Some new algebras of functions on topological groups arising from
... G. The structure of SU C(G) — in contrast to RU C(G) which is always huge for non-precompact groups — is “computable” for several large groups like: H+ [0, 1], Iso(U1 ) (the isometry group of the Urysohn space of diameter one U1 ), U (H) (the unitary group on an infinite dimensional Hilbert space), ...
... G. The structure of SU C(G) — in contrast to RU C(G) which is always huge for non-precompact groups — is “computable” for several large groups like: H+ [0, 1], Iso(U1 ) (the isometry group of the Urysohn space of diameter one U1 ), U (H) (the unitary group on an infinite dimensional Hilbert space), ...
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY
... (ii) If one is willing to assume Hilbert-Smith conjecture (or to apply conditions (Hr) or (QC), when relevant) then one could deduce from Corollary 3 that – The only non-discrete locally compact groups with a faithful boundary action on the sphere X = S 2 are Isom + (H3 ) and its double cover Isom ( ...
... (ii) If one is willing to assume Hilbert-Smith conjecture (or to apply conditions (Hr) or (QC), when relevant) then one could deduce from Corollary 3 that – The only non-discrete locally compact groups with a faithful boundary action on the sphere X = S 2 are Isom + (H3 ) and its double cover Isom ( ...
General Topology - Faculty of Physics University of Warsaw
... (3) Let x ∈ X. Then x ∈ B iff there exists a neighborhood U of x such that U ⊂ A; (4) For any x ∈ X choose a basis of neighborhoods of x, denoted Vx . Then x ∈ B iff there exists a V ∈ Vx such that V ⊂ A. (5) Let x ∈ X. Then x ∈ B iff for any net (xi )i∈I in X convergent to x there exists i0 ∈ I suc ...
... (3) Let x ∈ X. Then x ∈ B iff there exists a neighborhood U of x such that U ⊂ A; (4) For any x ∈ X choose a basis of neighborhoods of x, denoted Vx . Then x ∈ B iff there exists a V ∈ Vx such that V ⊂ A. (5) Let x ∈ X. Then x ∈ B iff for any net (xi )i∈I in X convergent to x there exists i0 ∈ I suc ...
26 - HKU
... at a general point. The hard part of Tsai’s proof then proceeded by exploiting the complex geometry of bounded symmetric domains with respect to the Bergman metric. In another completely different direction, in the context of Algebraic Geometry Hwang-Mok studied the geometry of uniruled projective m ...
... at a general point. The hard part of Tsai’s proof then proceeded by exploiting the complex geometry of bounded symmetric domains with respect to the Bergman metric. In another completely different direction, in the context of Algebraic Geometry Hwang-Mok studied the geometry of uniruled projective m ...
HYPERBOLIZATION OF POLYHEDRA
... The fact that polyhedral homology manifolds which are not PL manifolds have something to do with exotic universal covers was first recognized in [11], through the use of reflection groups. In the recent Ph.D. thesis of G. Moussong [24], it is shown that some of the results of [11] on reflection grou ...
... The fact that polyhedral homology manifolds which are not PL manifolds have something to do with exotic universal covers was first recognized in [11], through the use of reflection groups. In the recent Ph.D. thesis of G. Moussong [24], it is shown that some of the results of [11] on reflection grou ...
NOTES WEEK 13 DAY 1 Unassigned exercise: Let X and Y be
... Proof. We wish to show: For any a P pX ˆ Y qN , there exists a subsequence c‚ of a‚ such that c‚ is convergent in X ˆ Y . Let a P pX ˆ Y qN be given. We wish to show that there exists a subsequence c‚ of a‚ asuch that c‚ is convergent in X ˆ Y . Define p : X ˆ Y Ñ X and q : X ˆ Y Ñ Y by ppx, yq “ x ...
... Proof. We wish to show: For any a P pX ˆ Y qN , there exists a subsequence c‚ of a‚ such that c‚ is convergent in X ˆ Y . Let a P pX ˆ Y qN be given. We wish to show that there exists a subsequence c‚ of a‚ asuch that c‚ is convergent in X ˆ Y . Define p : X ˆ Y Ñ X and q : X ˆ Y Ñ Y by ppx, yq “ x ...
Elsevier Editorial System(tm) for Topology and its Applications
... the need for G-invariant compatible metrics which are proper in the sense that the balls are relatively compact. This is for instance the case when one wants to attack the Baum-Connes conjecture by considering proper affine isometric actions of G on various Banach spaces, following an idea of Gromov ...
... the need for G-invariant compatible metrics which are proper in the sense that the balls are relatively compact. This is for instance the case when one wants to attack the Baum-Connes conjecture by considering proper affine isometric actions of G on various Banach spaces, following an idea of Gromov ...
