Qualifying Exam in Topology January 2006
... (a) If f is a closed map, then g is continuous. (b) If f is not a closed map, then g may fail to be continuous. 2. Let X = [0, 1]/( 41 , 34 ) be the quotient space of the unit interval, where the open interval ( 41 , 34 ) is identified to a single point. Show that: (a) X is connected. (b) X is compa ...
... (a) If f is a closed map, then g is continuous. (b) If f is not a closed map, then g may fail to be continuous. 2. Let X = [0, 1]/( 41 , 34 ) be the quotient space of the unit interval, where the open interval ( 41 , 34 ) is identified to a single point. Show that: (a) X is connected. (b) X is compa ...
Problem Set 5 - Stony Brook Mathematics
... space is a homology n-manifold, then (a) X consists entirely of n-simplices and their faces, (b) Every (n − 1)-simplex is a face of precisely two n-simplices. Problem 2. Suppose that X is a compact triangulable homology n-manifold. (a) Show that if X is orientable, then Hn−1 (X) is torsion-free, and ...
... space is a homology n-manifold, then (a) X consists entirely of n-simplices and their faces, (b) Every (n − 1)-simplex is a face of precisely two n-simplices. Problem 2. Suppose that X is a compact triangulable homology n-manifold. (a) Show that if X is orientable, then Hn−1 (X) is torsion-free, and ...
FINAL EXAM
... 1. (a) Suppose Y is a closed subspace of a topological space X, and A is a closed subspace of Y . Show that A is a closed subspace of X. (b) Suppose A is a closed subspace of X, and B is a closed subspace of Y . Show that A × B is a closed subspace of X × Y . 2. Let Y be a compact space, and X an ar ...
... 1. (a) Suppose Y is a closed subspace of a topological space X, and A is a closed subspace of Y . Show that A is a closed subspace of X. (b) Suppose A is a closed subspace of X, and B is a closed subspace of Y . Show that A × B is a closed subspace of X × Y . 2. Let Y be a compact space, and X an ar ...
13b.pdf
... π1 T S(Õ). The image is trivial, since a reflection folds the fibers above its axis in half. Every easily producible simply connected closed three-manifold seems to be S 3 . We can draw the picture of T S(O) by piecing. ...
... π1 T S(Õ). The image is trivial, since a reflection folds the fibers above its axis in half. Every easily producible simply connected closed three-manifold seems to be S 3 . We can draw the picture of T S(O) by piecing. ...
Orbifolds and Wallpaper Patterns João Guerreiro LMAC Instituto Superior Técnico
... such that each connected component of p−1 (U ) is homeomorphic to Ũ /Γi for some subgroup Γi of Γ. In addition, the homeomorphism has to respect both projection (p and the canonic one between Ũ /Γi and Ũ /Γ). Definition 5.2 An orbifold is good if it is covered by a manifold. Otherwise, it is a ba ...
... such that each connected component of p−1 (U ) is homeomorphic to Ũ /Γi for some subgroup Γi of Γ. In addition, the homeomorphism has to respect both projection (p and the canonic one between Ũ /Γi and Ũ /Γ). Definition 5.2 An orbifold is good if it is covered by a manifold. Otherwise, it is a ba ...
Orbifolds and their cohomology.
... can be extended to orbifolds. Most importantly for us, there is a notion of an orbifold vector bundle (and in particular, of a tangent and cotangent bundle to an orbifold) and of de Rham cohomology. The general principle when defining the orbifold analogues of such concepts is that one should specif ...
... can be extended to orbifolds. Most importantly for us, there is a notion of an orbifold vector bundle (and in particular, of a tangent and cotangent bundle to an orbifold) and of de Rham cohomology. The general principle when defining the orbifold analogues of such concepts is that one should specif ...
Math 8301, Manifolds and Topology Homework 7
... 1. (CORRECTED) Let F = hx, yi be a free group on two generators. Show explicitly that the subgroup H ⊂ F generated by the elements zn = (y −n xy n ), as n ranges over the integers, is a free group on the elements zn . 2. If F and F 0 are free groups, show that F ∗ F 0 is also free. 3. Suppose f : H ...
... 1. (CORRECTED) Let F = hx, yi be a free group on two generators. Show explicitly that the subgroup H ⊂ F generated by the elements zn = (y −n xy n ), as n ranges over the integers, is a free group on the elements zn . 2. If F and F 0 are free groups, show that F ∗ F 0 is also free. 3. Suppose f : H ...
Math 8301, Manifolds and Topology Homework 8 1. Show that S
... 2. Give a description of the universal cover of the space S 2 ∨ S 1 , obtained by gluing together S 2 and S 1 at a single point. 3. Suppose X and Y are path-connected spaces, p : Y → X is a covering map, and y ∈ Y . Let the image of π1 (Y, y) in the fundamental group G = π1 (X, p(y)) be the subgroup ...
... 2. Give a description of the universal cover of the space S 2 ∨ S 1 , obtained by gluing together S 2 and S 1 at a single point. 3. Suppose X and Y are path-connected spaces, p : Y → X is a covering map, and y ∈ Y . Let the image of π1 (Y, y) in the fundamental group G = π1 (X, p(y)) be the subgroup ...