1300Y Geometry and Topology, Assignment 1 Exercise 1. Let Γ be a
... b) Show that M naturally inherits the structure of a smooth manifold. c) Let M̃ = S n and Let Γ = Z/2Z act via h · x = −x, where h is the generator of Γ. Show that the hypotheses above are satisfied, and identify the resulting quotient manifold. d) Let M̃ = Cn − {0} and let the generator of Γ = Z ac ...
... b) Show that M naturally inherits the structure of a smooth manifold. c) Let M̃ = S n and Let Γ = Z/2Z act via h · x = −x, where h is the generator of Γ. Show that the hypotheses above are satisfied, and identify the resulting quotient manifold. d) Let M̃ = Cn − {0} and let the generator of Γ = Z ac ...
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.