LOCAL MONODROMY OF BRANCHED COVERS AND DIMENSION
... f : X → M a proper branched cover and y ∈ f Bf . If the local dimension of f Bf at y ∈ f Bf is less than n − 2, then Gf (y) is a finite perfect group. We begin with the abelian case of the theorem. Theorem 6. Let X be a Fox-completion of an n-manifold, M an n-manifold, and f : X → M a proper branche ...
... f : X → M a proper branched cover and y ∈ f Bf . If the local dimension of f Bf at y ∈ f Bf is less than n − 2, then Gf (y) is a finite perfect group. We begin with the abelian case of the theorem. Theorem 6. Let X be a Fox-completion of an n-manifold, M an n-manifold, and f : X → M a proper branche ...
topologies on spaces of subsets
... better than the metric topology (see 2.4.1 and 4.10). §2 deals with fundamental properties of the topologies on 2X. What is probably the most interesting result of this section (Theorem 2.5) can be ...
... better than the metric topology (see 2.4.1 and 4.10). §2 deals with fundamental properties of the topologies on 2X. What is probably the most interesting result of this section (Theorem 2.5) can be ...
Transitive actions of locally compact groups on locally contractible
... Now ˛ and ˇ are injective and hence homeomorphisms onto their respective images. Moreover, the image of is contained in the image of ˛ and the image of ı is contained in the image of ˇ. Thus we can fill in the dotted map and obtain an upper bound .L; / of T . By Zorn’s lemma, P has maximal element ...
... Now ˛ and ˇ are injective and hence homeomorphisms onto their respective images. Moreover, the image of is contained in the image of ˛ and the image of ı is contained in the image of ˇ. Thus we can fill in the dotted map and obtain an upper bound .L; / of T . By Zorn’s lemma, P has maximal element ...
The Hilbert–Smith conjecture for three-manifolds
... (see Definition 2.4). For a quasicylinder M, we let S(M) denote the set of isotopy classes of incompressible surfaces in M which generate H2 (M). The main result of this section (suggested by Ian Agol [1]) is that S(M) is a lattice (see Lemma 2.19) under its natural partial order. Related ideas may ...
... (see Definition 2.4). For a quasicylinder M, we let S(M) denote the set of isotopy classes of incompressible surfaces in M which generate H2 (M). The main result of this section (suggested by Ian Agol [1]) is that S(M) is a lattice (see Lemma 2.19) under its natural partial order. Related ideas may ...
Topologies on Spaces of Subsets Ernest Michael Transactions of
... the Hausdorff metric. The finite topology 2T, on the other hand, agrees with this metric only if X is compact (see Proposition 3.5); this is no calamity, however, for in some important respects the finite topology behaves much better than the metric topology (see 2.4.1 and 4.10). $2 deals with funda ...
... the Hausdorff metric. The finite topology 2T, on the other hand, agrees with this metric only if X is compact (see Proposition 3.5); this is no calamity, however, for in some important respects the finite topology behaves much better than the metric topology (see 2.4.1 and 4.10). $2 deals with funda ...
Introduction to Topology
... if every covering of Y by sets open in X contains a finite subcollection covering Y . Proof (continued). Conversely, suppose every covering of Y by sets open in X contain a finite subcollection covering Y . Let A0 = {A0α } be an arbitrary covering of Y by sets open in Y . For each α, choose a set Aα ...
... if every covering of Y by sets open in X contains a finite subcollection covering Y . Proof (continued). Conversely, suppose every covering of Y by sets open in X contain a finite subcollection covering Y . Let A0 = {A0α } be an arbitrary covering of Y by sets open in Y . For each α, choose a set Aα ...
Linear operators between partially ordered Banach spaces and
... these results, together with the definitions is given in Chapter I. In C hapter II we look at the general case. we find conditions for the wedge to be normal, and certain cases in which it is generating. w e also investigate when the wedge is one of some special forms that are of interest, and final ...
... these results, together with the definitions is given in Chapter I. In C hapter II we look at the general case. we find conditions for the wedge to be normal, and certain cases in which it is generating. w e also investigate when the wedge is one of some special forms that are of interest, and final ...
Extensions of functions which preserve the continuity on the original
... Some instances of this question were already considered by Arhangel’skii in [1, pp. 91– 92], where he introduced the notion of a subset A being weakly C-embedded into a topological space X: this means that every f : A → R can be extended to a f˜ : X → R which is continuous at every point of A. Arhan ...
... Some instances of this question were already considered by Arhangel’skii in [1, pp. 91– 92], where he introduced the notion of a subset A being weakly C-embedded into a topological space X: this means that every f : A → R can be extended to a f˜ : X → R which is continuous at every point of A. Arhan ...
