Review & Final Exam Preparation – ma516
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Defintions
- Connected sum of surfaces
- Orientable vs non-orientable surfaces
- Genus of a orientable closed surface
- Retract
- Deformation retract
- path
- homotopy
- homotopy equivalence
- fundamental group
- simple closed curve
- essential simple closed curve
- non-separating simple closed curve
- separating simple closed curve
- homotopy class of a simple closed curve
- \(X\) is connected if …
- \(X\) is path connected if …
- \(X\) is simply connected if …
- \(X\) is homotopy equivalent to \(Y\) if …
- \(X\) is homeomorphic to \(Y\) if …
- \(X\) is homotopy equivalent to a point if …
- \(X\) is contractible if …
- \(X\) is compact if …
- \(X\) is a n-manifold if …
- \(X\) is a closed n-manifold if …
- \(X\) is a surface if …
Theorems
- State the Brouwer Fixed Point Theorem.
- State the classification of surfaces.
- Prove that homeomorphic spaces are homotopy equivalent.
- Prove that homotopy equivalent spaces have isomorphic fundamental groups.
- State a theoerm that relates deformation retracts and homotopy equivalences.
Examples
- Give an example of a non-orientable surface.
- Give an example of a non-separating simple closed curve on a torus.
- Give an example of a separating simple closed curve on a torus.
- Give an example of a compact subset of \(\mathbb{R}^2\).
- Give an example of a non-compact subset of \(\mathbb{R}^2\).
- The topologist’s sine curve is an example of …
Computations
- Compute \(\pi_1(T^2)\) (the torus)
- Compute \(\pi_1(S^1)\)
- Compute \(\pi_1(\mathbb{R}P^2)\)
- Connected sums
- \(\mathbb{R}P^2 \# \mathbb{R}P^2\)
- \(\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2\)
- \(\mathbb{R}P^2 \# T^2\).
- Compute a fundamental group using S.V.K.
- Compute a fundamental group using deformation retracts.
- Compute \(\pi_1\) of the punctured torus.
- Compute \(\pi_1\) of the thrice (3 times) punctured sphere (also draw picture and recognize it as a surface with boundary).
- Realize the Mobius band as a punctured projective plane.
