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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...