HYPERBOLIZATION OF POLYHEDRA
... Moussong proves that the natural contractible simplicial complex on which a Coxeter group W acts properly with compact quotient can be given a piecewise flat structure with nonpositive curvature. Sometimes this simplicial complex is a polyhedral homology manifold and one can use the results of §3 to ...
... Moussong proves that the natural contractible simplicial complex on which a Coxeter group W acts properly with compact quotient can be given a piecewise flat structure with nonpositive curvature. Sometimes this simplicial complex is a polyhedral homology manifold and one can use the results of §3 to ...
Part I : PL Topology
... For the class of C ∞ triangulations of a differentiable manifold, Whitehead proved an isotopy Haupvermutung in 1940, but in 1960 Milnor found a polyhedron of dimension six for which the generalised Hauptvermutung is false. This polyhedron is not a PL manifold and therefore the conjecture remained op ...
... For the class of C ∞ triangulations of a differentiable manifold, Whitehead proved an isotopy Haupvermutung in 1940, but in 1960 Milnor found a polyhedron of dimension six for which the generalised Hauptvermutung is false. This polyhedron is not a PL manifold and therefore the conjecture remained op ...
Smooth manifolds - University of Arizona Math
... Now, we have the notion of smooth coordinate chart (which must be in the atlas), and all of the corresponding terminology as in the topological manifold case. In the smooth setting, coordinate charts are often described in the following way. Suppose ~ = (U ) Rn : (U; ) is a smooth coordinate chart. ...
... Now, we have the notion of smooth coordinate chart (which must be in the atlas), and all of the corresponding terminology as in the topological manifold case. In the smooth setting, coordinate charts are often described in the following way. Suppose ~ = (U ) Rn : (U; ) is a smooth coordinate chart. ...
ALGEBRAIC TOPOLOGY NOTES, PART II: FUNDAMENTAL GROUP
... Exercise 1. Let X be connected and locally path-connected. Show that X is pathconnected. Exercise 2. Let W = {(0, y) | |y| ≤ 1} ∪ {(x, sin(1/x) | 0 < x ≤ π2 } be the topologist’s sine curve in R2 . Show that W is connected, but not path-connected. If we adjoin an arc from (0, 1) to ( π2 , 1) the res ...
... Exercise 1. Let X be connected and locally path-connected. Show that X is pathconnected. Exercise 2. Let W = {(0, y) | |y| ≤ 1} ∪ {(x, sin(1/x) | 0 < x ≤ π2 } be the topologist’s sine curve in R2 . Show that W is connected, but not path-connected. If we adjoin an arc from (0, 1) to ( π2 , 1) the res ...
Homotopy Theory of Finite Topological Spaces
... the homotopy groups of a topological space with only finitely many points? The naive answer would be to assume that these groups must be trivial. This is based, however, on a mistaken intuition, namely that a finite space is endowed with the discrete topology (as it would be, for example, if it were ...
... the homotopy groups of a topological space with only finitely many points? The naive answer would be to assume that these groups must be trivial. This is based, however, on a mistaken intuition, namely that a finite space is endowed with the discrete topology (as it would be, for example, if it were ...
The fundamental groupoid as a topological
... (b) For each x in X, the subspace Stn/Nx ofnjNis the regular covering space of X based at x and determined by the subgroup N{x) ofn(X, x). (c) The fundamental group of n/N at lx is isomorphic to the subgroup of n(X, x) x n(X, x) of pairs (a, b) such that aN{x) = bN{x}. For the proof of Theorem 1, we ...
... (b) For each x in X, the subspace Stn/Nx ofnjNis the regular covering space of X based at x and determined by the subgroup N{x) ofn(X, x). (c) The fundamental group of n/N at lx is isomorphic to the subgroup of n(X, x) x n(X, x) of pairs (a, b) such that aN{x) = bN{x}. For the proof of Theorem 1, we ...
topological group
... While every group can be made into a topological group, the same cannot be said of every topological space. In this section we mention some of the properties that the underlying topological space must have. Every topological group is bihomogeneous and completely regular. Note that our earlier claim ...
... While every group can be made into a topological group, the same cannot be said of every topological space. In this section we mention some of the properties that the underlying topological space must have. Every topological group is bihomogeneous and completely regular. Note that our earlier claim ...
An Introduction to Simplicial Sets
... In this section, we define simplicial sets without providing motivation, and we describe the combinatorial data necessary for specifying a simplicial set. We then try to build intuition by bringing in the geometric notion of simplices from algebraic topology. We first define the category ∆, a visual ...
