SHAPIRO`S LEMMA FOR TOPOLOGICAL K
... have the same K-theories. Thus, by checking that the various assembly maps are compatible with these isomorphisms we obtain Theorem 0.5. Let X, H, K and A be as above. Then the following are equivalent: (i) H × K satisfies BC for A. (ii) H satisfies BC for AK . (iii) K satisfies BC for AH . As a spe ...
... have the same K-theories. Thus, by checking that the various assembly maps are compatible with these isomorphisms we obtain Theorem 0.5. Let X, H, K and A be as above. Then the following are equivalent: (i) H × K satisfies BC for A. (ii) H satisfies BC for AK . (iii) K satisfies BC for AH . As a spe ...
What is a Group Representation?
... Definition 5 (Subrepresentations). A G-subrepresentation of V is a vector subspace W ⊆ V which is invariant by the action of G. Definition 6 (Irreducible representations). A representation V of G is irreducible if it has no proper subrepresentations. Example 3 (Non irreducible representation). Let S ...
... Definition 5 (Subrepresentations). A G-subrepresentation of V is a vector subspace W ⊆ V which is invariant by the action of G. Definition 6 (Irreducible representations). A representation V of G is irreducible if it has no proper subrepresentations. Example 3 (Non irreducible representation). Let S ...
On a coincidence theorem of FB Fuller
... by typewritten (double spaced). The first paragraph or two must be capable of being used separately as a synopsis of the entire paper. It should not contain references to the bibliography. No separate author's resume is required. Manuscripts may be sent to any one of the four editors. All other comm ...
... by typewritten (double spaced). The first paragraph or two must be capable of being used separately as a synopsis of the entire paper. It should not contain references to the bibliography. No separate author's resume is required. Manuscripts may be sent to any one of the four editors. All other comm ...
Quiz-1 Algebraic Topology 1. Show that for odd n, the antipodal map
... 2. Let f : S n → X be a continuous map which is homotopic to a constant map. Show that f can be extended to a continuous map F : B n+1 → X. 3. Find the fundamental group of R3 − E, where E is the union of x and y-axis. 4. A subspace Y of X is called a deformation retract of X if there exist a homoto ...
... 2. Let f : S n → X be a continuous map which is homotopic to a constant map. Show that f can be extended to a continuous map F : B n+1 → X. 3. Find the fundamental group of R3 − E, where E is the union of x and y-axis. 4. A subspace Y of X is called a deformation retract of X if there exist a homoto ...
My notes - Harvard Mathematics
... Now we want to make the collection of definable sets into a category. 2.7 Definition. A definable function between X, Y is a definable subset Z ⊂ X × Y such that for every model M (of the theory), Z(M ) ⊂ X(M ) × Y (M ) is the graph of a function X(M ) → Y (M ). ...
... Now we want to make the collection of definable sets into a category. 2.7 Definition. A definable function between X, Y is a definable subset Z ⊂ X × Y such that for every model M (of the theory), Z(M ) ⊂ X(M ) × Y (M ) is the graph of a function X(M ) → Y (M ). ...
Integers modulo N
... Notice that ZN is a set of size N, and that its distinct elements are precisely 0, 1, . . . , N − 1. We want to define addition of elements of ZN . We do it like this. Suppose x, y ∈ ZN . Choose a ∈ x, b ∈ y. Define x + y to be [a+b]. Notice that the plus sign in [a + b] indicates addition of intege ...
... Notice that ZN is a set of size N, and that its distinct elements are precisely 0, 1, . . . , N − 1. We want to define addition of elements of ZN . We do it like this. Suppose x, y ∈ ZN . Choose a ∈ x, b ∈ y. Define x + y to be [a+b]. Notice that the plus sign in [a + b] indicates addition of intege ...
Quotients of adic spaces by finite groups
... the larger category of ringed spaces, and the existence of a G-invariant covering by affines can be interpreted as a criterion for X/G to be a scheme. In this note we prove a similar result in the context of nonarchimedean analytic geometry. Due to its scope and generality, we work in the language of ...
... the larger category of ringed spaces, and the existence of a G-invariant covering by affines can be interpreted as a criterion for X/G to be a scheme. In this note we prove a similar result in the context of nonarchimedean analytic geometry. Due to its scope and generality, we work in the language of ...
