of the twisted interval bundle over S) is incompressible.
  2. Read about torus-semi-bundles. See Martelli: 11.4.3-11.4.5
  3. Read about the uniqueness of torus decomposition - Hatcher: pg 27-29, Martelli-11.5.2
  4. Walter Neumann's notes on JSJ decomposition - See Chapter 2 of Neumann's notes



Background image: London Tsai
John Baez on Hopf fibration