LECTURE 30: INDUCED MAPS BETWEEN CLASSIFYING SPACES

... by providing a natural transformation between the functors that these spaces rep resent: α∗ : {Gbundles over X} → {Hbundles over X}. Namely, send a Gbundle E → X to the Hbundle H ×G E → X where G acts on H through the homomorphism α. Thus B may be viewed as a functor Topological groups → homoto ...

... by providing a natural transformation between the functors that these spaces rep resent: α∗ : {Gbundles over X} → {Hbundles over X}. Namely, send a Gbundle E → X to the Hbundle H ×G E → X where G acts on H through the homomorphism α. Thus B may be viewed as a functor Topological groups → homoto ...

18.906 Problem Set 7 Due Friday, April 6 in class

... In particular, if the group G is path-connected, show that the action of G on H∗ (F ) is always trivial. 3. Suppose that ξ → X is a complex vector bundle with inner product associated to a principal U (n)-bundle P → X. There is then a unit sphere bundle S ⊂ ξ consisting of the unit vectors; this is ...

... In particular, if the group G is path-connected, show that the action of G on H∗ (F ) is always trivial. 3. Suppose that ξ → X is a complex vector bundle with inner product associated to a principal U (n)-bundle P → X. There is then a unit sphere bundle S ⊂ ξ consisting of the unit vectors; this is ...

Universal spaces in birational geometry

... Universal spaces in birational geometry — Fedor Bogomolov, October 8, 2010 I want to discuss our joint results with Yuri Tschinkel. The Bloch-Kato conjecture implies that cohomology elements with finite constant coefficients of an algebraic variety can be induced from abelian quotient of the fundame ...

... Universal spaces in birational geometry — Fedor Bogomolov, October 8, 2010 I want to discuss our joint results with Yuri Tschinkel. The Bloch-Kato conjecture implies that cohomology elements with finite constant coefficients of an algebraic variety can be induced from abelian quotient of the fundame ...

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 ...

Textbook definition Geometry example Real world

... A postulate, or axiom, is an accepted statement of fact. ...

... A postulate, or axiom, is an accepted statement of fact. ...

S1-Equivariant K-Theory of CP1

... 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 Hausdorff G -spaces to the category of abelian groups. ...

... 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 Hausdorff G -spaces to the category of abelian groups. ...

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 ...

Math 210B. Homework 4 1. (i) If X is a topological space and a

... of X admits a finite subcover) and that any subspace Y ⊂ X is noetherian. (iii) Conversely to (ii), if every subspace of a topological space X is quasi-compact then prove X is noetherian. 2. Over a field k = k with char(k) 6= 2, decompose Z(y 4 − x2 , y 4 − x2 y 2 + xy 2 − x3 ) ⊂ k 2 and Z(u2 + v 2 ...

... of X admits a finite subcover) and that any subspace Y ⊂ X is noetherian. (iii) Conversely to (ii), if every subspace of a topological space X is quasi-compact then prove X is noetherian. 2. Over a field k = k with char(k) 6= 2, decompose Z(y 4 − x2 , y 4 − x2 y 2 + xy 2 − x3 ) ⊂ k 2 and Z(u2 + v 2 ...

ON TAMAGAWA NUMBERS 1. Adele geometry Let X be an

... with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among them or connections with other invariants of X. The Tamagawa number x (X) is an example of such invariants which is, so far, definable only when X is a connected line ...

... with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among them or connections with other invariants of X. The Tamagawa number x (X) is an example of such invariants which is, so far, definable only when X is a connected line ...

索书号：O187 /C877 (2) (MIT) Ideals, Varieties, and Algorithms C

... Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solution of ...

... Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solution of ...

Algebraic Geometry

... with a and a radical, then the intersection W and W in the sense of schemes is Spec kŒX1 ; : : : ; XnCn0 =.a; a / while their intersection in the sense of varieties is Spec kŒX1 ; : : : ; XnCn0 =rad.a; a0 / (and their intersection in the sense of algebraic spaces is Spm kŒX1 ; : : : ; XnCn0 =.a; ...

... with a and a radical, then the intersection W and W in the sense of schemes is Spec kŒX1 ; : : : ; XnCn0 =.a; a / while their intersection in the sense of varieties is Spec kŒX1 ; : : : ; XnCn0 =rad.a; a0 / (and their intersection in the sense of algebraic spaces is Spm kŒX1 ; : : : ; XnCn0 =.a; ...

