UNIFORMIZATION OF SURFACES COMPLEX ANALYSIS 8702 1. Riemann surfaces; Summary
... fα fβ−1 : fβ (Uα ∩ Uβ ) → fα (uα ∩ Uβ ) is a conformal map of each component. Via the maps {fα } and the transition property, angles in C are carried up to well defined angles on R. 2. Denote by F = π1 (R) its fundamental group. As an abstract group, F is independent of basepoint; the fundamental gr ...
... fα fβ−1 : fβ (Uα ∩ Uβ ) → fα (uα ∩ Uβ ) is a conformal map of each component. Via the maps {fα } and the transition property, angles in C are carried up to well defined angles on R. 2. Denote by F = π1 (R) its fundamental group. As an abstract group, F is independent of basepoint; the fundamental gr ...
Padic Homotopy Theory
... There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is infinite dimensional, t ...
... There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is infinite dimensional, t ...
abs
... When discussing the concept of connectedness, we often come across the equivalent criterion that a space is connected if and only if any continuous map from it to the discrete space {0,1} is constant. It would be interesting to see what concept arises if the discrete space of two points is replaced ...
... When discussing the concept of connectedness, we often come across the equivalent criterion that a space is connected if and only if any continuous map from it to the discrete space {0,1} is constant. It would be interesting to see what concept arises if the discrete space of two points is replaced ...
Math 8301, Manifolds and Topology Homework 3
... then α ∗ β is homotopic to γ ∗ δ. 3. Suppose that a topological space X has a function m : X × X → X. Show that if α and β are any paths in X, the definition (α · β)(t) = m(α(t), β(t)) is homotopy invariant, in the sense that [α] ∗ [β] = [α ∗ β] is well-defined on homotopy classes of paths. 4. Show ...
... then α ∗ β is homotopic to γ ∗ δ. 3. Suppose that a topological space X has a function m : X × X → X. Show that if α and β are any paths in X, the definition (α · β)(t) = m(α(t), β(t)) is homotopy invariant, in the sense that [α] ∗ [β] = [α ∗ β] is well-defined on homotopy classes of paths. 4. Show ...
Topology I – Problem Set Five Fall 2011
... (b) Let G be any finitely presented group. Show that there is a 4-manifold that has G as its fundamental group (Hint: Let F be the free group on the generators, R the set of relations. Start with X, the 4-manifold from part a). Represent each relation in R by a loop in X. Go ahead and assume that th ...
... (b) Let G be any finitely presented group. Show that there is a 4-manifold that has G as its fundamental group (Hint: Let F be the free group on the generators, R the set of relations. Start with X, the 4-manifold from part a). Represent each relation in R by a loop in X. Go ahead and assume that th ...
. TOPOLOGY QUALIFYING EXAMINATION Time: Three hours.
... (b) The space G is a topological group meaning that G is a group and also a Hausdorff topological space such that the multiplication and map taking each element to its inverse are continuous operations. Given two loops based at the identity e in G, say α(s) and β(s), we have two ways to combine them ...
... (b) The space G is a topological group meaning that G is a group and also a Hausdorff topological space such that the multiplication and map taking each element to its inverse are continuous operations. Given two loops based at the identity e in G, say α(s) and β(s), we have two ways to combine them ...
