PRECOMPACT NONCOMPACT REFLEXIVE ABELIAN GROUPS 1
... |C 0 | ≥ 2ω . It follows from Proposition 2.1 that C0 is closed in G, so C 0 ∩ G = C0 . Hence C 0 \ C0 is an uncountable subset of %G \ G. Consider K = hx0 i for some x0 ∈ C 0 \ C0 and the quotient homomorphism π : %G → %G/K. Let us see that the subgroup H = π(G) of %G/K has the required properties. ...
... |C 0 | ≥ 2ω . It follows from Proposition 2.1 that C0 is closed in G, so C 0 ∩ G = C0 . Hence C 0 \ C0 is an uncountable subset of %G \ G. Consider K = hx0 i for some x0 ∈ C 0 \ C0 and the quotient homomorphism π : %G → %G/K. Let us see that the subgroup H = π(G) of %G/K has the required properties. ...
Topology
... overview of core parts, and a “potential user’s introduction.” 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 r ...
... overview of core parts, and a “potential user’s introduction.” 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 r ...
RATIONAL HOMOTOPY THEORY Contents 1. Introduction 1 2
... We claim now that X has the property that πi ( X ) = 0 for i < n. We have a long exact sequence πn+1 ( X, X (n) ) → πn ( X (n) ) → πn ( X ) → πn ( X, X (n) ) But, note that πn+1 ( X, X (n) ) = Z[r β ] and πn ( X (n) ) = Z[ gα ], πn ( X, X (n) ) = 0. So, the above exact sequence actually gives a pres ...
... We claim now that X has the property that πi ( X ) = 0 for i < n. We have a long exact sequence πn+1 ( X, X (n) ) → πn ( X (n) ) → πn ( X ) → πn ( X, X (n) ) But, note that πn+1 ( X, X (n) ) = Z[r β ] and πn ( X (n) ) = Z[ gα ], πn ( X, X (n) ) = 0. So, the above exact sequence actually gives a pres ...
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 ...
9.15 Group Structures on Homotopy Classes of Maps
... Group Structures on Homotopy Classes of Maps ...
... Group Structures on Homotopy Classes of Maps ...
Math 31 – Homework 5 Solutions
... 1. Determine if each mapping is a homomorphism. State why or why not. If it is a homomorphism, find its kernel, and determine whether it is one-to-one and onto. (a) Define ϕ : Z → R by ϕ(n) = n. (Both are groups under addition here.) (b) Let G be a group, and define ϕ : G → G by ϕ(a) = a−1 for all a ...
... 1. Determine if each mapping is a homomorphism. State why or why not. If it is a homomorphism, find its kernel, and determine whether it is one-to-one and onto. (a) Define ϕ : Z → R by ϕ(n) = n. (Both are groups under addition here.) (b) Let G be a group, and define ϕ : G → G by ϕ(a) = a−1 for all a ...
II.4. Compactness - Faculty
... paper he addresses the idea of Cauchy sequences in metric spaces and comments: “The need of uniformity in [metric space] M arises from the fact that the elements of a fundamental sequence are postulated to be ‘near to each other,’ and not near to any fixed point. As a general topological space . . . ...
... paper he addresses the idea of Cauchy sequences in metric spaces and comments: “The need of uniformity in [metric space] M arises from the fact that the elements of a fundamental sequence are postulated to be ‘near to each other,’ and not near to any fixed point. As a general topological space . . . ...
Section 3.2 - Cohomology of Sheaves
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...
Notes
... Proposition 2.3.3. The Yoneda embedding C → Psh(C) is universal among functors from C into cocomplete categories, i.e. any functor F : C → D to a cocomplete catgeory factors through Psh(C), and the map Psh(C) → D preserves colimits. We have also seen the co-Yoneda lemma, which tells us that every pr ...
... Proposition 2.3.3. The Yoneda embedding C → Psh(C) is universal among functors from C into cocomplete categories, i.e. any functor F : C → D to a cocomplete catgeory factors through Psh(C), and the map Psh(C) → D preserves colimits. We have also seen the co-Yoneda lemma, which tells us that every pr ...
GLOBALIZING LOCALLY COMPACT LOCAL GROUPS 1
... (3) if (x, y), (y, z) ∈ Ω and either (xy, z) ∈ Ω or (x, yz) ∈ Ω, then both (xy, z) and (x, yz) belong to Ω and (xy)z = x(yz). Also, “topological group” will stand for “hausdorff topological group”, so any topological group G is a local group with Ω = G × G. From now on G is a local group, and a, b, ...
... (3) if (x, y), (y, z) ∈ Ω and either (xy, z) ∈ Ω or (x, yz) ∈ Ω, then both (xy, z) and (x, yz) belong to Ω and (xy)z = x(yz). Also, “topological group” will stand for “hausdorff topological group”, so any topological group G is a local group with Ω = G × G. From now on G is a local group, and a, b, ...
![On the number of polynomials with coefficients in [n] Dorin Andrica](http://s1.studyres.com/store/data/023344871_1-db0421bb4ddb78289c0a283cc0aa29ca-300x300.png)
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.