Exercises 01 [1.1]

... [1.6] Prove that the following construction of a free group i : S → G on a finite set succeeds. [4] First, show that for any set map f : S → H, the subgroup hf (S)i of H generated by [5] the image f (S) is countable (either finite or countable infinite). Show that there are finitely-many (isomorphis ...

... [1.6] Prove that the following construction of a free group i : S → G on a finite set succeeds. [4] First, show that for any set map f : S → H, the subgroup hf (S)i of H generated by [5] the image f (S) is countable (either finite or countable infinite). Show that there are finitely-many (isomorphis ...

The equivariant spectral sequence and cohomology with local coefficients Alexander I. Suciu

... Particularly interesting is the case of a smooth manifold X fibering over the circle, with ν = p∗ : π Z the homomorphism induced by the projection map, p : X → S 1 . The homology of the resulting infinite cyclic cover was studied by J. Milnor in [7]. This led to another spectral sequence, introduc ...

... Particularly interesting is the case of a smooth manifold X fibering over the circle, with ν = p∗ : π Z the homomorphism induced by the projection map, p : X → S 1 . The homology of the resulting infinite cyclic cover was studied by J. Milnor in [7]. This led to another spectral sequence, introduc ...

OPERADS IN ALGEBRAIC TOPOLOGY II Contents The little

... the proofs) is that the free Dn -algebra on a space X is ⌦n ⌃n X. Remark. If n > 1, r 1, then Dn (r) is path connected. In particular, Dn (2) is path connected for all n > 1. This is the space of binary operations on a Dn -algebra. And thus, there exists a path from any multiplication m 2 Dn (2) to ...

... the proofs) is that the free Dn -algebra on a space X is ⌦n ⌃n X. Remark. If n > 1, r 1, then Dn (r) is path connected. In particular, Dn (2) is path connected for all n > 1. This is the space of binary operations on a Dn -algebra. And thus, there exists a path from any multiplication m 2 Dn (2) to ...

Lecture 1. Modules

... 1.3. Submodules, quotient modules and homomorphisms. Definition. Let M be an R-module. A subset N of M is called an R-submodule if (1) N is a subgroup of (M, +) (2) for any r ∈ R, n ∈ N we have rn ∈ N . Example: Let R be a ring, M = R (with action by left multiplication). Then submodules of R = left ...

... 1.3. Submodules, quotient modules and homomorphisms. Definition. Let M be an R-module. A subset N of M is called an R-submodule if (1) N is a subgroup of (M, +) (2) for any r ∈ R, n ∈ N we have rn ∈ N . Example: Let R be a ring, M = R (with action by left multiplication). Then submodules of R = left ...

Exercises 5 5.1. Let A be an abelian group. Set A ∗ = HomZ(A,Q/Z

... 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. Let K be a field, and A a K-linear space with a basis {xi }i∈I . Show that a bilinear map A × A → A, (a, b) 7→ a · b makes A an algebra (not necessa ...

... 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. Let K be a field, and A a K-linear space with a basis {xi }i∈I . Show that a bilinear map A × A → A, (a, b) 7→ a · b makes A an algebra (not necessa ...

The Exponent Problem in Homotopy Theory (Jie Wu) The

... S n to a point and pinching one line of longitude to the point. The space S n ∨ S n can be regarded as two spheres joining at the north pole. Let f, g : S n → X with f (N ) = g(N ) = x0 . We obtain a map φ : S n ∨ S n → X where φ restricted to the top sphere of S n ∨ S n is f and φ restricted to the ...

... S n to a point and pinching one line of longitude to the point. The space S n ∨ S n can be regarded as two spheres joining at the north pole. Let f, g : S n → X with f (N ) = g(N ) = x0 . We obtain a map φ : S n ∨ S n → X where φ restricted to the top sphere of S n ∨ S n is f and φ restricted to the ...

PDF

... Definition 1. Let V be an irreducible algebraic variety (we assume it to be integral and quasi-projective) over a field K with characteristic zero. We regard V as a topological space with the usual Zariski topology. 1. A subset A ⊂ V (K) is said to be of type C1 if there is a closed subset W ⊂ V , w ...

