Splitting of the Identity Component in Locally Compact Abelian Groups
... x QR/ZY' for some n&No and cardinal number m, and likewise these are exactly the connected LCA groups splitting in every LCA group in which they are contained as identity components (compare Ahem-Jewitt [1], Dixmier[2] and Fulp-GrifSth [4]). It is weU known that this statement remains true if «L(3A ...
... x QR/ZY' for some n&No and cardinal number m, and likewise these are exactly the connected LCA groups splitting in every LCA group in which they are contained as identity components (compare Ahem-Jewitt [1], Dixmier[2] and Fulp-GrifSth [4]). It is weU known that this statement remains true if «L(3A ...
A non-weakly amenable augementation ideal
... In this section we will consider a discrete group G acting as a group of bijections of a discrete space X. The bounded group cohomology of G is essentially the same as the `1 (G) algebra cohomology and we will express our results in terms of the former. We will write Hn (G, X) for Hn (`1 (G), `∞ (X) ...
... In this section we will consider a discrete group G acting as a group of bijections of a discrete space X. The bounded group cohomology of G is essentially the same as the `1 (G) algebra cohomology and we will express our results in terms of the former. We will write Hn (G, X) for Hn (`1 (G), `∞ (X) ...
PICARD GROUPS OF MODULI PROBLEMS
... (1) If any of the objects X we are parametrizing admits an automorphism of finite order, there exist non-isomorphic étale-locally trivial families, all of whose fibers are isomorphic to X, over, say, Gm . But these families would correspond to the same (constant) map to the moduli space. (2) There ...
... (1) If any of the objects X we are parametrizing admits an automorphism of finite order, there exist non-isomorphic étale-locally trivial families, all of whose fibers are isomorphic to X, over, say, Gm . But these families would correspond to the same (constant) map to the moduli space. (2) There ...
ON SOME DIFFERENTIALS IN THE MOTIVIC COHOMOLOGY
... algebra generator (Proposition 2.2). Finally, to check that the proportionality coefficient is not 0, I construct an example of a variety with non-trivial differentials (Theorem 4.1). The computation of the p-local Steenrod algebra is based on Voevodsky’s result on the structure of the motivic Steen ...
... algebra generator (Proposition 2.2). Finally, to check that the proportionality coefficient is not 0, I construct an example of a variety with non-trivial differentials (Theorem 4.1). The computation of the p-local Steenrod algebra is based on Voevodsky’s result on the structure of the motivic Steen ...
PDF
... A singular n-simplex in a topological space X is a continuous map f : ∆n → X. A singular n-chain is a formal linear combination (with integer coefficients) of a finite number of singular n-simplices. The n-chains in X form a group under formal addition, denoted Cn (X, Z). Next, we define a boundary ...
... A singular n-simplex in a topological space X is a continuous map f : ∆n → X. A singular n-chain is a formal linear combination (with integer coefficients) of a finite number of singular n-simplices. The n-chains in X form a group under formal addition, denoted Cn (X, Z). Next, we define a boundary ...
de Rham cohomology
... between the de Rham groups and topology was first proved by Georges de Rham himself in the 1931. In these notes we give a proof of de Rham’s Theorem, that states there exists an isomorphism between de Rham and singular cohomology groups of a smooth manifold. In the first chapter we recall some notio ...
... between the de Rham groups and topology was first proved by Georges de Rham himself in the 1931. In these notes we give a proof of de Rham’s Theorem, that states there exists an isomorphism between de Rham and singular cohomology groups of a smooth manifold. In the first chapter we recall some notio ...
A Grothendieck site is a small category C equipped with a
... 1) f ∗ is left adjoint to f∗, and 2) f ∗ preserves finite limits. The left adjoint f ∗ is called the inverse image functor, while f∗ is called the direct image. The inverse image f ∗ is left and right exact in the sense that it preserves all finite limits and colimits. The direct image f∗ is usuall ...
... 1) f ∗ is left adjoint to f∗, and 2) f ∗ preserves finite limits. The left adjoint f ∗ is called the inverse image functor, while f∗ is called the direct image. The inverse image f ∗ is left and right exact in the sense that it preserves all finite limits and colimits. The direct image f∗ is usuall ...
An Introduction to K-theory
... Grothendieck group of finitely generated projective R-modules for a (commutative) ring R if Spec R = X, of topological vector vector bundles over X if X is a finite dimensional C.W. complex, and of coherent, locally free OX -modules if X is a scheme. Without a doubt, a primary goal (if not the prima ...
... Grothendieck group of finitely generated projective R-modules for a (commutative) ring R if Spec R = X, of topological vector vector bundles over X if X is a finite dimensional C.W. complex, and of coherent, locally free OX -modules if X is a scheme. Without a doubt, a primary goal (if not the prima ...
