PDF
... topological space. In a different way, they also provide information about the “holes” of the space. • The free groups are important in algebraic topology. In a sense, they are the most general groups, having only those relations among their elements that are absolutely required by the group axioms. ...
... topological space. In a different way, they also provide information about the “holes” of the space. • The free groups are important in algebraic topology. In a sense, they are the most general groups, having only those relations among their elements that are absolutely required by the group axioms. ...
PH Kropholler Olympia Talelli
... Proof of the Proposition. For any set &? let [a, z] denote the set of functions a-, Z which take only finitely many distinct values. This is a subgroup of the additive group of all functions from 0 to E. We define 2 to be [G, m]. Then I is a ZG-submodule of the coinduced module CoindyH and contains ...
... Proof of the Proposition. For any set &? let [a, z] denote the set of functions a-, Z which take only finitely many distinct values. This is a subgroup of the additive group of all functions from 0 to E. We define 2 to be [G, m]. Then I is a ZG-submodule of the coinduced module CoindyH and contains ...
Class 43: Andrew Healy - Rational Homotopy Theory
... method for computing the first part for a wide class of spaces. The idea is to study πnQ (X) := πn (X) ⊗Z Q, which is a torsion-free abelian group with the same rank as πn (X). Of course, this definition is not useful as a computational tool; if we knew πn (X), we wouldn’t be interested in πnQ (X). ...
... method for computing the first part for a wide class of spaces. The idea is to study πnQ (X) := πn (X) ⊗Z Q, which is a torsion-free abelian group with the same rank as πn (X). Of course, this definition is not useful as a computational tool; if we knew πn (X), we wouldn’t be interested in πnQ (X). ...
THE BRAUER GROUP 0.1. Number theory. Let X be a Q
... where the limit is taken over all finite Galois extensions of k. This limit is actually a union. 0.5. Cohomology. Let K be a Galois extension of k, and G = Gal(K, k). The cohomology set H 1 (G, P GLn (K)) classifies central simple k-algebras of degree n which are K-split up to isomorphism. The exact ...
... where the limit is taken over all finite Galois extensions of k. This limit is actually a union. 0.5. Cohomology. Let K be a Galois extension of k, and G = Gal(K, k). The cohomology set H 1 (G, P GLn (K)) classifies central simple k-algebras of degree n which are K-split up to isomorphism. The exact ...
Functors and natural transformations A covariant functor F : C → D is
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
An Uncertainty Principle for Topological Sectors
... 1. What happens when X is noncompact? 2. Complete the quantum theory of the free self-dual field. 3. If one cannot measure the complete K-theory class of RR flux what about the D-brane charge? a.) If no, we need to make an important conceptual revision of the standard picture of a D-brane b.) If yes ...
... 1. What happens when X is noncompact? 2. Complete the quantum theory of the free self-dual field. 3. If one cannot measure the complete K-theory class of RR flux what about the D-brane charge? a.) If no, we need to make an important conceptual revision of the standard picture of a D-brane b.) If yes ...
1. Let G be a sheaf of abelian groups on a topological space. In this
... 1. Let G be a sheaf of abelian groups on a topological space. In this problem, we define H 1 (X, G) as the set of isomorphism classes of G-torsors on X. Let F be a G-torsor, and F 0 be a sheaf of sets with an action by G. Recall that we defined F ×G F 0 as the sheafification of the presheaf that ass ...
... 1. Let G be a sheaf of abelian groups on a topological space. In this problem, we define H 1 (X, G) as the set of isomorphism classes of G-torsors on X. Let F be a G-torsor, and F 0 be a sheaf of sets with an action by G. Recall that we defined F ×G F 0 as the sheafification of the presheaf that ass ...
OPERADS IN ALGEBRAIC TOPOLOGY II Contents The little
... Yesterday I gave you a very elementary introduction to operads. The examples I gave you weren’t very topological. Today we’re going to get into some topology. The little n-disks operad Goal. We’ll consider how to interpolate “up to homotopy” between As and Com. There is an operad morphism As ! Com. ...
... Yesterday I gave you a very elementary introduction to operads. The examples I gave you weren’t very topological. Today we’re going to get into some topology. The little n-disks operad Goal. We’ll consider how to interpolate “up to homotopy” between As and Com. There is an operad morphism As ! Com. ...
Lecture 1. Modules
... Similarly, xn v = An v for any n ∈ N, and finally for any p(x) ∈ F [x] we must have p(x)v = (p(A))v, that is, (cn xn + . . . + c0 )v = (cn An + . . . + c0 )(v) ...
... Similarly, xn v = An v for any n ∈ N, and finally for any p(x) ∈ F [x] we must have p(x)v = (p(A))v, that is, (cn xn + . . . + c0 )v = (cn An + . . . + c0 )(v) ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
Classifying spaces and spectral sequences
... between each pair of elements ofG, i.e. mor(G)=GxG. It is equivalent to the trivial category with one object and one morphism, so BG is contract! ble by (2.1). There is a functor G->G which takes the morphism (^1,^3) to Si1^^ 2in(^ xt induces a map BG ->BG. Now NG is (G, GxG, . . .), a semi-simplici ...
