Categories and functors
... This sections is a review of basic concepts in category theory. It will follow as an immediate consequence of basic definitions that all left adjoint functors are right exact and similarly for right adjoint functors. Definition 22.1. A category C is a collection Ob(C) of objects A and morphism sets ...
... This sections is a review of basic concepts in category theory. It will follow as an immediate consequence of basic definitions that all left adjoint functors are right exact and similarly for right adjoint functors. Definition 22.1. A category C is a collection Ob(C) of objects A and morphism sets ...
Algebraic Topology
... 3. If A, B and C are groups and i:C → A and j: C → B are injections induced by maps of spaces K(C,1) → K(A,1) and K(C,1) → K(B,1), then show that K(A,1) ∪K(C,1) K(B,1) is a K(A*CB, 1). Give a counterexample if they are not injections. 4. Show that any complex line bundle over Sn is trivial for n>2. ...
... 3. If A, B and C are groups and i:C → A and j: C → B are injections induced by maps of spaces K(C,1) → K(A,1) and K(C,1) → K(B,1), then show that K(A,1) ∪K(C,1) K(B,1) is a K(A*CB, 1). Give a counterexample if they are not injections. 4. Show that any complex line bundle over Sn is trivial for n>2. ...
Lecture 10 homotopy Consider continuous maps from a topological
... Homology groups are always abelian, but the fundamental group can be non-abelian. It is a good place to introduce the notion of free group. Start with a finite set of letters, X = {a, b, c, ...}. Words are ordered lists of letters of the form, ω = xn1 1 xn2 2 · · · xnNN with n1 , ..., nN ∈ Z and x1 ...
... Homology groups are always abelian, but the fundamental group can be non-abelian. It is a good place to introduce the notion of free group. Start with a finite set of letters, X = {a, b, c, ...}. Words are ordered lists of letters of the form, ω = xn1 1 xn2 2 · · · xnNN with n1 , ..., nN ∈ Z and x1 ...
Homotopy type of symplectomorphism groups of × S Geometry & Topology
... More precisely, let Mλ be the symplectic manifold (S 2 ×S 2 , ωλ = (1+λ)σ0 ⊕σ0 ) where 0 ≤ λ ∈ R and σ0 is the standard area form on S 2 with total area equal to 1. Denote by Gλ the group of symplectomorphisms of Mλ that act as the identity on H2 (S 2 × S 2 ; Z). Gromov proved that G0 is connected a ...
... More precisely, let Mλ be the symplectic manifold (S 2 ×S 2 , ωλ = (1+λ)σ0 ⊕σ0 ) where 0 ≤ λ ∈ R and σ0 is the standard area form on S 2 with total area equal to 1. Denote by Gλ the group of symplectomorphisms of Mλ that act as the identity on H2 (S 2 × S 2 ; Z). Gromov proved that G0 is connected a ...
Defining Gm and Yoneda and group objects
... certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, which give multiplication, taking the inverse, and picking out the identity ...
... certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, which give multiplication, taking the inverse, and picking out the identity ...
9 Direct products, direct sums, and free abelian groups
... 4) Recall: if S is a set then F (S) is the free group generated by S. A map of sets f : S → T defines a homomorphism f˜: F (S) → F (T ) given by f˜(xλ1 1 xλ2 2 · · · · · xλk k ) = f (x1 )λ1 f (x2 )λ2 · · · · · f (xk )λk . Check: the assignment S �→ F (S), ...
... 4) Recall: if S is a set then F (S) is the free group generated by S. A map of sets f : S → T defines a homomorphism f˜: F (S) → F (T ) given by f˜(xλ1 1 xλ2 2 · · · · · xλk k ) = f (x1 )λ1 f (x2 )λ2 · · · · · f (xk )λk . Check: the assignment S �→ F (S), ...
![arXiv:math/0302340v2 [math.AG] 7 Sep 2003](http://s1.studyres.com/store/data/014807023_1-69877329430c783f1dba3c51b4839685-300x300.png)
arXiv:math/0302340v2 [math.AG] 7 Sep 2003
... X is a complete smooth divisor with normal crossings then the weight filtration coincides with the Zeeman filtration given by ”codimension of cycles”. In general there is an inclusion (the terms of the weight filtration are bigger). It is a result of McCrory for hypersurfaces and of F. Guillen ([8]) ...
... X is a complete smooth divisor with normal crossings then the weight filtration coincides with the Zeeman filtration given by ”codimension of cycles”. In general there is an inclusion (the terms of the weight filtration are bigger). It is a result of McCrory for hypersurfaces and of F. Guillen ([8]) ...
ARITHMETIC OF CURVES OVER TWO DIMENSIONAL LOCAL
... H 1 (k, Ql /Zl ) −→ H 2 (k, Z/lZ) is injective. By assumption on k, we have H 2 (k, Z/lZ) ' H 2 (k, µl ) ' Z/` The last isomorphism is well-known for one-dimensional local field and was generalized to non archimedian and locally compact fields by Shatz in [7]. The proof is now reduced to the fact th ...
... H 1 (k, Ql /Zl ) −→ H 2 (k, Z/lZ) is injective. By assumption on k, we have H 2 (k, Z/lZ) ' H 2 (k, µl ) ' Z/` The last isomorphism is well-known for one-dimensional local field and was generalized to non archimedian and locally compact fields by Shatz in [7]. The proof is now reduced to the fact th ...
A parametrized Borsuk-Ulam theorem for a product of - Icmc-Usp
... The definition of the Stiefel-Whitney polynomials for vector bundles with the antipodal involution was introduced by Dold in [1] and it is an useful tool in studying parametrized Borsuk-Ulam type problem. Dold used the Stiefel-Whitney polynomials to prove that if p = 2 and if m and k are the dimensi ...
