¾ - Hopf Topology Archive
... topology. In algebra they often arise as the stable category of a Frobenius category ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99, 7.1]. The triangulated categories which ...
... topology. In algebra they often arise as the stable category of a Frobenius category ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99, 7.1]. The triangulated categories which ...
Topology
... For a very theoretical part of mathematics such as topology it may sound strange to talk about applications. However, topology is not only one fo the most theoretical and most highly developed areas of so-called “pure mathematics” with remarkable successes and results in this subject. In the course ...
... For a very theoretical part of mathematics such as topology it may sound strange to talk about applications. However, topology is not only one fo the most theoretical and most highly developed areas of so-called “pure mathematics” with remarkable successes and results in this subject. In the course ...
Finite Spaces Handouts 1
... Proposition 1.2.5. On a set X, let U and U 0 be the topologies generated by the bases B and B 0 , respectively. Then U ⊆ U 0 if and only if for every x ∈ B ∈ B, there exists B 0 ∈ B 0 such that x ∈ B 0 ⊆ B. Exercise 1.2.6. In R2 , one might consider the set of open disks (balls) {Br (x, y)} as a bas ...
... Proposition 1.2.5. On a set X, let U and U 0 be the topologies generated by the bases B and B 0 , respectively. Then U ⊆ U 0 if and only if for every x ∈ B ∈ B, there exists B 0 ∈ B 0 such that x ∈ B 0 ⊆ B. Exercise 1.2.6. In R2 , one might consider the set of open disks (balls) {Br (x, y)} as a bas ...
A convenient category - VBN
... (fi : X → UAi )i∈I in Set has a unique U-initial lift (f¯i : A → Ai )i∈I (see [2]). It means that (1) UA = X and U f¯i = fi for each i ∈ I and (2) given h : UB → X with fi h = U h̄i , h̄i : B → Ai for each i ∈ I then h = U h̄ for h̄ : B → A. Each topological functor is faithful and thus the pair (K, ...
... (fi : X → UAi )i∈I in Set has a unique U-initial lift (f¯i : A → Ai )i∈I (see [2]). It means that (1) UA = X and U f¯i = fi for each i ∈ I and (2) given h : UB → X with fi h = U h̄i , h̄i : B → Ai for each i ∈ I then h = U h̄ for h̄ : B → A. Each topological functor is faithful and thus the pair (K, ...
Tibor Macko
... Theorem 1.1 ([Ran92], Theorem 17.4). Let X be a finite Poincaré complex of formal dimension n ≥ 5. Then X is homotopy equivalent to a closed n-dimensional topological manifold if and only if 0 = s(X) ∈ Sn (X). By choosing a simplicial complex homotopy equivalent to X we can assume that X is a simpl ...
... Theorem 1.1 ([Ran92], Theorem 17.4). Let X be a finite Poincaré complex of formal dimension n ≥ 5. Then X is homotopy equivalent to a closed n-dimensional topological manifold if and only if 0 = s(X) ∈ Sn (X). By choosing a simplicial complex homotopy equivalent to X we can assume that X is a simpl ...
Notes from Craigfest - University of Melbourne
... Theorem 2.11. For a fibration F → E → B, we have a long exact sequence in homotopy given by . . . → πn (F ) → πn (E) → πn (B) → πn−1 (F ) → . . . → π0 (F ) → π0 (E) → π0 (B) Proof. Putting Z = S 0 in the Puppe sequence gives . . . → [S 0 , Ωn F ] → [S 0 , Ωn E] → [S 0 , Ωn B] → . . . . Then using th ...
... Theorem 2.11. For a fibration F → E → B, we have a long exact sequence in homotopy given by . . . → πn (F ) → πn (E) → πn (B) → πn−1 (F ) → . . . → π0 (F ) → π0 (E) → π0 (B) Proof. Putting Z = S 0 in the Puppe sequence gives . . . → [S 0 , Ωn F ] → [S 0 , Ωn E] → [S 0 , Ωn B] → . . . . Then using th ...
ALGEBRAIC TOPOLOGY NOTES, PART II: FUNDAMENTAL GROUP
... (c) Give examples to show that each of the three bold-face assumptions is needed for part (b). [Hint for (b): for each h ∈ H consider the function sending g ∈ G to ghg −1 and apply each of the bold-face assumptions to show that this function is constant.] 2. the basic result: π1 (S 1 , 1) ∼ =Z This ...
... (c) Give examples to show that each of the three bold-face assumptions is needed for part (b). [Hint for (b): for each h ∈ H consider the function sending g ∈ G to ghg −1 and apply each of the bold-face assumptions to show that this function is constant.] 2. the basic result: π1 (S 1 , 1) ∼ =Z This ...
On Colimits in Various Categories of Manifolds
... to P is an initial map in the category of Froelicher spaces (which sits between Mn f d and T op). The details only take one paragraph, but would take us away from our goal of understanding pushouts in Mn f d so we’ll omit them. The point of this example is that none of the following categories are c ...