... between each pair of elements ofG, i.e. mor(G)=GxG. It is equivalent to the trivial category with one object and one morphism, so BG is contract! ble by (2.1). There is a functor G->G which takes the morphism (^1,^3) to Si1^^ 2in(^ xt induces a map BG ->BG. Now NG is (G, GxG, . . .), a semi-simplici ...
0.1 A lemma of Kempf
... have assumed inductively that the result is valid for n − 1. So α maps to some β ∈ H n−1 (X, H); this means there is an open cover of X by various V ∈ A such that β maps to zero in H n−1 (X, V H). This means that α maps to zero in these H n (X, V F) by naturality. This completes the proof of the ind ...
... have assumed inductively that the result is valid for n − 1. So α maps to some β ∈ H n−1 (X, H); this means there is an open cover of X by various V ∈ A such that β maps to zero in H n−1 (X, V H). This means that α maps to zero in these H n (X, V F) by naturality. This completes the proof of the ind ...
Graduate Algebra Homework 3
... (c) A function φ : ModR → A (where A is an abelian group) is said to be additive if φ(M ) = φ(M 0 ) + φ(M 00 ) for exact sequences 0 → M 0 → M → M 00 → 0. Show that φ extends to a homomorphism of abelian groups φ : G(R) → A. 3. Let R be a ring. Let Z[ProjR ] be the free abelian group generated by is ...
... (c) A function φ : ModR → A (where A is an abelian group) is said to be additive if φ(M ) = φ(M 0 ) + φ(M 00 ) for exact sequences 0 → M 0 → M → M 00 → 0. Show that φ extends to a homomorphism of abelian groups φ : G(R) → A. 3. Let R be a ring. Let Z[ProjR ] be the free abelian group generated by is ...
Cohomology jump loci of quasi-projective varieties Botong Wang June 27 2013
... Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will not use this at all in this talk. Instead, we need to introduce our main player, the cohomology jump loci in MB (X ), def ...
... Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will not use this at all in this talk. Instead, we need to introduce our main player, the cohomology jump loci in MB (X ), def ...
S1-Equivariant K-Theory of CP1
... If f : X → Y is a continuous map which commutes with the action of G , then one can define a map f ∗ : KG (Y ) → KG (X ). If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdo ...
... If f : X → Y is a continuous map which commutes with the action of G , then one can define a map f ∗ : KG (Y ) → KG (X ). If f is a homotopy equivalence, then f ∗ is a group isomorphism, and id∗X = idKG (X ) . Thus, KG is a homotopy invariant contravariant functor from the category of compact Hausdo ...
850 Oberwolfach Report 15 Equivariant Sheaves on Flag Varieties
... • The perverse t-structure on DbB,c (X) corresponds to a t-structure on the perfect derived category Perf(Ext(IC)) that can be described for a more general class of dg algebras (see [Sch08a]). This yields an algebraic description of the category of B-equivariant perverse sheaves on X. • The algebra ...
... • The perverse t-structure on DbB,c (X) corresponds to a t-structure on the perfect derived category Perf(Ext(IC)) that can be described for a more general class of dg algebras (see [Sch08a]). This yields an algebraic description of the category of B-equivariant perverse sheaves on X. • The algebra ...
PDF
... When studying algebraic topology, the fact that we have a diagonal embedding for any space X lets us define a bit of extra structure in cohomology, called the cup product. This makes cohomology into a ring, so that we can bring additional algebraic muscle to bear on topological questions. ∗ hDiagona ...
... When studying algebraic topology, the fact that we have a diagonal embedding for any space X lets us define a bit of extra structure in cohomology, called the cup product. This makes cohomology into a ring, so that we can bring additional algebraic muscle to bear on topological questions. ∗ hDiagona ...
9. Sheaf Cohomology Definition 9.1. Let X be a topological space
... Theorem 9.6 (Serre vanishing). Let X be a projective variety over a Noetherian ring and let OX (1) be a very ample line bundle on X. Let F be a coherent sheaf. (1) H i (X, F) are finitely generated A-modules. (2) There is an integer n0 such that H i (X, F(n)) = 0 for all n ≥ n0 and i > 0. Proof. By ...
... Theorem 9.6 (Serre vanishing). Let X be a projective variety over a Noetherian ring and let OX (1) be a very ample line bundle on X. Let F be a coherent sheaf. (1) H i (X, F) are finitely generated A-modules. (2) There is an integer n0 such that H i (X, F(n)) = 0 for all n ≥ n0 and i > 0. Proof. By ...
(pdf)
... If we require g ◦ f = 0, i.e., Im(f ) ⊂ ker(g), then we have a chain complex and homology groups are defined. We say the sequence is exact (at B) if we have ker(g) = Im(f ). This condition is to equivalent to saying the homology group at B is trivial, and thus homology groups of a chain complex meas ...
... If we require g ◦ f = 0, i.e., Im(f ) ⊂ ker(g), then we have a chain complex and homology groups are defined. We say the sequence is exact (at B) if we have ker(g) = Im(f ). This condition is to equivalent to saying the homology group at B is trivial, and thus homology groups of a chain complex meas ...