... Definition 1. Let V be an irreducible algebraic variety (we assume it to be integral and quasi-projective) over a field K with characteristic zero. We regard V as a topological space with the usual Zariski topology. 1. A subset A ⊂ V (K) is said to be of type C1 if there is a closed subset W ⊂ V , w ...

June 2007 901-902

... A. Groups and Character Theory 1. Consider the collection of groups G satisfying |G| = 56 = 23 · 7 and there is a subgroup H of G that is isomorphic to Z/2 × Z/2 × Z/2. (a) Prove there are at least three such groups which are not isomorphic to each other. (b) Prove there are exactly two such groups ...

... A. Groups and Character Theory 1. Consider the collection of groups G satisfying |G| = 56 = 23 · 7 and there is a subgroup H of G that is isomorphic to Z/2 × Z/2 × Z/2. (a) Prove there are at least three such groups which are not isomorphic to each other. (b) Prove there are exactly two such groups ...

An algebraic topological proof of the fundamental theorem of al

... Gouri Shankar Seal (3rd year Integrated M.Sc student) Indian Institute of Science Education and Research, Block HC-VII, Sector-III, Kolkata-700106 Abstract. Several proofs of the fundamental theorem of algebra, using purely algebraic and complex analytic (via Liouville’s theorem) methods are well kn ...

... Gouri Shankar Seal (3rd year Integrated M.Sc student) Indian Institute of Science Education and Research, Block HC-VII, Sector-III, Kolkata-700106 Abstract. Several proofs of the fundamental theorem of algebra, using purely algebraic and complex analytic (via Liouville’s theorem) methods are well kn ...

LECTURES MATH370-08C 1. Groups 1.1. Abstract groups versus

... Q, R, C and Z/(pZ) for p prime are fields; H is a skew-field. The notion of a subring of a ring R is defined naturally: it is a subset of R, closed under both ring operations. A subring I ⊂ R is called a left ideal, if I · R ⊂ I; a right ideal, if R · I ⊂ I; a two-sided ideal, if I · R ⊂ I & R · I ⊂ ...

... Q, R, C and Z/(pZ) for p prime are fields; H is a skew-field. The notion of a subring of a ring R is defined naturally: it is a subset of R, closed under both ring operations. A subring I ⊂ R is called a left ideal, if I · R ⊂ I; a right ideal, if R · I ⊂ I; a two-sided ideal, if I · R ⊂ I & R · I ⊂ ...

Exercises for Math535. 1 . Write down a map of rings that gives the

... 10∗ . Recall that for algebraic groups, if G is connected, then the commutator subgroup [G, G] is a closed algebraic subgroup. For Lie groups over C a similar statement doesn’t hold. Find a better example, or show that the following example works: Let H be the group of 3 × 3 upper triangular martice ...

... 10∗ . Recall that for algebraic groups, if G is connected, then the commutator subgroup [G, G] is a closed algebraic subgroup. For Lie groups over C a similar statement doesn’t hold. Find a better example, or show that the following example works: Let H be the group of 3 × 3 upper triangular martice ...

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 ...

Digression: Microbundles (Lecture 33)

... bundle TE/B is a vector bundle over E whose pullback s∗ TE/B can be regarded as a vector bundle over B. This construction determines a map { smooth microbundles over B}/ equivalence → { vector bundles over B}/ isomorphism . It is easy to see that this construction is left inverse to the construction ...

... bundle TE/B is a vector bundle over E whose pullback s∗ TE/B can be regarded as a vector bundle over B. This construction determines a map { smooth microbundles over B}/ equivalence → { vector bundles over B}/ isomorphism . It is easy to see that this construction is left inverse to the construction ...

characteristic classes in borel cohomology

... H,*(B(17; Z-)) = H,*((EZ-07). Of course, (ET)/l7 is a G-space of the same underlying homotopy type as BZ7. Note next that EC x ET is a free contractible r-space, so that its projection to ET is a r-homotopy ...

... H,*(B(17; Z-)) = H,*((EZ-07). Of course, (ET)/l7 is a G-space of the same underlying homotopy type as BZ7. Note next that EC x ET is a free contractible r-space, so that its projection to ET is a r-homotopy ...