... The definition of the Stiefel-Whitney polynomials for vector bundles with the antipodal involution was introduced by Dold in [1] and it is an useful tool in studying parametrized Borsuk-Ulam type problem. Dold used the Stiefel-Whitney polynomials to prove that if p = 2 and if m and k are the dimensi ...
Relation to the de Rham cohomology of Lie groups
... In other words, a vector field X is left-invariant if and only if it is Lg -related to itselft for all g ∈ G. ...
... In other words, a vector field X is left-invariant if and only if it is Lg -related to itselft for all g ∈ G. ...
The Weil-étale topology for number rings
... (b) H q .T .G/; G X; E/ is isomorphic to H q .X; ˛ E/. Proof. To prove (a), let fUi ! Y g be a local-section cover of Y . Then fG Ui ! G Y g is a local-section G-cover of G Y and G .Ui Y Uj / is naturally isomorphic to .G Ui / GY .G Uj /, so ˛ F is a sheaf. We note that ˛ is clea ...
... (b) H q .T .G/; G X; E/ is isomorphic to H q .X; ˛ E/. Proof. To prove (a), let fUi ! Y g be a local-section cover of Y . Then fG Ui ! G Y g is a local-section G-cover of G Y and G .Ui Y Uj / is naturally isomorphic to .G Ui / GY .G Uj /, so ˛ F is a sheaf. We note that ˛ is clea ...
IV.2 Homology
... and B(x) = x. B is the identity and therefore has degree 1. If f has no fixed point then A is well defined and has degree 0 because it extends a map from the ball to the sphere. We now construct H : Sd × [0, 1] → Sd defined by H(x, t) = (x − tf (x))/kx − tf (x)k. For t = 1 we have x 6= f (x) because ...
... and B(x) = x. B is the identity and therefore has degree 1. If f has no fixed point then A is well defined and has degree 0 because it extends a map from the ball to the sphere. We now construct H : Sd × [0, 1] → Sd defined by H(x, t) = (x − tf (x))/kx − tf (x)k. For t = 1 we have x 6= f (x) because ...
Derived funcors, Lie algebra cohomology and some first applications
... Lemma 3.1. The objects Li F (A) are well defined up to natural isomorphism. That is, if Q → A is a second projective resolution, there is an isomorphism Hi F (P ) ∼ = Hi F (Q). Proof. Let P → A and Q → A be projective resolutions of A. By the comparison theorem there is a chain map f : P → Q lifting ...
... Lemma 3.1. The objects Li F (A) are well defined up to natural isomorphism. That is, if Q → A is a second projective resolution, there is an isomorphism Hi F (P ) ∼ = Hi F (Q). Proof. Let P → A and Q → A be projective resolutions of A. By the comparison theorem there is a chain map f : P → Q lifting ...
CW Complexes and the Projective Space
... and that πk (Sn ) = 0 if k < n. The long exact sequence of homotopy groups induced by the fiber bundle S1 ,→ S2n+1 −→ n CP is · · · −→ πk (S1 ) −→ πk (S2n+1 ) −→ πk (CP n ) −→ πk−1 (S1 ) −→ · · · When k 6= 1, 2, it follows that πk (CP n ) ∼ = πk (S2n+1 ). ...
... and that πk (Sn ) = 0 if k < n. The long exact sequence of homotopy groups induced by the fiber bundle S1 ,→ S2n+1 −→ n CP is · · · −→ πk (S1 ) −→ πk (S2n+1 ) −→ πk (CP n ) −→ πk−1 (S1 ) −→ · · · When k 6= 1, 2, it follows that πk (CP n ) ∼ = πk (S2n+1 ). ...
MATH 6280 - CLASS 2 Contents 1. Categories 1 2. Functors 2 3
... (6) Homology Hn : Top → Ab which sends a space S to the n’th simplicial homology group of S. (7) Cohomology H n : Topop → Ab which sends a space S to the n’th simplicial cohomology group of S. (8) The homotopy groups functors: πn : Topop → Ab (9) If F : G → H is a group homomorphism, then it gives r ...
... (6) Homology Hn : Top → Ab which sends a space S to the n’th simplicial homology group of S. (7) Cohomology H n : Topop → Ab which sends a space S to the n’th simplicial cohomology group of S. (8) The homotopy groups functors: πn : Topop → Ab (9) If F : G → H is a group homomorphism, then it gives r ...
Math 210B. Absolute Galois groups and fundamental groups 1
... in X of H1 (X, Z), and so likewise the contravariant functoriality in X of H1 (X, Z). In a similar spirit, we will see later that Galois cohomology Hj (Gal(ks /k), ·) is actually functorial in k alone, without reference to ks . One might wonder why the correct analogue of Gal(ks /k)-cohomology isn’t ...
... in X of H1 (X, Z), and so likewise the contravariant functoriality in X of H1 (X, Z). In a similar spirit, we will see later that Galois cohomology Hj (Gal(ks /k), ·) is actually functorial in k alone, without reference to ks . One might wonder why the correct analogue of Gal(ks /k)-cohomology isn’t ...
A Prelude to Obstruction Theory - WVU Math Department
... Example. The usual topology in Rn is Hausdorff, as is the subspace topology on any X ⊆ Rn . One must be careful, however, not to assume that all spaces are Hausdorff. The finite complement topology (where a set U is open if and only if X − U is a finite set) is very rarely Hausdorff. We will require ...
... Example. The usual topology in Rn is Hausdorff, as is the subspace topology on any X ⊆ Rn . One must be careful, however, not to assume that all spaces are Hausdorff. The finite complement topology (where a set U is open if and only if X − U is a finite set) is very rarely Hausdorff. We will require ...