Metric geometry of locally compact groups
... (fp) finitely presented groups. This will serve as a guideline below, in the more general setting of locally compact groups. Every group Γ has left-invariant metrics which induce the discrete topology, for example that defined by d(γ, γ ′ ) = 1 whenever γ, γ ′ are distinct. The three other classes can ...
... (fp) finitely presented groups. This will serve as a guideline below, in the more general setting of locally compact groups. Every group Γ has left-invariant metrics which induce the discrete topology, for example that defined by d(γ, γ ′ ) = 1 whenever γ, γ ′ are distinct. The three other classes can ...
THE GEOMETRIES OF 3
... situation for 3-manifolds is similar to that for surfaces. All the evidence points to this being the case, but it is a long way from being proved. For example, the Poincare Conjecture is a very special case of Thurston's conjecture. However, Thurston's most important result in this area asserts that ...
... situation for 3-manifolds is similar to that for surfaces. All the evidence points to this being the case, but it is a long way from being proved. For example, the Poincare Conjecture is a very special case of Thurston's conjecture. However, Thurston's most important result in this area asserts that ...
Exotic spheres and curvature - American Mathematical Society
... five, any smooth manifold with the homotopy type of a sphere must be homeomorphic to a sphere. This is the Generalised Poincaré Conjecture, proved by Smale in [Sm1]. Thus in these dimensions the set of diffeomorphism classes of homotopy spheres is precisely the union of the diffeomorphism class of the ...
... five, any smooth manifold with the homotopy type of a sphere must be homeomorphic to a sphere. This is the Generalised Poincaré Conjecture, proved by Smale in [Sm1]. Thus in these dimensions the set of diffeomorphism classes of homotopy spheres is precisely the union of the diffeomorphism class of the ...
SYMMETRIC SPECTRA Contents Introduction 2 1
... Symmetric spectra have already proved useful. In [GH97], symmetric spectra are used to extend the definition of topological cyclic homology from rings to schemes. Similarly, in [Shi97], Bökstedt’s approach to topological Hochschild homology [Bök85] is extended to symmetric ring spectra, without co ...
... Symmetric spectra have already proved useful. In [GH97], symmetric spectra are used to extend the definition of topological cyclic homology from rings to schemes. Similarly, in [Shi97], Bökstedt’s approach to topological Hochschild homology [Bök85] is extended to symmetric ring spectra, without co ...
Limit Spaces with Approximations
... A constructive theory of limit spaces is not elaborated so far. ...
... A constructive theory of limit spaces is not elaborated so far. ...
pdf
... Proof: Let X be an Rcl -locally connected space and let U be an rcl -open set in X. Then in view of Theorem 4.4 any component of U is open in U and hence in X. Thus every quasicomponent of U , being a union of components, is open. Since every open quasicomponent is a component, the result is immedia ...
... Proof: Let X be an Rcl -locally connected space and let U be an rcl -open set in X. Then in view of Theorem 4.4 any component of U is open in U and hence in X. Thus every quasicomponent of U , being a union of components, is open. Since every open quasicomponent is a component, the result is immedia ...
Slides for Nov. 12, 2014, lecture
... needs three spatial lengths (coordinates, say, in a fixed coordinate system) to fix one of its points; manifold a collection of points or elements (objects, entities) that has the structure of a multiply extended magnitude, i.e., has a fixed number of associated modes of specification of magnitude-c ...
... needs three spatial lengths (coordinates, say, in a fixed coordinate system) to fix one of its points; manifold a collection of points or elements (objects, entities) that has the structure of a multiply extended magnitude, i.e., has a fixed number of associated modes of specification of magnitude-c ...
ISOSPECTRAL AND ISOSCATTERING MANIFOLDS: A SURVEY
... isospectral. For compact manifolds with boundary, one may refer to Dirichlet isospectral or Neumann isospectral manifolds. For the constructions we will consider, the manifolds will be both Dirichlet and Neumann isospectral, so we will sometimes simply say “isospectral”. (There is, however, one exam ...
... isospectral. For compact manifolds with boundary, one may refer to Dirichlet isospectral or Neumann isospectral manifolds. For the constructions we will consider, the manifolds will be both Dirichlet and Neumann isospectral, so we will sometimes simply say “isospectral”. (There is, however, one exam ...
On characterizations of Euclidean spaces
... The two measures µa and µl are identical for all angles of some Minkowski plane M2 iff its unit ball is equiframed, i.e., if each point of the unit circle belongs to the boundary of some circumscribed parallelogram of minimal area. ...
... The two measures µa and µl are identical for all angles of some Minkowski plane M2 iff its unit ball is equiframed, i.e., if each point of the unit circle belongs to the boundary of some circumscribed parallelogram of minimal area. ...