The Tangent Space of a Lie Group – Lie Algebras • We will see that
... – By use of certain diffeomorphisms on the Lie group, namely left or right translations, we will see that it is enough to study the tangent space of a Lie group of a Lie group at the identity element e. – The tangent space at that point is not only a vector space but it is isomorphic to what is defi ...
... – By use of certain diffeomorphisms on the Lie group, namely left or right translations, we will see that it is enough to study the tangent space of a Lie group of a Lie group at the identity element e. – The tangent space at that point is not only a vector space but it is isomorphic to what is defi ...
Algebra part - Georgia Tech Math
... of order 5. Solution. (a) Let G act on X = G by conjugation. The size of the conjugacy class of g ∈ G is |G|/|C(g)| where C(g) is the centralizer of g in G. Thus, the size of the conjugacy class is 5k for k = 1, 2, 3. k = 1 iff g ∈ Z and |Z| divides 125. Since 125 = |X| and 5 divides the size of eve ...
... of order 5. Solution. (a) Let G act on X = G by conjugation. The size of the conjugacy class of g ∈ G is |G|/|C(g)| where C(g) is the centralizer of g in G. Thus, the size of the conjugacy class is 5k for k = 1, 2, 3. k = 1 iff g ∈ Z and |Z| divides 125. Since 125 = |X| and 5 divides the size of eve ...
Algebras
... Remark also that A has, a priori, no unit. Proposition & Definition 1.1.3 Let A be an algebra, the vector space A together with the multiplication defined by (x, y) 7→ yx is again an algebra called the opposite algebra and denoted Aop . Proposition & Definition 1.1.4 A subvector space B of A which i ...
... Remark also that A has, a priori, no unit. Proposition & Definition 1.1.3 Let A be an algebra, the vector space A together with the multiplication defined by (x, y) 7→ yx is again an algebra called the opposite algebra and denoted Aop . Proposition & Definition 1.1.4 A subvector space B of A which i ...
What is a Vector Space?
... A recipe isn’t complete if you only know the set of ingredients, or if you only know the instructions: if someone asked you “does 1 kilo dry spaghetti, 1 jar pasta sauce, and 4 quarts of water constitute a recipe,” you’d say “you haven’t given me all the information yet – what are the instructions f ...
... A recipe isn’t complete if you only know the set of ingredients, or if you only know the instructions: if someone asked you “does 1 kilo dry spaghetti, 1 jar pasta sauce, and 4 quarts of water constitute a recipe,” you’d say “you haven’t given me all the information yet – what are the instructions f ...
Cell-Like Maps (Lecture 5)
... many purposes, it is the conclusions of Proposition 17 which are important. For us, this will be irrelevant: we will only be interested in the case where X is a compact Hausdorff space. Definition 19. Let f : X → Y be a map of Hausdorff spaces. We will say that f is cell-like if it satisfies the fol ...
... many purposes, it is the conclusions of Proposition 17 which are important. For us, this will be irrelevant: we will only be interested in the case where X is a compact Hausdorff space. Definition 19. Let f : X → Y be a map of Hausdorff spaces. We will say that f is cell-like if it satisfies the fol ...
The classification of algebraically closed alternative division rings of
... compatible with its ring operations. This is equivalent to say that, if a finite sum Pn ...
... compatible with its ring operations. This is equivalent to say that, if a finite sum Pn ...
arXiv:math.OA/9901094 v1 22 Jan 1999
... cocycle functor ZΓ : Ab(Γ) → Ab, where Ab(Γ) is the category of Γ-sheaves and Ab is the category of abelian groups, is defined as follows: Given a Γ-sheaf A, the abelian group ZΓ (A) consists of all continuous functions f : Γ → A such that f (γ) ∈ Ar(γ) (i.e. f is a continuous section of r∗ (A)) and ...
... cocycle functor ZΓ : Ab(Γ) → Ab, where Ab(Γ) is the category of Γ-sheaves and Ab is the category of abelian groups, is defined as follows: Given a Γ-sheaf A, the abelian group ZΓ (A) consists of all continuous functions f : Γ → A such that f (γ) ∈ Ar(γ) (i.e. f is a continuous section of r∗ (A)) and ...