... to P is an initial map in the category of Froelicher spaces (which sits between Mn f d and T op). The details only take one paragraph, but would take us away from our goal of understanding pushouts in Mn f d so we’ll omit them. The point of this example is that none of the following categories are c ...
RATIONAL HOMOTOPY THEORY Contents 1. Introduction 1 2
... (1) We’ll see how working rationally greatly simplifies many computations. In particular, computations with Eilenberg-MacLane spaces become easy, rather than a combinatorial nightmare producing Steenrod operations. (2) In fact, we’ll show that if we are computing rationally, homotopy and homology do ...
... (1) We’ll see how working rationally greatly simplifies many computations. In particular, computations with Eilenberg-MacLane spaces become easy, rather than a combinatorial nightmare producing Steenrod operations. (2) In fact, we’ll show that if we are computing rationally, homotopy and homology do ...
HOMOLOGICAL PROPERTIES OF NON
... where Cf is the cone of f and where H0 (hoP− Z) is the free abelian group generated by the path-connected components of the homotopy branching space of the flow Z. By now, this homological result does not have any known computer scientific interpretation. But it sheds some light on the potential of ...
... where Cf is the cone of f and where H0 (hoP− Z) is the free abelian group generated by the path-connected components of the homotopy branching space of the flow Z. By now, this homological result does not have any known computer scientific interpretation. But it sheds some light on the potential of ...
HIGHER CATEGORIES 1. Introduction. Categories and simplicial
... to X. The faces and the degeneracies are defined via maps δ i : ∆[n − 1] → ∆[n] and σ i : ∆[n] → ∆[n − 1] where δ i inserts 0 at the place i and σ i puts xi + xi+1 at the place i. 1.5.2. Nerve of a category. There is a very similar construction in the world of categories — this is something that all ...
... to X. The faces and the degeneracies are defined via maps δ i : ∆[n − 1] → ∆[n] and σ i : ∆[n] → ∆[n − 1] where δ i inserts 0 at the place i and σ i puts xi + xi+1 at the place i. 1.5.2. Nerve of a category. There is a very similar construction in the world of categories — this is something that all ...
Here
... 1. Prove that any convext subspace Y ⊂ En is homotopy equivalent to the space consisting of a single point. 2. Prove that the complex plane minus the origin C \ {0} is homotopy equivalent to the circle S 1 . 3. Prove that homotopy equivalence of maps f ' g is an equivalence relation on the set of co ...
... 1. Prove that any convext subspace Y ⊂ En is homotopy equivalent to the space consisting of a single point. 2. Prove that the complex plane minus the origin C \ {0} is homotopy equivalent to the circle S 1 . 3. Prove that homotopy equivalence of maps f ' g is an equivalence relation on the set of co ...
SMSTC (2014/15) Geometry and Topology www.smstc.ac.uk
... We will also refer to A. Hatcher, Algebraic Topology, Cambridge University Press (2002), freely downloadable from the author’s website. Calculus gives us the concepts of open sets in Euclidean space and continuous functions, as well as properties such as connectedness, compactness and others. In thi ...
... We will also refer to A. Hatcher, Algebraic Topology, Cambridge University Press (2002), freely downloadable from the author’s website. Calculus gives us the concepts of open sets in Euclidean space and continuous functions, as well as properties such as connectedness, compactness and others. In thi ...
4.2 Simplicial Homology Groups
... • We say that two p-cycles are homologous if they differ by a boundary. • Homology is an equivalence relation. • The equivalence classes of the homology are called homology classes. • The homology groups are the sets of homology classes. • Theorem. Homology groups are topological invariants. In parti ...
... • We say that two p-cycles are homologous if they differ by a boundary. • Homology is an equivalence relation. • The equivalence classes of the homology are called homology classes. • The homology groups are the sets of homology classes. • Theorem. Homology groups are topological invariants. In parti ...
Section 07
... that the definition depends on the chosen order on the index set of the cover U. It can be shown, however, that the resulting cohomology groups do not, see Exercises.) This definition of the Čech cohomology groups may strike you as somewhat involved, but these groups are often very easy to calculat ...
... that the definition depends on the chosen order on the index set of the cover U. It can be shown, however, that the resulting cohomology groups do not, see Exercises.) This definition of the Čech cohomology groups may strike you as somewhat involved, but these groups are often very easy to calculat ...
Structure theory of manifolds
... A topological bundle is similarly a fiber bundle, but with the larger structure group Topn of all homeomorphisms (Rn , 0) ←-, in the compact-open topology. A P L bundle is defined by a projection p : E → B, which is assumed to be a P L map between P L spaces, and local trivializations Φα : p−1 (Uα ) ...
... A topological bundle is similarly a fiber bundle, but with the larger structure group Topn of all homeomorphisms (Rn , 0) ←-, in the compact-open topology. A P L bundle is defined by a projection p : E → B, which is assumed to be a P L map between P L spaces, and local trivializations Φα : p−1 (Uα ) ...
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.