THE EULER CLASS OF A SUBSET COMPLEX 1. Introduction Let G
... In the rest of the paper, we consider abelian p-groups. First we show that ζG is zero when G is isomorphic to Z/8 or (Z/4)2 ×Z/2, and hence conclude that any group which has a subquotient isomorphic to one of these groups has zero Euler class. This shows that if G is a 2-group with ζG 6= 0, then G i ...
... In the rest of the paper, we consider abelian p-groups. First we show that ζG is zero when G is isomorphic to Z/8 or (Z/4)2 ×Z/2, and hence conclude that any group which has a subquotient isomorphic to one of these groups has zero Euler class. This shows that if G is a 2-group with ζG 6= 0, then G i ...
Topological Extensions of Linearly Ordered Groups
... A topological group G is a topological space with continuous group operation and inversion. A topological semigroup is a Hausdorff topological space with continuous semigroup operation. A topological inverse semigroup is an inverse topological semigroup with continuous inversion. A linearly ordered ...
... A topological group G is a topological space with continuous group operation and inversion. A topological semigroup is a Hausdorff topological space with continuous semigroup operation. A topological inverse semigroup is an inverse topological semigroup with continuous inversion. A linearly ordered ...
Topological groups: local versus global
... quotient space G/H is locally perfect. This turns out to be a key result in the proof that a number of topological properties are transfered from the quotient space G/H to the topological group G, provided that H is locally compact. One of such statements is the classical result of J.-P. Serre that ...
... quotient space G/H is locally perfect. This turns out to be a key result in the proof that a number of topological properties are transfered from the quotient space G/H to the topological group G, provided that H is locally compact. One of such statements is the classical result of J.-P. Serre that ...
3. The Sheaf of Regular Functions
... as in the case of continuous or differentiable functions, we should not only aim for a definition of functions on all of X, but also on an arbitrary open subset U of X. In contrast to the coordinate ring A(X) of polynomial functions on the whole space X, this allows us to consider quotients gf of po ...
... as in the case of continuous or differentiable functions, we should not only aim for a definition of functions on all of X, but also on an arbitrary open subset U of X. In contrast to the coordinate ring A(X) of polynomial functions on the whole space X, this allows us to consider quotients gf of po ...
functors of artin ringso
... which is easily seen to determine, for each -qe F(A), a group action of tFI on
the subset F(p)'1(iq) of F(A') (provided that subset is not empty). (Hx) implies
that this action is "transitive," while (H4) is precisely the condition that this action
makes F(p)-1(^) a (formally) principal homogene ...
... which is easily seen to determine, for each -qe F(A), a group action of tF
Summer School Topology Midterm
... IV.2. Let K be a compact subset of X and a ∈ X \ K. Show that there are disjoint open subsets U and V such that K ⊆ U and a ∈ V. IV.3. Let K1 and K2 be two disjoint compact subsets. Show that there are two disjoint open subsets U1 and U2 of X such that Ki ⊆ Ui. IV.4. Assume now X is a metric space w ...
... IV.2. Let K be a compact subset of X and a ∈ X \ K. Show that there are disjoint open subsets U and V such that K ⊆ U and a ∈ V. IV.3. Let K1 and K2 be two disjoint compact subsets. Show that there are two disjoint open subsets U1 and U2 of X such that Ki ⊆ Ui. IV.4. Assume now X is a metric space w ...
Orbifolds and their cohomology.
... Example 1.0.6. In the definition of weighted projective space given in Example 1.0.3, allowing the integers c0 , . . . , cn to have a common factor d produces an orbifold in which every chart (Ũ , G) has a subgroup Zd ⊂ G acting ineffectively; in other words, every point has Zd isotropy, and the co ...
... Example 1.0.6. In the definition of weighted projective space given in Example 1.0.3, allowing the integers c0 , . . . , cn to have a common factor d produces an orbifold in which every chart (Ũ , G) has a subgroup Zd ⊂ G acting ineffectively; in other words, every point has Zd isotropy, and the co ...
Exponential laws for topological categories, groupoids
... k-spaces and continuous maps has been considered by a number ...
... k-spaces and continuous maps has been considered by a number ...
on the shape of torus-like continua and compact connected
... then it has the same shape as a compact connected abelian topological group. It is shown, in fact, that X has the same shape as char HX(X) where Hn(X) is «-dimensional Cech cohomology over the integers. Using our knowledge of the shape properties of compact connected abelian topological groups conta ...
... then it has the same shape as a compact connected abelian topological group. It is shown, in fact, that X has the same shape as char HX(X) where Hn(X) is «-dimensional Cech cohomology over the integers. Using our knowledge of the shape properties of compact connected abelian topological groups conta ...
