Math 8301, Manifolds and Topology Homework 7

... be the space obtained from the disjoint union Dn ∪ X by identifying any point v ∈ S n−1 ⊂ Dn with its image φ(v) ∈ X. We say that Y is obtained from X by attaching an n-cell. Use the Seifert-van Kampen theorem to describe the fundamental group of Y in terms of the fundamental group of X and the map ...

... be the space obtained from the disjoint union Dn ∪ X by identifying any point v ∈ S n−1 ⊂ Dn with its image φ(v) ∈ X. We say that Y is obtained from X by attaching an n-cell. Use the Seifert-van Kampen theorem to describe the fundamental group of Y in terms of the fundamental group of X and the map ...

. 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 ...

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 ...

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 ...

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 ...