Introduction to Sheaves
... 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of ...
... 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of ...
Abelian topological groups and (A/k)C ≈ k 1. Compact
... Given non-trivial ψ ∈ (A/k)b, the k-vectorspace k · ψ inside (A/k)b injects to a copy of k · ψ inside A Assuming for a moment that the image in A is essentially the same as the diagonal copy of k, the quotient (A/k)b/k injects to the compact A/k. The topology of (A/k)b is discrete, and the quotient ...
... Given non-trivial ψ ∈ (A/k)b, the k-vectorspace k · ψ inside (A/k)b injects to a copy of k · ψ inside A Assuming for a moment that the image in A is essentially the same as the diagonal copy of k, the quotient (A/k)b/k injects to the compact A/k. The topology of (A/k)b is discrete, and the quotient ...
slides - Math User Home Pages
... Examples 4) Simplicial abelian groups. A simplicial abelian group is a simplicial object in the category Ab of abelian groups. i) Let F : Sets → Ab be a functor assigning a to a set X the free group generated on X. It induces a functor F∗ : sSets → sAb Σ 7→ F ◦ Σ Intuitively, one can think of F∗ (Σ ...
... Examples 4) Simplicial abelian groups. A simplicial abelian group is a simplicial object in the category Ab of abelian groups. i) Let F : Sets → Ab be a functor assigning a to a set X the free group generated on X. It induces a functor F∗ : sSets → sAb Σ 7→ F ◦ Σ Intuitively, one can think of F∗ (Σ ...
IV.2 Homology
... Brouwer’s Fixed Point Theorem. A continuous map f : Bd+1 → Bd+1 has at least one fixed point x = f (x). Proof. Let A, B : Sd → Sd be maps defined by A(x) = (x − f (x))/kx − f (x)k and B(x) = x. B is the identity and therefore has degree 1. If f has no fixed point then A is well defined and has degre ...
... Brouwer’s Fixed Point Theorem. A continuous map f : Bd+1 → Bd+1 has at least one fixed point x = f (x). Proof. Let A, B : Sd → Sd be maps defined by A(x) = (x − f (x))/kx − f (x)k and B(x) = x. B is the identity and therefore has degree 1. If f has no fixed point then A is well defined and has degre ...
An Intuitive Introduction - University of Chicago Math Department
... Proof. By the definition of a covering space, p is globally surjective and locally homeomorphic. Thus it suffices to show that p is one to one globally—that is, given any x1 , x2 ∈ X̃ with x1 6= x2 , p(x1 ) 6= p(x2 ). (Note that this fails for the previous example: S 1 is not simply connected since ...
... Proof. By the definition of a covering space, p is globally surjective and locally homeomorphic. Thus it suffices to show that p is one to one globally—that is, given any x1 , x2 ∈ X̃ with x1 6= x2 , p(x1 ) 6= p(x2 ). (Note that this fails for the previous example: S 1 is not simply connected since ...
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY 1
... Given points x and y of the space X, a path in X from x to y is a continuous map f : [a, b] → X of some closed interval in the real line into X such that f (a) = x and f (b) = y. X is called path connected if every pair of points X can be joined by a path in X. X is called locally path connected if, ...
... Given points x and y of the space X, a path in X from x to y is a continuous map f : [a, b] → X of some closed interval in the real line into X such that f (a) = x and f (b) = y. X is called path connected if every pair of points X can be joined by a path in X. X is called locally path connected if, ...
STRATIFIED SPACES TWIGS 1. Introduction These
... still being useful, and so we will use it. 3. Examples There are many examples of such spaces, and in fact this is the main point about them: they arise naturally in many situations where manifolds just won’t do. Example 3.1. A manifold with boundary M is a stratified space. One stratum is the bound ...
... still being useful, and so we will use it. 3. Examples There are many examples of such spaces, and in fact this is the main point about them: they arise naturally in many situations where manifolds just won’t do. Example 3.1. A manifold with boundary M is a stratified space. One stratum is the bound ...
Since Lie groups are topological groups (and manifolds), it is useful
... For example, manifolds are locally compact. ...
... For example, manifolds are locally compact. ...
THE COARSE HAWAIIAN EARRING: A COUNTABLE SPACE WITH
... that the results should be shared so a year after his graduation, he collected these results and put together this manuscript with the student’s permission. 1. Introduction The homotopy theory of finite spaces, i.e. topological spaces with finitely many points is a well-studied topic [5, 8]. See [1, ...
