topological group
... A topological group is a group G endowed with a topology such that the multiplication and inverse operations of G are continuous. That is, the map G × G → G defined by (x, y) 7→ xy is continuous, where the topology on G×G is the product topology, and the map G → G defined by x 7→ x−1 is also continu ...
... A topological group is a group G endowed with a topology such that the multiplication and inverse operations of G are continuous. That is, the map G × G → G defined by (x, y) 7→ xy is continuous, where the topology on G×G is the product topology, and the map G → G defined by x 7→ x−1 is also continu ...
Fundamental groups and finite sheeted coverings
... If X is a variety over an algebraically closed field and p : Y → X is finite 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 ...
... If X is a variety over an algebraically closed field and p : Y → X is finite 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 ...
INTRODUCTION TO ALGEBRAIC TOPOLOGY 1.1. Topological
... ⊲ morphisms: continuous maps, equipped with composition of continuous maps. Explicitly HomTop (X, Y ) is the set of continuous maps from X to Y and composition is a set map ◦ : HomTop (Y, Z) × HomTop (X, Y ) → HomTop (X, Z). These satisfy the Axioms of a category: the existence and properties of ide ...
... ⊲ morphisms: continuous maps, equipped with composition of continuous maps. Explicitly HomTop (X, Y ) is the set of continuous maps from X to Y and composition is a set map ◦ : HomTop (Y, Z) × HomTop (X, Y ) → HomTop (X, Z). These satisfy the Axioms of a category: the existence and properties of ide ...
Lecture Notes
... seen to be continuous, whence ṡi (x) = kssii (x)k is also continuous. Thus ṡi (x) ∈ Sn and ṡi (x) ⊥ x. We now observe: Proposition 4.3. If there is a continous map v : Sn → Sn such that v(x) ⊥ x for all x ∈ Sn , then the antpodal map is homotopic to the identity. The required homotopy is given by ...
... seen to be continuous, whence ṡi (x) = kssii (x)k is also continuous. Thus ṡi (x) ∈ Sn and ṡi (x) ⊥ x. We now observe: Proposition 4.3. If there is a continous map v : Sn → Sn such that v(x) ⊥ x for all x ∈ Sn , then the antpodal map is homotopic to the identity. The required homotopy is given by ...
WHAT IS A TOPOLOGICAL STACK? 1. introduction Stacks were
... orbits. For instance, [X/G] in some sense remembers all the stabilizer groups of the action, while X/G is completely blind to them. There is a natural morphism πmod : [X/G] → X/G enabling us to compare the stacky quotient [X/G] with the coarse quotient X/G. We will encounter πmod again later in this ...
... orbits. For instance, [X/G] in some sense remembers all the stabilizer groups of the action, while X/G is completely blind to them. There is a natural morphism πmod : [X/G] → X/G enabling us to compare the stacky quotient [X/G] with the coarse quotient X/G. We will encounter πmod again later in this ...
THE EXACT SEQUENCE OF A SHAPE FIBRATION Q. Haxhibeqiri
... given in [5] we show when a restriction of shape fibration is again a shape fibration (Theorem 4.1) and when a shape fibration induces an isomorphism of homotopy pro-groups (Theorem 5.7) obtaining also the exaet sequence of shape fibration (Theorem 5.9). ...
... given in [5] we show when a restriction of shape fibration is again a shape fibration (Theorem 4.1) and when a shape fibration induces an isomorphism of homotopy pro-groups (Theorem 5.7) obtaining also the exaet sequence of shape fibration (Theorem 5.9). ...
CONGRUENCES BETWEEN MODULAR FORMS GIVEN BY THE
... Organization of the paper. In Section 2 we summarize the chromatic spectral sequence. We also recall Morava’s change of rings theorem, which relates the terms of the chromatic spectral sequence to the cohomology of the Morava stabilizer groups Sn . In Section 3 we explain how to associate a p-comple ...
... Organization of the paper. In Section 2 we summarize the chromatic spectral sequence. We also recall Morava’s change of rings theorem, which relates the terms of the chromatic spectral sequence to the cohomology of the Morava stabilizer groups Sn . In Section 3 we explain how to associate a p-comple ...
A CONVENIENT CATEGORY FOR DIRECTED HOMOTOPY
... (2’) given h : X → U B with hfi = U h̄i , h̄i : Ai → B for each i ∈ I then h = U h̄ for a unique h̄ : A → B. 3.1. Example. (1) A preordered set (A, ≤) is a set A equipped with a reflexive and transitive relation ≤. It means that it satisfies the formulas (∀x)(x ≤ x) and (∀x, y, z)(x ≤ y ∧ y ≤ z → x ...
... (2’) given h : X → U B with hfi = U h̄i , h̄i : Ai → B for each i ∈ I then h = U h̄ for a unique h̄ : A → B. 3.1. Example. (1) A preordered set (A, ≤) is a set A equipped with a reflexive and transitive relation ≤. It means that it satisfies the formulas (∀x)(x ≤ x) and (∀x, y, z)(x ≤ y ∧ y ≤ z → x ...
Generalized Normal Bundles for Locally
... An n-hpb ( <, +o) is G-orientableif ( +, <,o) is G-orientable as a fibered pair. DEFINITION (3.2). A generalized n-plane bundle (n-gpb) (E, 0) (E, Eo, p, B) is a fibered pair with fiber (F, Fo) and the following additional properties: (i) There exists a cross section v: B -4 E such that Eo = E - v(B ...
