The Exponent Problem in Homotopy Theory (Jie Wu) The
... uncountable) but by taking homotopy equivalence relation πn+2 (S n ) has only two elements for n ≥ 2. A space X is called simply connected if X is path connected and the fundamental group π1 (X) is trivial. For instance S n is simply connected for n ≥ 2. Since it is difficult to determine the homoto ...
... uncountable) but by taking homotopy equivalence relation πn+2 (S n ) has only two elements for n ≥ 2. A space X is called simply connected if X is path connected and the fundamental group π1 (X) is trivial. For instance S n is simply connected for n ≥ 2. Since it is difficult to determine the homoto ...
Problem 1. Let C be a category. Recall that: • a morphism f : X → Y is
... f g = idy and gf = idX . • a morphism f : X → Y is a monomorphism if for any object Z, the function morC (Z, X) 3 h 7→ f h ∈ morC (Z, Y ) is 1 to 1. • a morphism f : X → Y is an epimorphism if for any object Z, the function morC (Y, Z) 3 h 7→ hf ∈ morC (X, Z) is 1 to 1. (1) Show that in the categori ...
... f g = idy and gf = idX . • a morphism f : X → Y is a monomorphism if for any object Z, the function morC (Z, X) 3 h 7→ f h ∈ morC (Z, Y ) is 1 to 1. • a morphism f : X → Y is an epimorphism if for any object Z, the function morC (Y, Z) 3 h 7→ hf ∈ morC (X, Z) is 1 to 1. (1) Show that in the categori ...
SOLUTIONS FOR THE TRAINING FINAL Remember : the final exam
... questions on the last third of the peogram. This is a set of training exercises on that last third (Groups action, rings and fields). 1.– Let R be a ring with unity, R∗ the groups of unit. a.– Show that the application R∗ × R → R, (g, x) 7→ gx is an action of the group R∗ on the set R. b.– Let x in ...
... questions on the last third of the peogram. This is a set of training exercises on that last third (Groups action, rings and fields). 1.– Let R be a ring with unity, R∗ the groups of unit. a.– Show that the application R∗ × R → R, (g, x) 7→ gx is an action of the group R∗ on the set R. b.– Let x in ...
a reciprocity theorem for certain hypergeometric series
... In Entry 2 of Chapter 14 in his second notebook [4], Ramanujan states a beautiful reciprocity theorem (with no hypotheses or proof) for certain hypergeometric series. In his notebooks [4], Ramanujan recorded many “reciprocity theorems” or “modular relations” for infinite series, but we are unaware o ...
... In Entry 2 of Chapter 14 in his second notebook [4], Ramanujan states a beautiful reciprocity theorem (with no hypotheses or proof) for certain hypergeometric series. In his notebooks [4], Ramanujan recorded many “reciprocity theorems” or “modular relations” for infinite series, but we are unaware o ...
Homework sheet 6
... Y 2 Z = X 2 (X + Z). Write down a non-constant morphism from P1 to C; carefully check that what you’ve written down is a morphism. 4. If X is a topological space, {Ui }i∈I is an open cover of X, and Z is a subset of X, prove that Z is closed if and only if Z ∩ Ui is closed in Ui (when Ui is given th ...
... Y 2 Z = X 2 (X + Z). Write down a non-constant morphism from P1 to C; carefully check that what you’ve written down is a morphism. 4. If X is a topological space, {Ui }i∈I is an open cover of X, and Z is a subset of X, prove that Z is closed if and only if Z ∩ Ui is closed in Ui (when Ui is given th ...
Exercise Sheet 4 - D-MATH
... ϕpxq “ x, and consider also the differentiable structure induced by the chart ψ : R Ñ R, ψpxq “ x3 . Show that the two differentiable structures are not equal, but that nevertheless the two differentiable manifolds thus defined are diffeomorphic. 4. (Review of Quaternions) Let Q denote the vector sp ...
... ϕpxq “ x, and consider also the differentiable structure induced by the chart ψ : R Ñ R, ψpxq “ x3 . Show that the two differentiable structures are not equal, but that nevertheless the two differentiable manifolds thus defined are diffeomorphic. 4. (Review of Quaternions) Let Q denote the vector sp ...
Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.