Notes on Topological Dimension Theory
... (a) (Reflexive Property) For all x ∈ A we have x ≺ x. (b) (Transitive Property) If x, y, z ∈ A are such that x ≺ y and y ≺ z, then x ≺ z. (c) (Lower Bound Property) For all x, y ∈ A there is some w ∈ A such that w ≺ x and w ≺ y. These are similar to the defining conditions for a partially ordered se ...
... (a) (Reflexive Property) For all x ∈ A we have x ≺ x. (b) (Transitive Property) If x, y, z ∈ A are such that x ≺ y and y ≺ z, then x ≺ z. (c) (Lower Bound Property) For all x, y ∈ A there is some w ∈ A such that w ≺ x and w ≺ y. These are similar to the defining conditions for a partially ordered se ...
Topology
... Topology is an important, classical part of mathematics. It deals with interesting objects (the Klein bottle, Bing’s house, manifolds, lens spaces, knots, . . . ). To study it in detail is a considerable enterprise (a huge subject with many subtle sub-disciplines and methods); here, however, we are ...
... Topology is an important, classical part of mathematics. It deals with interesting objects (the Klein bottle, Bing’s house, manifolds, lens spaces, knots, . . . ). To study it in detail is a considerable enterprise (a huge subject with many subtle sub-disciplines and methods); here, however, we are ...
PRODUCTIVE PROPERTIES IN TOPOLOGICAL GROUPS
... type topological groups. We recall that the spaces X and Y in Novák’s example were defined as subspaces of the compact space βN, and all infinite closed subsets of the latter space have cardinality 2c . In particular, βN does not contain non-trivial convergent sequences. Compact topological groups ...
... type topological groups. We recall that the spaces X and Y in Novák’s example were defined as subspaces of the compact space βN, and all infinite closed subsets of the latter space have cardinality 2c . In particular, βN does not contain non-trivial convergent sequences. Compact topological groups ...
Stratified Morse Theory
... independent of V , δ , and ε. In fact, by a miracle of complex geometry, the topological type of the normal Morse data is also independent of f . Theorems A’ and B’ are essential to what follows, but we do not prove them. And for good reason: although they emerge intuitively from examples (see for i ...
... independent of V , δ , and ε. In fact, by a miracle of complex geometry, the topological type of the normal Morse data is also independent of f . Theorems A’ and B’ are essential to what follows, but we do not prove them. And for good reason: although they emerge intuitively from examples (see for i ...
Ultrafilters and Independent Systems - KTIML
... One branch of inquiry asks what can be proven in ZFC. Following this line of research, in 1967 Z. Frolı́k was able to show in ZFC, using an ingenious combinatorial argument, that ω ∗ is not homogeneous ([Fro67a], [Fro67b]). In fact, there are 2c pairwise “topologically different” points (i.e. there ...
... One branch of inquiry asks what can be proven in ZFC. Following this line of research, in 1967 Z. Frolı́k was able to show in ZFC, using an ingenious combinatorial argument, that ω ∗ is not homogeneous ([Fro67a], [Fro67b]). In fact, there are 2c pairwise “topologically different” points (i.e. there ...
introduction to algebraic topology and algebraic geometry
... topics, singular homology (Chapter 2) and sheaf theory, including their cohomology (Chapter 3). Chapter 1 assembles some basics fact in homological algebra and develops the first rudiments of de Rham cohomology, with the aim of providing an example to the various abstract constructions. Chapter 5 is ...
... topics, singular homology (Chapter 2) and sheaf theory, including their cohomology (Chapter 3). Chapter 1 assembles some basics fact in homological algebra and develops the first rudiments of de Rham cohomology, with the aim of providing an example to the various abstract constructions. Chapter 5 is ...
Vector Bundles and K
... Definition A family E of vector spaces over X is said to be locally trivial if every x ∈ X has a neighbourhood U ⊂ X such that E|U is trivial : any isomorphism between E|U and a product family over U will be called a trivialisation of E over U . In this case the family will be called a vector bundle ...
... Definition A family E of vector spaces over X is said to be locally trivial if every x ∈ X has a neighbourhood U ⊂ X such that E|U is trivial : any isomorphism between E|U and a product family over U will be called a trivialisation of E over U . In this case the family will be called a vector bundle ...
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE, FMedSci FAA, HonFREng (born 22 April 1929) is a British mathematician specialising in geometry.Atiyah grew up in Sudan and Egypt and spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study. He has been president of the Royal Society (1990–1995), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and president of the Royal Society of Edinburgh (2005–2008). Since 1997, he has been an honorary professor at the University of Edinburgh.Atiyah's mathematical collaborators include Raoul Bott, Friedrich Hirzebruch and Isadore Singer, and his students include Graeme Segal, Nigel Hitchin and Simon Donaldson. Together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is widely used in counting the number of independent solutions to differential equations. Some of his more recent work was inspired by theoretical physics, in particular instantons and monopoles, which are responsible for some subtle corrections in quantum field theory. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, and the Abel Prize in 2004.