Thom Spectra that Are Symmetric Spectra
... Theorem 1.4. If (X, f ) → (Y, g) is an I-equivalence of T -good I-spaces over BF , then the induced map T (f ) → T (g) is a stable equivalence of symmetric spectra. Here stable equivalence refers to the stable model structure on SpΣ defined in [16] and [27]. It is a subtle property of this model str ...
... Theorem 1.4. If (X, f ) → (Y, g) is an I-equivalence of T -good I-spaces over BF , then the induced map T (f ) → T (g) is a stable equivalence of symmetric spectra. Here stable equivalence refers to the stable model structure on SpΣ defined in [16] and [27]. It is a subtle property of this model str ...
Rational homotopy theory
... It follows from Exercise 3.2 that Hk (Xn ) ∈ C for all k > 0. Theorem 3.4. Let C be a Serre’ class of abelian groups satisfying (ii) and (iii), and let X be a simply connected space. If Hk (X) ∈ C for all k > 0, then πk (X) ∈ C for all k > 0. Proof. Again, let πn := πn (X) and let {Xn } be a Postnik ...
... It follows from Exercise 3.2 that Hk (Xn ) ∈ C for all k > 0. Theorem 3.4. Let C be a Serre’ class of abelian groups satisfying (ii) and (iii), and let X be a simply connected space. If Hk (X) ∈ C for all k > 0, then πk (X) ∈ C for all k > 0. Proof. Again, let πn := πn (X) and let {Xn } be a Postnik ...
homotopy types of topological stacks
... one to transport homotopical information back and forth between the diagram and its homotopy type. The above theorem has various applications. For example, it implies an equivariant version of Theorem 1.1 for the (weak) action of a discrete group. It also allows one to define homotopy types of pairs ...
... one to transport homotopical information back and forth between the diagram and its homotopy type. The above theorem has various applications. For example, it implies an equivariant version of Theorem 1.1 for the (weak) action of a discrete group. It also allows one to define homotopy types of pairs ...
NOTES ON NON-ARCHIMEDEAN TOPOLOGICAL GROUPS
... Studying the properties of the Heisenberg group HX , we get a unified approach to several (mostly known) equivalent characterizations of the class N A of nonarchimedean groups (Lemma 3.2 and Theorem 5.1). In particular, we show that the class of all topological subgroups of Aut(K), for compact abeli ...
... Studying the properties of the Heisenberg group HX , we get a unified approach to several (mostly known) equivalent characterizations of the class N A of nonarchimedean groups (Lemma 3.2 and Theorem 5.1). In particular, we show that the class of all topological subgroups of Aut(K), for compact abeli ...
Homology Theory - Section de mathématiques
... Lemma. (Braid Lemma) If we have a braid diagram as above in any abelian category (where the homologies are replaced by arbitrary objects of this category), where three of the sequences are exact and the fourth is a chain complex then this will be exact, ...
... Lemma. (Braid Lemma) If we have a braid diagram as above in any abelian category (where the homologies are replaced by arbitrary objects of this category), where three of the sequences are exact and the fourth is a chain complex then this will be exact, ...
Introduction to Combinatorial Homotopy Theory
... Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to di ...
... Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to di ...
Chapter VII. Covering Spaces and Calculation of Fundamental Groups
... 34.B Path Lifting Theorem. Let p : X → B be a covering, x0 ∈ X, b0 ∈ B be points such that p(x0 ) = b0 . Then for any path s : I → B starting at b0 there exists a unique path s̃ : I → X starting at x0 and being a lifting of s. (In other words, there exists a unique path s̃ : I → X with s̃(0) = x0 an ...
... 34.B Path Lifting Theorem. Let p : X → B be a covering, x0 ∈ X, b0 ∈ B be points such that p(x0 ) = b0 . Then for any path s : I → B starting at b0 there exists a unique path s̃ : I → X starting at x0 and being a lifting of s. (In other words, there exists a unique path s̃ : I → X with s̃(0) = x0 an ...
Arithmetic fundamental groups and moduli of curves
... see x3 below. In topology theory, it is often more convenient to consider the monodromy of the bration p : F ! B , as follows. The rough idea is as follows: take a closed path 2 1 (B; b). Then, because of the local triviality of the bration, we can consider the family Fc := p 1 (c) where c move ...
... see x3 below. In topology theory, it is often more convenient to consider the monodromy of the bration p : F ! B , as follows. The rough idea is as follows: take a closed path 2 1 (B; b). Then, because of the local triviality of the bration, we can consider the family Fc := p 1 (c) where c move ...
A Crash Course in Topological Groups
... Any group taken with the discrete topology. Any (arbitrary) direct product of these with the product topology. Note that, for example, (Z/2Z)ω is not discrete with the product topology (it is homeomorphic to the Cantor set). ...
... Any group taken with the discrete topology. Any (arbitrary) direct product of these with the product topology. Note that, for example, (Z/2Z)ω is not discrete with the product topology (it is homeomorphic to the Cantor set). ...
FIBRATIONS OF TOPOLOGICAL STACKS Contents 1. Introduction 2
... weak Hurewicz fibration then it is a weak Serre fibration. In §4 we provide some general classes of examples of fibrations of stacks. Throughout the paper, we also prove various results which can be used to produce new fibrations out of old ones. This way we can produce plenty of examples of fibrati ...
