Algebraic topology exam

... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. Prove the Zig-Zag lemma: let 0 C D E 0 be a short exact sequence of chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequenc ...

... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. Prove the Zig-Zag lemma: let 0 C D E 0 be a short exact sequence of chain complexes with the above maps being f: C D, g : D E. Show that there is a long exact sequenc ...

PDF

... construct a stable homotopy theory, so that the homotopy category of spectra is canonically defined in the classical manner. Therefore, for any given construction of an Ω–spectrum one is able to canonically define an associated cohomology theory; thus, one defines the cohomology groups of a CW-compl ...

... construct a stable homotopy theory, so that the homotopy category of spectra is canonically defined in the classical manner. Therefore, for any given construction of an Ω–spectrum one is able to canonically define an associated cohomology theory; thus, one defines the cohomology groups of a CW-compl ...

Problem Set 5 - Stony Brook Mathematics

... (b) Show that if X is non-orientable, then the torsion subgroup of Hn−1 (X) is Z/2, Hn (X; G) = ker(G →2 G), and H n (X; G) = G/2G. In particular, Hn (X) = 0, H n (X) = Z/2. Problem 3. Let X be a homology n-manifold (not necessarily compact) that is triagulated by a locally finite simplicial complex ...

... (b) Show that if X is non-orientable, then the torsion subgroup of Hn−1 (X) is Z/2, Hn (X; G) = ker(G →2 G), and H n (X; G) = G/2G. In particular, Hn (X) = 0, H n (X) = Z/2. Problem 3. Let X be a homology n-manifold (not necessarily compact) that is triagulated by a locally finite simplicial complex ...

Topology Qual Winter 2000

... 1. a) Let G and H be functors from a category C to a category D. Define a natural transformation from G to H. b) For an admissible pair of topological spaces (X,A) define functors G and H by G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map * is a natural transformation of G to H. Define and give ...

... 1. a) Let G and H be functors from a category C to a category D. Define a natural transformation from G to H. b) For an admissible pair of topological spaces (X,A) define functors G and H by G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map * is a natural transformation of G to H. Define and give ...

Complex Bordism (Lecture 5)

... Remark 5. In the setting of Definition 2, it suffices to check the condition at one point x in each connected component of X. Our next goal is to show that if E is a complex-oriented cohomology theory, then all complex vector bundles have a canonical E-orientation. To prove this, it suffices to cons ...

... Remark 5. In the setting of Definition 2, it suffices to check the condition at one point x in each connected component of X. Our next goal is to show that if E is a complex-oriented cohomology theory, then all complex vector bundles have a canonical E-orientation. To prove this, it suffices to cons ...

Schnabl

... Therefore in this class of solutions, the trivial ones are those for which F2(0) ≠ 1. Tachyon vacuum solutions are those for which F2(0) = 1 but the zero of 1-F2 is first order When the order of zero of 1-F2 at K=0 is of higher order the solution is not quite well defined, but it has been conjecture ...

... Therefore in this class of solutions, the trivial ones are those for which F2(0) ≠ 1. Tachyon vacuum solutions are those for which F2(0) = 1 but the zero of 1-F2 is first order When the order of zero of 1-F2 at K=0 is of higher order the solution is not quite well defined, but it has been conjecture ...

RIGID RATIONAL HOMOTOPY THEORY AND

... The existence of such a ’resolution’ X‚ Ñ X in the second step is a straightforward and very standard application of de Jong’s theorem - the isomorphism RΓrig pX{Kq » holim∆ RΓrig pX‚ {Kq then follows more or less immediately from cohomological descent for rigid cohomology. To explain the third step ...

... The existence of such a ’resolution’ X‚ Ñ X in the second step is a straightforward and very standard application of de Jong’s theorem - the isomorphism RΓrig pX{Kq » holim∆ RΓrig pX‚ {Kq then follows more or less immediately from cohomological descent for rigid cohomology. To explain the third step ...

Section 07

... (a) Let ϕ : A → B be a morphism of presheaves on X. If for any open U ⊆ X one takes the kernel ker(ϕU ) ⊆ A(U ), then these groups together form another presheaf on X, denoted ker(ϕ), and the evident inclusion is a morphism of presheaves ker(ϕ) A. In exactly the same way, by taking images and coke ...

... (a) Let ϕ : A → B be a morphism of presheaves on X. If for any open U ⊆ X one takes the kernel ker(ϕU ) ⊆ A(U ), then these groups together form another presheaf on X, denoted ker(ϕ), and the evident inclusion is a morphism of presheaves ker(ϕ) A. In exactly the same way, by taking images and coke ...

(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 ...

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

... 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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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) ...