Thom Spectra that Are Symmetric Spectra
... can be realized as diagram ring spectra in this way. It is often convenient to replace the D-Thom spectrum associated to a map of D-spaces f : X → BF by a symmetric spectrum, and our preferred way of doing this is to first transform f to a map of I-spaces and then evaluate the symmetric Thom spectru ...
... can be realized as diagram ring spectra in this way. It is often convenient to replace the D-Thom spectrum associated to a map of D-spaces f : X → BF by a symmetric spectrum, and our preferred way of doing this is to first transform f to a map of I-spaces and then evaluate the symmetric Thom spectru ...
COUNTABLY S-CLOSED SPACES ∗
... Proof. 1) ⇔ 2) ⇔ 3) : This is obvious since the closure of every semi-open set is regular closed. Furthermore, every regular closed set is regular semi-open and thus semi-open. 1) ⇒ 4) : This is trivial. 4) ⇒ 1) : Suppose that (X, τ ) is not countably S-closed. Then there exists a countable S regula ...
... Proof. 1) ⇔ 2) ⇔ 3) : This is obvious since the closure of every semi-open set is regular closed. Furthermore, every regular closed set is regular semi-open and thus semi-open. 1) ⇒ 4) : This is trivial. 4) ⇒ 1) : Suppose that (X, τ ) is not countably S-closed. Then there exists a countable S regula ...
(core) compactly generated spaces
... the family {pi : Ci → X}i∈I slightly, if necessary, by including all constant maps from some non-empty generating space. Since I is a set (rather than just a proper class) we can take the disjoint sum of the spaces Ci . Call it S and, for each i ∈ I, let ji : Ci → S be the inclusion. By the universa ...
... the family {pi : Ci → X}i∈I slightly, if necessary, by including all constant maps from some non-empty generating space. Since I is a set (rather than just a proper class) we can take the disjoint sum of the spaces Ci . Call it S and, for each i ∈ I, let ji : Ci → S be the inclusion. By the universa ...
1 Comparing cartesian closed categories of (core) compactly
... the family {pi : Ci → X}i∈I slightly, if necessary, by including all constant maps from some non-empty generating space. Since I is a set (rather than just a proper class) we can take the disjoint sum of the spaces Ci . Call it S and, for each i ∈ I, let ji : Ci → S be the inclusion. By the universa ...
... the family {pi : Ci → X}i∈I slightly, if necessary, by including all constant maps from some non-empty generating space. Since I is a set (rather than just a proper class) we can take the disjoint sum of the spaces Ci . Call it S and, for each i ∈ I, let ji : Ci → S be the inclusion. By the universa ...
Powerpoint - Microsoft Research
... J:M→K is a family tr A,B : K ( A U , B U ) K ( A, B) satisfying the usual trace axioms except ...
... J:M→K is a family tr A,B : K ( A U , B U ) K ( A, B) satisfying the usual trace axioms except ...
On uniformly locally compact quasi
... It is well known that a Tychonoff space is paracompact if and only if its fine uniformity is cofinally complete ([5], [7], [8]). In our context a topological space X will be called locally compact if each point of X has a neighborhood whose closure is compact. A (quasi-)uniformity U on a set X is s ...
... It is well known that a Tychonoff space is paracompact if and only if its fine uniformity is cofinally complete ([5], [7], [8]). In our context a topological space X will be called locally compact if each point of X has a neighborhood whose closure is compact. A (quasi-)uniformity U on a set X is s ...
![PFA(S)[S] and Locally Compact Normal Spaces](http://s1.studyres.com/store/data/001741952_1-aeadbdb88f4d36d94e029d346a059e2c-300x300.png)
PFA(S)[S] and Locally Compact Normal Spaces
... first evident in [3] is that “normal plus L ≤ ℵ1 ” can often substitute for “hereditarily normal” in this area of investigation. In fact, an even further weakening is possible: Definition. Let U be an open cover of a space X and let x ∈ X. Ord (x, U) = |{U ∈ U : xS∈ U}|. X is submeta-ℵ1 -Lindelöf i ...
... first evident in [3] is that “normal plus L ≤ ℵ1 ” can often substitute for “hereditarily normal” in this area of investigation. In fact, an even further weakening is possible: Definition. Let U be an open cover of a space X and let x ∈ X. Ord (x, U) = |{U ∈ U : xS∈ U}|. X is submeta-ℵ1 -Lindelöf i ...
Factorization homology of stratified spaces
... map, one can then construct a link homology theory, via factorization homology with coefficients in this triple. This promises to provide a new source of such knot homology theories, similar to Khovanov homology. Khovanov homology itself does not fit into this structure, for a very simple reason: an ...