... An n-hpb ( <, +o) is G-orientableif ( +, <,o) is G-orientable as a fibered pair. DEFINITION (3.2). A generalized n-plane bundle (n-gpb) (E, 0) (E, Eo, p, B) is a fibered pair with fiber (F, Fo) and the following additional properties: (i) There exists a cross section v: B -4 E such that Eo = E - v(B ...
Homotopy theories and model categories
... We have tried to minimize the prerequisites needed for understanding this paper; it should be enough to have some familiarity with CW-complexes, with chain complexes, and with the basic terminology associated with categories. Almost all of the material we present is due to Quillen [22], but we have ...
... We have tried to minimize the prerequisites needed for understanding this paper; it should be enough to have some familiarity with CW-complexes, with chain complexes, and with the basic terminology associated with categories. Almost all of the material we present is due to Quillen [22], but we have ...
FiniteSpaces.pdf
... If f : X −→ Y is a homeomorphism, then f determines a bijection from the basis for X to the basis for Y that preserves inclusions and the number of elements that determine corresponding basic sets, hence X and Y determine the same element of M . Conversely, suppose that X and Y have minimal bases {U ...
... If f : X −→ Y is a homeomorphism, then f determines a bijection from the basis for X to the basis for Y that preserves inclusions and the number of elements that determine corresponding basic sets, hence X and Y determine the same element of M . Conversely, suppose that X and Y have minimal bases {U ...
FINITE TOPOLOGICAL SPACES 1. Introduction: finite spaces and
... Lemma 4.6. Each Ux is connected. If X is connected and x, y ∈ X, there is a sequence of points zi , 1 ≤ i ≤ s, such that z1 = x, zs = y and either zi ≤ zi+1 or zi+1 ≤ zi for i < s. Proof. If Ux = A q B, A and B open, say x ∈ A, then Ux ⊂ A and therefore B = ∅. Fix x and consider the set A of points ...
... Lemma 4.6. Each Ux is connected. If X is connected and x, y ∈ X, there is a sequence of points zi , 1 ≤ i ≤ s, such that z1 = x, zs = y and either zi ≤ zi+1 or zi+1 ≤ zi for i < s. Proof. If Ux = A q B, A and B open, say x ∈ A, then Ux ⊂ A and therefore B = ∅. Fix x and consider the set A of points ...
Aalborg University - VBN
... (2’) given h : X → U B with hfi = U h̄i , h̄i : Ai → B for each i ∈ I then h = U h̄ for a unique h̄ : A → B. 3.1. Example. (1) A preordered set (A, ≤) is a set A equipped with a reflexive and transitive relation ≤. It means that it satisfies the formulas (∀x)(x ≤ x) and (∀x, y, z)(x ≤ y ∧ y ≤ z → x ...
... (2’) given h : X → U B with hfi = U h̄i , h̄i : Ai → B for each i ∈ I then h = U h̄ for a unique h̄ : A → B. 3.1. Example. (1) A preordered set (A, ≤) is a set A equipped with a reflexive and transitive relation ≤. It means that it satisfies the formulas (∀x)(x ≤ x) and (∀x, y, z)(x ≤ y ∧ y ≤ z → x ...
The bordism version of the h
... MR follows closely the argument of Galatius-Madsen-Tillman-Weiss [16], and relies on the construction of classifying Γ-spaces by Segal [41]. Here we introduce and use category valued partial Γ-sheaves in order to show that BMR is equivalent to BMR not only as a space, but also as an H -space with co ...
... MR follows closely the argument of Galatius-Madsen-Tillman-Weiss [16], and relies on the construction of classifying Γ-spaces by Segal [41]. Here we introduce and use category valued partial Γ-sheaves in order to show that BMR is equivalent to BMR not only as a space, but also as an H -space with co ...
Homotopy Theory of Topological Spaces and Simplicial Sets
... and if f g = ida and gf = idb we will call the objects a and b isomorphic, as usual. Isomorphic objects will have some common properties, depending on the category. For example in the setting of sets as before isomorphisms are just bijections and the sets that are isomorphic will have the same cardi ...
... and if f g = ida and gf = idb we will call the objects a and b isomorphic, as usual. Isomorphic objects will have some common properties, depending on the category. For example in the setting of sets as before isomorphisms are just bijections and the sets that are isomorphic will have the same cardi ...
Contents - Columbia Math
... Definition 1.10. Let (X, τ ) and (Y, σ) be topological spaces. A map f : X → Y is called continuous if for each open subset V of Y , the preimage f −1 (V ) is an open subset of X. This definition coincides with the usual definition of continuity for maps between Euclidean spaces. We invite the inter ...
... Definition 1.10. Let (X, τ ) and (Y, σ) be topological spaces. A map f : X → Y is called continuous if for each open subset V of Y , the preimage f −1 (V ) is an open subset of X. This definition coincides with the usual definition of continuity for maps between Euclidean spaces. We invite the inter ...
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as Sn — generalizes the familiar circle (S1) and the ordinary sphere (S2). The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin. The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An ""addition"" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n. For 0 < i < n, any mapping from Si to Sn is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of Si to a single point of Sn. When i = n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies πn(Sn) with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping. However, the most interesting and surprising results occur when i > n. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, and so is not equivalent to a one-point mapping.The question of computing the homotopy group πn+k(Sn) for positive k turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups πn+k(Sn) are independent of n for n ≥ k + 2. These are called the stable homotopy groups of spheres and have been computed for values of k up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for n < k + 2) are more erratic; nevertheless, they have been tabulated for k < 20. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet much remains unknown and unexplained.