Boundary manifolds of projective hypersurfaces Daniel C. Cohen Alexander I. Suciu
... There are many ways to understand the topology of a homogeneous polynomial f : Cℓ+1 → C. The most direct approach is to study the hypersurface V in CPℓ defined as the zero locus of f . Another approach is to view the complement, X = CPℓ \ V , as the primary object of study. And perhaps the most thor ...
... There are many ways to understand the topology of a homogeneous polynomial f : Cℓ+1 → C. The most direct approach is to study the hypersurface V in CPℓ defined as the zero locus of f . Another approach is to view the complement, X = CPℓ \ V , as the primary object of study. And perhaps the most thor ...
Equivariant cohomology and equivariant intersection theory
... which associates with any χ ∈ Ξ(T ) the Chern class of the corresponding line bundle on G/T . The map S ⊗S W S → HT∗ (G/T ) is a lift of this characteristic homomorphism to T -equivariant cohomology. For G and X as before, consider the restriction map ρ : HG∗ (X)/(S+W ) → H ∗(X). We already saw that ...
... which associates with any χ ∈ Ξ(T ) the Chern class of the corresponding line bundle on G/T . The map S ⊗S W S → HT∗ (G/T ) is a lift of this characteristic homomorphism to T -equivariant cohomology. For G and X as before, consider the restriction map ρ : HG∗ (X)/(S+W ) → H ∗(X). We already saw that ...
Cohomology of cyro-electron microscopy
... [37, 36], and has relations to profound problems in computational complexity [1] and operator theory [2]. This article examines the problem from an algebraic topological angle — we will show that the problem of cryo-EM is a problem of cohomology, or, more specifically, the Čech cohomology of a simp ...
... [37, 36], and has relations to profound problems in computational complexity [1] and operator theory [2]. This article examines the problem from an algebraic topological angle — we will show that the problem of cryo-EM is a problem of cohomology, or, more specifically, the Čech cohomology of a simp ...
Sheaf Theory (London Mathematical Society Lecture Note Series)
... has been used with great success as a tool in the solution of several longstanding problems. In this course we build enough of the foundations of sheaf theory to give a broad definition of manifold, covering as special cases the algebraic geometer's schemes as well as the topological, differentiable ...
... has been used with great success as a tool in the solution of several longstanding problems. In this course we build enough of the foundations of sheaf theory to give a broad definition of manifold, covering as special cases the algebraic geometer's schemes as well as the topological, differentiable ...
Since Lie groups are topological groups (and manifolds), it is useful
... Proposition 2.13. If G is a topological group and H is a subgroup of G then the following properties hold: (1) The map p : G ! G/H is an open map, which means that p(V ) is open in G/H whenever V is open in G. (2) The space G/H is Hausdor↵ i↵ H is closed in G. (3) If H is open, then H is closed and ...
... Proposition 2.13. If G is a topological group and H is a subgroup of G then the following properties hold: (1) The map p : G ! G/H is an open map, which means that p(V ) is open in G/H whenever V is open in G. (2) The space G/H is Hausdor↵ i↵ H is closed in G. (3) If H is open, then H is closed and ...
Categories and functors
... More generally, a presheaf on a category A is a functor Aop Functors are the structure-preserving maps of categories; they can be composed, so there is a (large) category Cat consisting of small categories and functors. Informally, there is also a (huge) category CAT consisting of all categories and ...
... More generally, a presheaf on a category A is a functor Aop Functors are the structure-preserving maps of categories; they can be composed, so there is a (large) category Cat consisting of small categories and functors. Informally, there is also a (huge) category CAT consisting of all categories and ...
arXiv:0903.2024v3 [math.AG] 9 Jul 2009
... and its algebraic structure as a monoid did not play any role. One of the goals of the present paper is to promote this additional structure by pointing out how and where it provides a precious guide. In §5, we consider the particular case of the Mo-scheme P1F1 describing a projective line over F1 ...
... and its algebraic structure as a monoid did not play any role. One of the goals of the present paper is to promote this additional structure by pointing out how and where it provides a precious guide. In §5, we consider the particular case of the Mo-scheme P1F1 describing a projective line over F1 ...
REVIEW OF POINT-SET TOPOLOGY I WOMP 2006 The
... Proposition 4.5. All closed subsets of a compact space are compact. The converse holds if the space is also Hausdorff. Proposition 4.6. Let f : X → Y be a continuous bijection, and suppose X is compact and Y is Hausdorff. Then f is a homeomorphism; i.e., f −1 is also continuous. Definition 4.7. In t ...
... Proposition 4.5. All closed subsets of a compact space are compact. The converse holds if the space is also Hausdorff. Proposition 4.6. Let f : X → Y be a continuous bijection, and suppose X is compact and Y is Hausdorff. Then f is a homeomorphism; i.e., f −1 is also continuous. Definition 4.7. In t ...