Exam 2 Study Guide
... Draw pictures to illustrate terms and theorems Explain what needs to be shown to prove something is one of the terms listed Explain what conditions must hold to use a particular theorem Describe what results or conditions are guaranteed by a particular theorem Identify which conditions will imply th ...
... Draw pictures to illustrate terms and theorems Explain what needs to be shown to prove something is one of the terms listed Explain what conditions must hold to use a particular theorem Describe what results or conditions are guaranteed by a particular theorem Identify which conditions will imply th ...
Topology/Geometry Aug 2011
... 2. Let D2 denote the unit disc in R2 with the unit circle S 1 its boundary. If f : D2 → D2 is a homeomorphism, show that the restriction f |S 1 is a homeomorphism onto S 1 . (Hint: one way to do this is to assume it is not and obtain a contradiction by considering fundamental groups.) 3. Consider th ...
... 2. Let D2 denote the unit disc in R2 with the unit circle S 1 its boundary. If f : D2 → D2 is a homeomorphism, show that the restriction f |S 1 is a homeomorphism onto S 1 . (Hint: one way to do this is to assume it is not and obtain a contradiction by considering fundamental groups.) 3. Consider th ...
Practice Exam 5: Topology
... 1. Prove that if F : M → N is a smooth map of smooth manifolds and Q is a regular value of F then F −1 (Q) is a smooth submanifold of M . 2. Let V and W be vector spaces and A : V → W be a linear map. Prove that if ω, η ∈ Λ∗ (W ) then A∗ (ω ∧ η) = A∗ (ω) ∧ A∗ (η). 3. Produce a C ∞ -compatible atlas ...
... 1. Prove that if F : M → N is a smooth map of smooth manifolds and Q is a regular value of F then F −1 (Q) is a smooth submanifold of M . 2. Let V and W be vector spaces and A : V → W be a linear map. Prove that if ω, η ∈ Λ∗ (W ) then A∗ (ω ∧ η) = A∗ (ω) ∧ A∗ (η). 3. Produce a C ∞ -compatible atlas ...
5 The hyperbolic plane
... of sheets. By counting vertices, edges and faces it is clear that χ(X̃) = kχ(X). Since χ(S) = 2, we must have k = 1 or 2, but if the latter χ(X) = 1 which is not of the allowable form 2 − 2g for an orientable surface and a Riemann surface is orientable. So it is only the Riemann sphere in this case. ...
... of sheets. By counting vertices, edges and faces it is clear that χ(X̃) = kχ(X). Since χ(S) = 2, we must have k = 1 or 2, but if the latter χ(X) = 1 which is not of the allowable form 2 − 2g for an orientable surface and a Riemann surface is orientable. So it is only the Riemann sphere in this case. ...
Lecture 7
... to ∂H, and hence the angle at this vertex is zero. By applying a Möbius transformation, we can map this vertex to ∞ without altering the area or the angles. By applying the Möbius transformation z 7→ z + b for a suitable b we can assume that the circle joining the other two vertices is centred at ...
... to ∂H, and hence the angle at this vertex is zero. By applying a Möbius transformation, we can map this vertex to ∞ without altering the area or the angles. By applying the Möbius transformation z 7→ z + b for a suitable b we can assume that the circle joining the other two vertices is centred at ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.