... weak Hurewicz fibration then it is a weak Serre fibration. In §4 we provide some general classes of examples of fibrations of stacks. Throughout the paper, we also prove various results which can be used to produce new fibrations out of old ones. This way we can produce plenty of examples of fibrati ...
A model structure for quasi-categories
... be studied via quasi-categories. The nerve functor will be described in more detail below. A modern introduction to quasi-categories must note that they also serve as a model for the “homotopy theory of homotopy theories.” In some sense a “homotopy theory” can be regarded as a category with some cla ...
... be studied via quasi-categories. The nerve functor will be described in more detail below. A modern introduction to quasi-categories must note that they also serve as a model for the “homotopy theory of homotopy theories.” In some sense a “homotopy theory” can be regarded as a category with some cla ...
On s-Topological Groups
... if (G, ∗) is a group, (G, τ ) is a topological space, and (a) the multiplication mapping m : G × G → G defined by m(x, y) = x ∗ y, x, y ∈ G, is semi-continuous, (b) the inverse mapping i : G → G defined by i(x) = x−1 , x ∈ G, is semi-continuous. (b) ([3]) An s-topological group is a group (G, ∗) wit ...
... if (G, ∗) is a group, (G, τ ) is a topological space, and (a) the multiplication mapping m : G × G → G defined by m(x, y) = x ∗ y, x, y ∈ G, is semi-continuous, (b) the inverse mapping i : G → G defined by i(x) = x−1 , x ∈ G, is semi-continuous. (b) ([3]) An s-topological group is a group (G, ∗) wit ...
- Iranian Journal of Fuzzy Systems
... equivalence classes are called fuzzy homotopy classes and the set of all fuzzy homotopy classes of the fuzzy continuous functions from (X, τ ) to (Y, τ 0 ) is denoted by [(X, τ ); (Y, τ 0 )]. The fuzzy homotopy class of a function f is denoted by [f ]. Definition 2.3. Let (X, τ ), (Y, τ 0 ) be fuzzy ...
... equivalence classes are called fuzzy homotopy classes and the set of all fuzzy homotopy classes of the fuzzy continuous functions from (X, τ ) to (Y, τ 0 ) is denoted by [(X, τ ); (Y, τ 0 )]. The fuzzy homotopy class of a function f is denoted by [f ]. Definition 2.3. Let (X, τ ), (Y, τ 0 ) be fuzzy ...
COMBINATORIAL HOMOTOPY. I 1. Introduction. This is the first of a
... Let K be the universal covering complex of a CW-complex K. Since TI(K) = 1, irn(K) ^Tn(K) if n> 1 and since L is a Jm-complex if 7TV(L) = 0 for r = 1, • • • , m — 1 it follows from Theorem 10 that K is a J m -complex if 7r n (i£)=0 for n = 2, • • • , m — 1 . In particular if is a J^-complex if its u ...
... Let K be the universal covering complex of a CW-complex K. Since TI(K) = 1, irn(K) ^Tn(K) if n> 1 and since L is a Jm-complex if 7TV(L) = 0 for r = 1, • • • , m — 1 it follows from Theorem 10 that K is a J m -complex if 7r n (i£)=0 for n = 2, • • • , m — 1 . In particular if is a J^-complex if its u ...
minimalrevised.pdf
... of these two kinds of reductions. In section 2 of this paper, we exhibit an example which shows that the answer to his question is negative. Therefore, his methods of reduction are not always effective and could not be applied to prove Theorems 2.13 and 4.7 mentioned above. We think that the methods ...
... of these two kinds of reductions. In section 2 of this paper, we exhibit an example which shows that the answer to his question is negative. Therefore, his methods of reduction are not always effective and could not be applied to prove Theorems 2.13 and 4.7 mentioned above. We think that the methods ...
AN INTRODUCTION TO ∞-CATEGORIES Contents 1. Introduction 1
... 1.1.1. Finding an appropriate language for such categories. Perusing through Maclane [Maclane72], you see immediately that there’s a ton of useful category theory for usual categories. And we know from previous talks that things like the Barr-Beck theorem help us say great things about categories. ( ...
... 1.1.1. Finding an appropriate language for such categories. Perusing through Maclane [Maclane72], you see immediately that there’s a ton of useful category theory for usual categories. And we know from previous talks that things like the Barr-Beck theorem help us say great things about categories. ( ...
Topology A chapter for the Mathematics++ Lecture Notes
... a metric space is a pair (X, dX ), where X is a set and dX : X × X → R is a metric satisfying several natural axioms (x, y, z are arbitrary points of X): dX (x, y) ≥ 0, dX (x, x) = 0, dX (x, y) > 0 for x 6= y, dX (y, x) = dX (x, y), and dX (x, y) + dX (y, z) ≥ dX (x, z) (the triangle inequality). Th ...
... a metric space is a pair (X, dX ), where X is a set and dX : X × X → R is a metric satisfying several natural axioms (x, y, z are arbitrary points of X): dX (x, y) ≥ 0, dX (x, x) = 0, dX (x, y) > 0 for x 6= y, dX (y, x) = dX (x, y), and dX (x, y) + dX (y, z) ≥ dX (x, z) (the triangle inequality). Th ...
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.