Categorically proper homomorphisms of topological groups
... on the categorical Tychonoff Theorem [7] that had already been used to affirm product stability of c-compactness for topological groups: see Example 9.5 of [5]. In fact, we not only extend but slightly generalize the known object-level results since, unlike the authors of [12, 19] and of most papers ...
... on the categorical Tychonoff Theorem [7] that had already been used to affirm product stability of c-compactness for topological groups: see Example 9.5 of [5]. In fact, we not only extend but slightly generalize the known object-level results since, unlike the authors of [12, 19] and of most papers ...
subgroups of free topological groups and free
... Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup th ...
... Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup th ...
Universitat Jaume I Departament de Matem` atiques BOUNDED SETS IN TOPOLOGICAL
... for important classes of groups. This translation, which is well-known in the Pontryagin-van Kampen duality theory of locally compact abelian groups, is often very useful and has been extended by many authors to more general classes of topological groups. In this work we follow basically the pattern ...
... for important classes of groups. This translation, which is well-known in the Pontryagin-van Kampen duality theory of locally compact abelian groups, is often very useful and has been extended by many authors to more general classes of topological groups. In this work we follow basically the pattern ...
A study of remainders of topological groups
... provide some new characterizations of a few popular classes of topological groups by certain properties of their remainders. In particular, we improve a theorem from [6] in the following way: a non-locally compact topological group G is separable and metrizable if (and only if) some remainder Y of G ...
... provide some new characterizations of a few popular classes of topological groups by certain properties of their remainders. In particular, we improve a theorem from [6] in the following way: a non-locally compact topological group G is separable and metrizable if (and only if) some remainder Y of G ...
Lecture notes
... Weak FLT. For any n > 4 there are at most finitely many integer solutions to an + bn = cn with hcf(a, b, c) = 1. We prove neither of these results in this course3. Instead we focus on one of the most basic objects endowed with both geometric and algebraic structure, a topological group. The definiti ...
... Weak FLT. For any n > 4 there are at most finitely many integer solutions to an + bn = cn with hcf(a, b, c) = 1. We prove neither of these results in this course3. Instead we focus on one of the most basic objects endowed with both geometric and algebraic structure, a topological group. The definiti ...
Locally ringed spaces and manifolds
... Definition 0.1. Let X be a topological space. A k-differentiable atlas AkX on X is a family {(Ui , hi , Xi , ni )}i∈I of quadruples, where I is a set and for each i ∈ I Xi is an open subset of X, ni ≥ 0 is an integer, Ui is an open subset of Rni and hi : Ui → Xi is a homeomorphism such that X = ∪i X ...
... Definition 0.1. Let X be a topological space. A k-differentiable atlas AkX on X is a family {(Ui , hi , Xi , ni )}i∈I of quadruples, where I is a set and for each i ∈ I Xi is an open subset of X, ni ≥ 0 is an integer, Ui is an open subset of Rni and hi : Ui → Xi is a homeomorphism such that X = ∪i X ...
MAT1360: Complex Manifolds and Hermitian Differential Geometry
... independent of the choice of charts because overlap maps are biholomorphic. A basic consequence of the maximum principle Proposition 1.3 is that every holomorphic function on a connected, compact complex manifold is constant; the absolute value must have a maximum value by compactness, so the functi ...
... independent of the choice of charts because overlap maps are biholomorphic. A basic consequence of the maximum principle Proposition 1.3 is that every holomorphic function on a connected, compact complex manifold is constant; the absolute value must have a maximum value by compactness, so the functi ...
A Note on Free Topological Groupoids
... = a(b)}, making I' into a category in the usual sense. Finally such a topological category is a topological groupoid if it is abstractly a groupoid (that is, a, category with inverses, [7,11) and the inverse map a-a-1 is continuous. Morphisnis of topological graphs, categories and groupoids are defi ...
... = a(b)}, making I' into a category in the usual sense. Finally such a topological category is a topological groupoid if it is abstractly a groupoid (that is, a, category with inverses, [7,11) and the inverse map a-a-1 is continuous. Morphisnis of topological graphs, categories and groupoids are defi ...
PRODUCTIVE PROPERTIES IN TOPOLOGICAL GROUPS
... there exists a homeomorphism ϕ of G onto itself such that ϕ(a) = b and ϕ(b) = a. It is worth noting that the long line is a homogeneous locally compact space which does not have this property! Let us mention the following special features of topological groups (to list a few): a) T0 ⇐⇒ T3.5 (Pontrya ...