... In this section, we define simplicial sets without providing motivation, and we describe the combinatorial data necessary for specifying a simplicial set. We then try to build intuition by bringing in the geometric notion of simplices from algebraic topology. We first define the category ∆, a visual ...
Chapter VII. Covering Spaces and Calculation of Fundamental Groups
... x ∈ F(a1 , . . . , aq ) is the length of x. Certainly, this number is not well defined unless the generators are fixed. 35.5. Show that an automorphism of Fq can map x ∈ Fq to an element with different length. For what value of q does such an example not exist? Is it possible to change the length in ...
... x ∈ F(a1 , . . . , aq ) is the length of x. Certainly, this number is not well defined unless the generators are fixed. 35.5. Show that an automorphism of Fq can map x ∈ Fq to an element with different length. For what value of q does such an example not exist? Is it possible to change the length in ...
Introduction to symmetric spectra I
... category of spectra. All other categories of spectra defined in a similar manner share this disadvantage. This has caused algebraic topologists a lot of frustration: for example, the absence of a monoidal structure means that one does not have a good notion of a ring spectrum. (This problem was solv ...
... category of spectra. All other categories of spectra defined in a similar manner share this disadvantage. This has caused algebraic topologists a lot of frustration: for example, the absence of a monoidal structure means that one does not have a good notion of a ring spectrum. (This problem was solv ...
NOTES ON NON-ARCHIMEDEAN TOPOLOGICAL GROUPS
... is quite large. Besides the results of this section see Theorem 5.1 below. The prodiscrete (in particular, the profinite) groups are in N A. All N A groups are totally disconnected and for every locally compact totally disconnected group G both G and Aut(G) are N A (see Theorems 7.7 and 26.8 in [16] ...
... is quite large. Besides the results of this section see Theorem 5.1 below. The prodiscrete (in particular, the profinite) groups are in N A. All N A groups are totally disconnected and for every locally compact totally disconnected group G both G and Aut(G) are N A (see Theorems 7.7 and 26.8 in [16] ...
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY
... group containing a closed amenable subgroup P so that G/P is compact, then the universal G-boundary B(G) is a Gequivariant image of G/P by a unique continuous surjective map π : G/P → B(G), and every G-boundary X is obtained as a G-equivariant quotient G/Q where Q ⊇ P is a closed subgroup. In partic ...
... group containing a closed amenable subgroup P so that G/P is compact, then the universal G-boundary B(G) is a Gequivariant image of G/P by a unique continuous surjective map π : G/P → B(G), and every G-boundary X is obtained as a G-equivariant quotient G/Q where Q ⊇ P is a closed subgroup. In partic ...
Here
... Physically that describes a particle that moves |k| times around the circle, going counterclockwise for k > 0 and clockwise for k < 0. We will prove that fk and f` are homotopic if and only if k = `. Moreover, we will show that any map f : S 1 → S 1 is homotopic to fk for some k ∈ Z. In other words, ...
... Physically that describes a particle that moves |k| times around the circle, going counterclockwise for k > 0 and clockwise for k < 0. We will prove that fk and f` are homotopic if and only if k = `. Moreover, we will show that any map f : S 1 → S 1 is homotopic to fk for some k ∈ Z. In other words, ...
A BORDISM APPROACH TO STRING TOPOLOGY 1. Introduction
... Plan of the paper and results. In this paper we adopt a quite different approach to string topology, namely we use a geometric version of singular homology introduced by M. Jakob [26]. And we show how it is possible to define Gysin morphisms, exterior products and intersection type products (such as ...
... Plan of the paper and results. In this paper we adopt a quite different approach to string topology, namely we use a geometric version of singular homology introduced by M. Jakob [26]. And we show how it is possible to define Gysin morphisms, exterior products and intersection type products (such as ...
Chapter Three
... Let p ∈ E. For every point q ∈ F , we can find an open neighborhood U (q) of p and and open neighborhood V (q) of q which don’t intersect. Since F is closed, it is compact, so we can cover F with finitely many such V (qi ), i = 1 . . . n. Let Vp = ∪ni=1 V (qi ) and Up = ∩ni=1 U (qi ). Then F ⊆ Vp an ...
... Let p ∈ E. For every point q ∈ F , we can find an open neighborhood U (q) of p and and open neighborhood V (q) of q which don’t intersect. Since F is closed, it is compact, so we can cover F with finitely many such V (qi ), i = 1 . . . n. Let Vp = ∪ni=1 V (qi ) and Up = ∩ni=1 U (qi ). Then F ⊆ Vp an ...