... that the results should be shared so a year after his graduation, he collected these results and put together this manuscript with the student’s permission. 1. Introduction The homotopy theory of finite spaces, i.e. topological spaces with finitely many points is a well-studied topic [5, 8]. See [1, ...
Lecture Notes 2
... The following theorem is not so hard to prove, though it is a bit tediuos, specially in the noncompact case: Theorem 1.6.1. Every connected 1-dimensional manifold is homeomorphic to either S1 , if it is compact, and to R otherwise. To describe the classification of 2-manifolds, we need to introduce t ...
... The following theorem is not so hard to prove, though it is a bit tediuos, specially in the noncompact case: Theorem 1.6.1. Every connected 1-dimensional manifold is homeomorphic to either S1 , if it is compact, and to R otherwise. To describe the classification of 2-manifolds, we need to introduce t ...
oi(a) = 5>(0,C,). - American Mathematical Society
... The complex manifold X is said to be hyperbolic if kx is an actual distance (i.e., kx(z, to) = 0 implies z = to ). In this case, the Kobayashi distance induces the original manifold topology on X [B2]. There are many examples of hyperbolic manifolds; for instance, bounded domains in C" , hermitian m ...
... The complex manifold X is said to be hyperbolic if kx is an actual distance (i.e., kx(z, to) = 0 implies z = to ). In this case, the Kobayashi distance induces the original manifold topology on X [B2]. There are many examples of hyperbolic manifolds; for instance, bounded domains in C" , hermitian m ...
Fuchsian Groups: Intro
... 2) The G orbit of any point is discrete and the stabilizer of that point is finite. 3) For any point, there is a neighborhood of that point, V , for which only finitely many T ∈ G satisfy T (V ) ∩ V 6= ∅. To review, by the G orbit of a point x ∈ X, we mean the set {T x ∈ X : T ∈ G}. By the stabilize ...
... 2) The G orbit of any point is discrete and the stabilizer of that point is finite. 3) For any point, there is a neighborhood of that point, V , for which only finitely many T ∈ G satisfy T (V ) ∩ V 6= ∅. To review, by the G orbit of a point x ∈ X, we mean the set {T x ∈ X : T ∈ G}. By the stabilize ...
PROPERTIES OF FINITE-DIMENSIONAL GROUPS Topological
... Even when G is a Lie group and M is a differentiable or analytic manifold and if f(g\ x) is assumed simultaneously differentiable or analytic, then many problems remain. At first glance this might seem unlikely after the great work of Lie and his followers. But much of their work is concerned with l ...
... Even when G is a Lie group and M is a differentiable or analytic manifold and if f(g\ x) is assumed simultaneously differentiable or analytic, then many problems remain. At first glance this might seem unlikely after the great work of Lie and his followers. But much of their work is concerned with l ...
Seminar in Topology and Actions of Groups. Topological Groups
... said to be compatible if they satisfy (i) and (ii). Example 1. 1.The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 2. The trivial topology on G is compatible with the group structure of G. 3. Every no ...
... said to be compatible if they satisfy (i) and (ii). Example 1. 1.The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 2. The trivial topology on G is compatible with the group structure of G. 3. Every no ...
K - CIS @ UPenn
... K, of dimension d to be realized in Em, the dimension of the “ambient space”, m, must be big enough. For example, there are 2-complexes that can’t be realized in E3 or even in E4. There has to be enough room in order for condition (2) to be satisfied. It is not hard to prove that m = 2d+1 is always s ...
... K, of dimension d to be realized in Em, the dimension of the “ambient space”, m, must be big enough. For example, there are 2-complexes that can’t be realized in E3 or even in E4. There has to be enough room in order for condition (2) to be satisfied. It is not hard to prove that m = 2d+1 is always s ...
Subdivide.pdf
... unital products) can be identified with categories with a single object. Analogously, posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the p ...
... unital products) can be identified with categories with a single object. Analogously, posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the p ...
SUBDIVISIONS OF SMALL CATEGORIES Let A be a
... unital products) can be identified with categories with a single object. Analogously, posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the p ...
... unital products) can be identified with categories with a single object. Analogously, posets can be identified with those categories A with at most one arrow between any two objects by defining x ≤ y if there is an arrow x −→ y between the objects x and y of A . In particular, we write [n] for the p ...
(pdf)
... Theorem 2.10 (Simplicial Embedding Theorem). An abstract simplicial complex K with dimension n can be realized in R2n+1 . Proof. This theorem is mainly a statement about the number of independent vectors needed to uniquely represent points in any two pairs of simplices. By definition, we need to fin ...
