Low-Dimensional Manifolds

• Given a 4-manifold made up of a 0-handle and some 2-handles, what does the determinant of the intersection form tell you about the boundary?
• What can you say about the components of the complements of a PL-embedded 2-sphere in S³?
• How about of a torus in S³?
• Prove that if one component U of the complement of the torus in S³ has fundamental group , the closure of U is homeomorphic to a solid torus.
• How can you weaken the condition on U?
• Sketch a proof of the Reidemeister-Singer Theorem.
• Sketch a proof of Milnor's theorem on unique decomposition of 3-manifolds into prime factors.
• What is a spin structure? (Kirby)
• If a manifold is spin, how many different spin structures does it have? (Casson)
• Give an example of a 4-manifold which is not spin. (Kirby)
• Draw your favorite knot (not the unknot) K and compute π₁(S³ − K). (Kirby)
• Why does the Wirtinger presentation method of computing π₁(S³ − K) work? (Kirby)
• Why is π₁(S³ − K) not if K is the trefoil? (Stallings)
• Why does π₁(S³ − K), with K the trefoil, have a subgroup isomorphic to ? (Casson)
• Define Heegaard splittings.
• Define handlebody.
• Do Heegaard splittings always exist? Why?
• Consider + k surgery on a trefoil. Can you relate this to a Heegaard splitting? (Kirby)
• How do you define Dehn surgery?
• Why is it enough to specify a homotopy class of essential curve (in Dehn surgery)? (Casson)
• Define Haken manifold.
• Define incompressible surface.
• Define -irreducible.
• Can you give an example of a Haken manifold with a separating incompressible surface? (Casson)
• One without boundary?
• Suppose M is non-Haken, specifically, that it is closed, orientable, irreducible, but does not contain an incompressible surface. Does every embedded torus in M necessarily bound a solid torus? (Casson)
• Suppose that M is an orientable compact irreducible manifold with π₁(M) free. What is M?
• Consider a + k surgery on a trefoil. How can this be related to a Heegard splitting?
• Why is it sufficient to specify a homotopy class of essential curves in a torus to define a Dehn surgery?
• Give a surgery presentation for a homology 3-sphere.
• Give a surgery on a knot which gives a reducible manifold.
• Give an example of a Haken manifold with a separating incompressible surface.
• Show that a manifold M is Haken iff it is incompressible and its fundamental group can be written non-trivially as π₁(M) = A * _BC or as π₁(M) = A * _B.
• Give an example of a non-Haken manifold.
• Show that if M³ is prime and orientable then π₂(M) = 0 or M³ is an S² bundle over S¹.
• Show that any Solv manifold is virtually Haken.
• Show that any knot has a Seifert surface.
• If two 3-manifolds have the same π₁ and π₂ is zero, are they homotopy equivalent?
• Show that any closed compact hyperbolic 3-manifold contains no subgroups.
• What is the Euler number of the S¹-bundle determined by the natural Seifert fibered structure on the Poincaré homology sphere?
• Let be a fibered knot. What surgeries on K give a 3-manifold which fibers over S¹?
• Give an example of a non-prime 3-manifold admitting a geometric structure.
• Give an example of a homeomorphism from the punctured torus to itself whose mapping torus does not admit a hyperbolic structure.
• What is a Dehn surgery presentation for the quotient of S³ by the group of order 48 giving the truncated cube space?
• Why does the Mostow Rigidity theorem fail for infinite volume manifolds?
• Give an explicit construction of a closed hyperbolic 3-manifold.
• Suppose that α and β are simple closed non-intersecting curves on a hyperbolic surface S. Show that their geodesic representatives α' and β' do not intersect.
• Show that no closed compact surface in can have strictly negative curvature everywhere.
• Show that there are hyperbolic structures on a pair of pants with geodesic boundary.
• Which one of Moise's theorem did you mean? (Kirby)
• What is Waldhausen's theorem? (Kirby)
• What about one with an infinite fundamental group? (Kirby)
• Explain why any 3-manifold with an infinite H₁ is sufficiently large. (i.e. Haken) (Kirby)
• Why does an element of the first cohomology of M correspond to a map from M to S¹? (Kirby)
• Is there a similar result for higher cohomology groups? (Kirby)
• List all double covers of (Casson)
• Compute (geometrically) the Alexander polynomial for the figure eight knot and relate it to the homology of (finite) cyclic covers of the knot complement. (McMullen?)
• Let M be a closed, oriented 3-manifold. Show irreducible implies prime. What if M is not oriented?
• Let be an embedded torus. Show that this torus compresses. (Stallings)

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