... map, one can then construct a link homology theory, via factorization homology with coefficients in this triple. This promises to provide a new source of such knot homology theories, similar to Khovanov homology. Khovanov homology itself does not fit into this structure, for a very simple reason: an ...
Topological properties
... (1) X = {0, 1} with the discrete topology is not connected. Indeed, U = {0}, V = {1} are disjoint non-empty opens (in X) whose union is X. (2) Similarly, X = [0, 1) ∪ [2, 3] is not connected (take U = [0, 1), V = [2, 3]). More generally, if X ⊂ R is connected, then X must be an interval. Indeed, if ...
... (1) X = {0, 1} with the discrete topology is not connected. Indeed, U = {0}, V = {1} are disjoint non-empty opens (in X) whose union is X. (2) Similarly, X = [0, 1) ∪ [2, 3] is not connected (take U = [0, 1), V = [2, 3]). More generally, if X ⊂ R is connected, then X must be an interval. Indeed, if ...
LOCALLY COMPACT PERFECTLY NORMAL SPACES MAY ALL
... sequential, then S forces that every locally countable subset of K of size ℵ1 is σdiscrete. The proof will appear in [Fi1]. A weaker version is proved in [To2]. Using this he got: Theorem 6. (PFA(S)) If K̇ is an S-name for a compact countably tight space, then Ẏ is S-forced to be a hereditarily sep ...
... sequential, then S forces that every locally countable subset of K of size ℵ1 is σdiscrete. The proof will appear in [Fi1]. A weaker version is proved in [To2]. Using this he got: Theorem 6. (PFA(S)) If K̇ is an S-name for a compact countably tight space, then Ẏ is S-forced to be a hereditarily sep ...
ON COUNTABLE CONNECTED HAUSDORFFSPACES IN WHICH
... an open connected neighbourhood U, of such that jr(U,) C_ UI(,). If/(U,) is not empty, it follows that there consider the set 24 {a X:/(a) =/()}. Since the set A \ and a connected open neishbourhood U, of a, such that /(U,) C_ exist a point a ] \ and 1"(o) {/(z)}. Therefore the component 6’j,(,) of/ ...
... an open connected neighbourhood U, of such that jr(U,) C_ UI(,). If/(U,) is not empty, it follows that there consider the set 24 {a X:/(a) =/()}. Since the set A \ and a connected open neishbourhood U, of a, such that /(U,) C_ exist a point a ] \ and 1"(o) {/(z)}. Therefore the component 6’j,(,) of/ ...
Homework05 Solutions
... perpendicular to BC and AD . 4. Show that the diagonals are perpendicular if and only if all four sides are congruent, and in that case, , ABCD has an inscribed circle with center M. ( ⇐ ) Assume that all four sides are congruent. By SSS ΔDMC ≅ ΔBMC which implies that ∠DMC ≅ ∠BMC and since they are ...
... perpendicular to BC and AD . 4. Show that the diagonals are perpendicular if and only if all four sides are congruent, and in that case, , ABCD has an inscribed circle with center M. ( ⇐ ) Assume that all four sides are congruent. By SSS ΔDMC ≅ ΔBMC which implies that ∠DMC ≅ ∠BMC and since they are ...
Connectedness - GMU Math 631 Spring 2011
... (And of course any more than one point discrete space is locally connected but not connected.) Proposition 29. The following conditions are equivalent: (1) X is locally connected; (2) Components of open subspaces of X are open. Proof. (1) ⇒ (2) Let U be an open set in X. Then X is locally connected. ...
... (And of course any more than one point discrete space is locally connected but not connected.) Proposition 29. The following conditions are equivalent: (1) X is locally connected; (2) Components of open subspaces of X are open. Proof. (1) ⇒ (2) Let U be an open set in X. Then X is locally connected. ...
On the equivalence of Alexandrov curvature and
... It is well known that the curvature bounded above (resp. below) in the sense of Alexandrov is stronger than the curvature bounded above (resp. below) in the sense of Busemann (see, e.g., [7, p. 107] or [9, p. 57]). The classical example that shows that the converse statement does not hold is the fin ...
... It is well known that the curvature bounded above (resp. below) in the sense of Alexandrov is stronger than the curvature bounded above (resp. below) in the sense of Busemann (see, e.g., [7, p. 107] or [9, p. 57]). The classical example that shows that the converse statement does not hold is the fin ...
Symplectic Topology
... can’t be reduced to volume-preserving geometry, and will come at the end of the course. ...
... can’t be reduced to volume-preserving geometry, and will come at the end of the course. ...