... Theorem 2.10 (Simplicial Embedding Theorem). An abstract simplicial complex K with dimension n can be realized in R2n+1 . Proof. This theorem is mainly a statement about the number of independent vectors needed to uniquely represent points in any two pairs of simplices. By definition, we need to fin ...
Quotient Spaces and Quotient Maps
... group: the operation is composition (The composition of two homeomorphisms is a homeomorphism; functional composition is associative; the identity map I : X → X is the identity element in the group of homeomorphisms.) The group of all self-homeomorphisms of X may have interesting subgroups. When we ...
... group: the operation is composition (The composition of two homeomorphisms is a homeomorphism; functional composition is associative; the identity map I : X → X is the identity element in the group of homeomorphisms.) The group of all self-homeomorphisms of X may have interesting subgroups. When we ...
(pdf)
... Proof. Let hSi denote the group generated by F , and let ι : S → hSi denote the inclusion map. By Def. 1.1, ι extends to a unique homomorphism ι̂ : F → hSi. Another homomorphism ϕ with ϕ|S = ι is the identity. By uniqueness, ι̂ = ϕ. Proposition 1.3. Let F and G be free groups, freely generated by ...
... Proof. Let hSi denote the group generated by F , and let ι : S → hSi denote the inclusion map. By Def. 1.1, ι extends to a unique homomorphism ι̂ : F → hSi. Another homomorphism ϕ with ϕ|S = ι is the identity. By uniqueness, ι̂ = ϕ. Proposition 1.3. Let F and G be free groups, freely generated by ...
(pdf)
... The construction is simple: Out of all the maximal simplicies in M , pick the one, say α(d) , such that f (α(d) ) is maximized. If f (α(d) ) ≤ n − 1 then we must have M = K(n − 1), by property 2 and the fact that every simplex in M is a face of some maximal simplex of M . If f (α(d) ) > n then α(d) ...
... The construction is simple: Out of all the maximal simplicies in M , pick the one, say α(d) , such that f (α(d) ) is maximized. If f (α(d) ) ≤ n − 1 then we must have M = K(n − 1), by property 2 and the fact that every simplex in M is a face of some maximal simplex of M . If f (α(d) ) > n then α(d) ...
WHAT IS A TOPOLOGICAL STACK? 1. introduction Stacks were
... in bijection with the isomorphism classes of principal G-bundles over T . If in the above situation G is a discrete group, the quotient map ∗ → BG becomes the universal cover of BG. This implies that π1 BG ∼ = G. 1Maps between two given stacks naturally form a groupoid. By an equivalence class we si ...
... in bijection with the isomorphism classes of principal G-bundles over T . If in the above situation G is a discrete group, the quotient map ∗ → BG becomes the universal cover of BG. This implies that π1 BG ∼ = G. 1Maps between two given stacks naturally form a groupoid. By an equivalence class we si ...
Chapter 2: Manifolds
... It is not possible to cover the sphere by a single chart, but it is possible to do so by two charts.1 For the two charts, we will construct what is called the stereographic projection. It is most convenient to draw this for a circle in the plane, i.e. S 1 in R2 , for which the equatorial ‘plane’ is ...
... It is not possible to cover the sphere by a single chart, but it is possible to do so by two charts.1 For the two charts, we will construct what is called the stereographic projection. It is most convenient to draw this for a circle in the plane, i.e. S 1 in R2 , for which the equatorial ‘plane’ is ...
Products, Quotients and Manifolds
... Please solve all three problems in §6.1, and four more problems, including at least one problem from each of § 6.2, § 6.3 and § 6.4. Starred parts of problems are optional. A topological space X is locally Euclidean if every x in X has an open neighborhood which is homeomorphic to RN . (The value of ...
... Please solve all three problems in §6.1, and four more problems, including at least one problem from each of § 6.2, § 6.3 and § 6.4. Starred parts of problems are optional. A topological space X is locally Euclidean if every x in X has an open neighborhood which is homeomorphic to RN . (The value of ...
(pdf)
... is a finite set with a topology. For each x ∈ X, let Ux be the intersection of all open sets containing x. (Note that since X is finite, Ux is open.) Then we can put a preorder on X (that is, we can establish a relation ≤ that is reflexive and transitive) by having x ≤ y if and only if Ux ⊂ Uy . Con ...
... is a finite set with a topology. For each x ∈ X, let Ux be the intersection of all open sets containing x. (Note that since X is finite, Ux is open.) Then we can put a preorder on X (that is, we can establish a relation ≤ that is reflexive and transitive) by having x ≤ y if and only if Ux ⊂ Uy . Con ...