... there exists a homeomorphism ϕ of G onto itself such that ϕ(a) = b and ϕ(b) = a. It is worth noting that the long line is a homogeneous locally compact space which does not have this property! Let us mention the following special features of topological groups (to list a few): a) T0 ⇐⇒ T3.5 (Pontrya ...
LECTURE NOTES IN TOPOLOGICAL GROUPS 1
... (d) The integers Z with the cofinite topology. Exercise 2.9. Let A and B are subsets of G and g ∈ G. Prove that: (1) If A is open then gA and AB are open in G. (2) If A and B are compact then AB is also compact. (3) If A and B are connected then AB is also connected. (4) If A and B are closed then A ...
... (d) The integers Z with the cofinite topology. Exercise 2.9. Let A and B are subsets of G and g ∈ G. Prove that: (1) If A is open then gA and AB are open in G. (2) If A and B are compact then AB is also compact. (3) If A and B are connected then AB is also connected. (4) If A and B are closed then A ...
Spectra for commutative algebraists.
... over them, but some constructions make it natural to extend further to considering chain complexes of R-modules; the need to consider robust, homotopy invariant properties leads to the derived category D(R). Once we admit chain complexes, it is natural to consider the corresponding multiplicative ob ...
... over them, but some constructions make it natural to extend further to considering chain complexes of R-modules; the need to consider robust, homotopy invariant properties leads to the derived category D(R). Once we admit chain complexes, it is natural to consider the corresponding multiplicative ob ...
Spectra for commutative algebraists.
... over them, but some constructions make it natural to extend further to considering chain complexes of R-modules; the need to consider robust, homotopy invariant properties leads to the derived category D(R). Once we admit chain complexes, it is natural to consider the corresponding multiplicative ob ...
... over them, but some constructions make it natural to extend further to considering chain complexes of R-modules; the need to consider robust, homotopy invariant properties leads to the derived category D(R). Once we admit chain complexes, it is natural to consider the corresponding multiplicative ob ...
A QUICK INTRODUCTION TO FIBERED CATEGORIES AND
... A QUICK INTRODUCTION TO FIBERED CATEGORIES AND TOPOLOGICAL STACKS ...
... A QUICK INTRODUCTION TO FIBERED CATEGORIES AND TOPOLOGICAL STACKS ...
Topological Algebra
... the spaces themselves are groups in a nice way (so that all the maps coming from group theory are continuous), or groups act on topological spaces and can be thought of as consisting of homeomorphisms. This material has interdisciplinary character. Although it plays important roles in many areas of ...
... the spaces themselves are groups in a nice way (so that all the maps coming from group theory are continuous), or groups act on topological spaces and can be thought of as consisting of homeomorphisms. This material has interdisciplinary character. Although it plays important roles in many areas of ...
The Hilbert–Smith conjecture for three-manifolds
... 2. The action of Zp on H1 (F ) is nontrivial. This means we have a cyclic subgroup Z/p ⊆ MCG(F ) such that H1 (F )Z/p has a submodule on which the intersection form is given by (1.2). The Nielsen classification of cyclic subgroups of the mapping class group shows that this is impossible. This contra ...
... 2. The action of Zp on H1 (F ) is nontrivial. This means we have a cyclic subgroup Z/p ⊆ MCG(F ) such that H1 (F )Z/p has a submodule on which the intersection form is given by (1.2). The Nielsen classification of cyclic subgroups of the mapping class group shows that this is impossible. This contra ...
ON THE OPPOSITE OF THE CATEGORY OF RINGS
... of open sets of X. The Alexandrov topology is always T0 (meaning that for any two points a, b ∈ X, there exists an open set containing one of a, b but not the other). If (X, ≤X ) and (Y, ≤Y ) are posets, then a function f : X → Y is continuous for the Alexandrov topology if and only if it is order-p ...
... of open sets of X. The Alexandrov topology is always T0 (meaning that for any two points a, b ∈ X, there exists an open set containing one of a, b but not the other). If (X, ≤X ) and (Y, ≤Y ) are posets, then a function f : X → Y is continuous for the Alexandrov topology if and only if it is order-p ...
WHAT IS A TOPOLOGICAL STACK? 1. introduction Stacks were
... We begin with the following not-so-illuminating definition. The reader unfamiliar with stack theory jargon may want to skip this definition and the paragraph that comes after it. A quick reading through would not be too harmful though! Definition 2.1. A topological stack X is a stack on the site of ...
... We begin with the following not-so-illuminating definition. The reader unfamiliar with stack theory jargon may want to skip this definition and the paragraph that comes after it. A quick reading through would not be too harmful though! Definition 2.1. A topological stack X is a stack on the site of ...