A Crash Course on Kleinian Groups
... discrete. However the G-orbits on Ĉ ‘pile up’, as one can see by looking at the orbit of 0 under the subgroup SL(2, Z). It is easy to see this consists of the extended rational numbers Q ∪ ∞.1 Nevertheless, we have the following crucial theorem: Theorem 1.12 G ⊂ P SL(2, C) is Kleinian iff it acts p ...
... discrete. However the G-orbits on Ĉ ‘pile up’, as one can see by looking at the orbit of 0 under the subgroup SL(2, Z). It is easy to see this consists of the extended rational numbers Q ∪ ∞.1 Nevertheless, we have the following crucial theorem: Theorem 1.12 G ⊂ P SL(2, C) is Kleinian iff it acts p ...
Existence of covering topological R-modules
... discrete and obtain a more general result than the one for the topological group case. So this generalized result guarantees that the R-module structure of a topological R-module lifts to the universal cover. Recently, in [1], some results on the covering morphisms of R-module objects in the categor ...
... discrete and obtain a more general result than the one for the topological group case. So this generalized result guarantees that the R-module structure of a topological R-module lifts to the universal cover. Recently, in [1], some results on the covering morphisms of R-module objects in the categor ...
Abstract Simplicial Complexes
... defined by the conditions (Def. 1). (Def. 1)(i) X ⊆ the subset-closure of X, and (ii) for every Y such that X ⊆ Y and Y is subset-closed holds the subsetclosure of X ⊆ Y. The following proposition is true (2) x ∈ the subset-closure of X iff there exists y such that x ⊆ y and y ∈ X. Let us consider X ...
... defined by the conditions (Def. 1). (Def. 1)(i) X ⊆ the subset-closure of X, and (ii) for every Y such that X ⊆ Y and Y is subset-closed holds the subsetclosure of X ⊆ Y. The following proposition is true (2) x ∈ the subset-closure of X iff there exists y such that x ⊆ y and y ∈ X. Let us consider X ...
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 ...
Fundamental groups and finite sheeted coverings
... and etale, then each fibre of p has exactly the same number of points. Thus, a finite etale map is a natural analogue of a finite covering space. We define FEt/X to be the category whose objects are the finite etale maps p : Y → X (sometimes referred to as finite etale coverings of X) and whose arro ...
... and etale, then each fibre of p has exactly the same number of points. Thus, a finite etale map is a natural analogue of a finite covering space. We define FEt/X to be the category whose objects are the finite etale maps p : Y → X (sometimes referred to as finite etale coverings of X) and whose arro ...
STRONG HOMOTOPY TYPES, NERVES AND COLLAPSES 1
... of this subject, since we are more interested in understanding the difference between the various notions of collapses from a geometric point of view. One of the most significant questions related to this concept is the so-called Evasiveness conjecture for simplicial complexes which asserts that a n ...
... of this subject, since we are more interested in understanding the difference between the various notions of collapses from a geometric point of view. One of the most significant questions related to this concept is the so-called Evasiveness conjecture for simplicial complexes which asserts that a n ...
Properties of topological groups and Haar measure
... (ii)A subset S of a group G is called symmetric if it satisfies S −1 = S. The group theoretic structure of a topological group allows us to pick a basis consisted from nicer sets than the general open sets and this is shown in the first two sentences of the following Lemma. The rest of the sentences ...
... (ii)A subset S of a group G is called symmetric if it satisfies S −1 = S. The group theoretic structure of a topological group allows us to pick a basis consisted from nicer sets than the general open sets and this is shown in the first two sentences of the following Lemma. The rest of the sentences ...
Sum theorems for topological spaces
... (Σ): Let X be a topological space and let {Fa} be a locally finite closed cover of X such that each Fa is in Q. Then X is in Q. It is known that (Σ) holds when Q is the class of regular spaces [14], normal spaces [13], collectionwise normal spaces [13], paracompact spaces [11], stratiίiable spaces [ ...
... (Σ): Let X be a topological space and let {Fa} be a locally finite closed cover of X such that each Fa is in Q. Then X is in Q. It is known that (Σ) holds when Q is the class of regular spaces [14], normal spaces [13], collectionwise normal spaces [13], paracompact spaces [11], stratiίiable spaces [ ...
Topological constructors
... Algebraic topology is concerned by any kind of topological space, but using the algebraic tool soon leads to favour the constructors producing spaces which can be conveniently so analyzed. Two important constructors are considered here, the first one giving the CW-complexes, the second one the simpl ...
... Algebraic topology is concerned by any kind of topological space, but using the algebraic tool soon leads to favour the constructors producing spaces which can be conveniently so analyzed. Two important constructors are considered here, the first one giving the CW-complexes, the second one the simpl ...