An Introduction to Topological Groups
... Definition 2.10. Let (X, τ ) be a topological space and S a subset of X. The closure of S, denoted S̄, is defined to be S̄ = ∩ {F : F ⊂ X is closed and contains S }. In symbols the closure can be written as S̄ = {x ∈ X : N ∩ S 6= ∅ for all N ∈ Nx }. It isn’t obvious that these two sets are the same, ...
... Definition 2.10. Let (X, τ ) be a topological space and S a subset of X. The closure of S, denoted S̄, is defined to be S̄ = ∩ {F : F ⊂ X is closed and contains S }. In symbols the closure can be written as S̄ = {x ∈ X : N ∩ S 6= ∅ for all N ∈ Nx }. It isn’t obvious that these two sets are the same, ...
PhD and MPhil Thesis Classes
... and apply it to vector- and particularly, line bundles. The isomorphism class of the latter is the Picard group. We show that under some assumptions on our monoid scheme X, if k is an integral domain (resp. PID), then the induced map Pic(X) → Pic(Xk ) from X to its realisation is a monomorphism (res ...
... and apply it to vector- and particularly, line bundles. The isomorphism class of the latter is the Picard group. We show that under some assumptions on our monoid scheme X, if k is an integral domain (resp. PID), then the induced map Pic(X) → Pic(Xk ) from X to its realisation is a monomorphism (res ...
HAUSDORFF TOPOLOGIES ON GROUPS
... large classes of topological spaces could be a fruitful approach to Markov's question. ...
... large classes of topological spaces could be a fruitful approach to Markov's question. ...
The Brauer group of a locally compact groupoid - MUSE
... (What follows is amplified in [5], which is based in turn on [46].) The quickest way to say that a C -algebra A, say, is continuous trace is to say that it has Hausdorff spectrum T and that there is a bundle A of C -algebras over T , each fibre A(t) of which is isomorphic to the algebra of compact ...
... (What follows is amplified in [5], which is based in turn on [46].) The quickest way to say that a C -algebra A, say, is continuous trace is to say that it has Hausdorff spectrum T and that there is a bundle A of C -algebras over T , each fibre A(t) of which is isomorphic to the algebra of compact ...
The bordism version of the h
... In some cases invariants of solutions of R are fairly complicated and a direct computation of these invariants seems improbable. On the other hand, it turns out that invariants of MR can be computed by means of stable homotopy theory. Indeed, we will see that the moduli space of solutions MR of an o ...
... In some cases invariants of solutions of R are fairly complicated and a direct computation of these invariants seems improbable. On the other hand, it turns out that invariants of MR can be computed by means of stable homotopy theory. Indeed, we will see that the moduli space of solutions MR of an o ...
Topology
... so-called “pure mathematics” with remarkable successes and results in this subject. In the course of the twentieth century it has also 1. provided notions and concepts that are of core importance for all of mathematics, such as the notion of “compactness”, 2. contributed a great variety of important ...
... so-called “pure mathematics” with remarkable successes and results in this subject. In the course of the twentieth century it has also 1. provided notions and concepts that are of core importance for all of mathematics, such as the notion of “compactness”, 2. contributed a great variety of important ...
Topological realizations of absolute Galois groups
... For algebraically closed fields F , the space XFM would have to be constructed in such a way as to freely adjoin the Steinberg relation on its cohomology groups; the general case should reduce to this case by descent. Descent along the cyclotomic extension. So far, all of our results were assuming t ...
... For algebraically closed fields F , the space XFM would have to be constructed in such a way as to freely adjoin the Steinberg relation on its cohomology groups; the general case should reduce to this case by descent. Descent along the cyclotomic extension. So far, all of our results were assuming t ...
Free Topological Groups - Universidad Complutense de Madrid
... slightly different. Again, one can show that if x1 , . . . , xn are pairwise distinct elements of X and k1 , . . . , kn are arbitrary integers, then the equality k1 x1 + k2 x2 + · · · + kn xn = 0A(X ) implies that k1 = k2 = · · · = kn = 0. Therefore, the group A(X ) is torsion-free and, again, X is ...
... slightly different. Again, one can show that if x1 , . . . , xn are pairwise distinct elements of X and k1 , . . . , kn are arbitrary integers, then the equality k1 x1 + k2 x2 + · · · + kn xn = 0A(X ) implies that k1 = k2 = · · · = kn = 0. Therefore, the group A(X ) is torsion-free and, again, X is ...