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Universal Coefficient Theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': :H_i(X,\Z) completely determine its ''homology groups with coefficients in'' , for any abelian group : :H_i(X,A) Here H_i might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients may be used, at the cost of using a Tor functor. For example, it is common to take A to be \Z/2\Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2- torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b_i of X and the Betti numbers b_ with coefficients in a field F. These can differ, but o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Exact Sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics), image of one morphism equals the kernel (algebra), kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Finitely Generated Abelian Groups
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_s. In this case, we say that the set \ is a '' generating set'' of G or that x_1,\dots, x_s ''generate'' G. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples * The integers, \left(\mathbb,+\right), are a finitely generated abelian group. * The integers modulo n, \left(\mathbb/n\mathbb,+\right), are a finite (hence finitely generated) abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW Complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k- bal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups ''H''''k''(''X'';''R'') on ''X'' with coefficients in a commutative ring ''R'' (typically ''R'' is Z''n'', Z, Q, R, or C) one can define the cup product, which takes the form :H^k(X;R) \times H^\ell(X;R) \to H^(X; R). The cup product gives a multiplication on the direct sum of the cohomology groups :H^\bullet(X;R) = \bigoplus_ H^k(X; R). This multiplication turns ''H''•(''X'';''R'') into a ring. In fact, it is naturally an N-graded r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction As with all projective spaces, is formed by taking the quotient of \R^\setminus \ under the equivalence relation for all real numbers . For all in \R^\setminus \ one can always find a such that has norm 1. There are precisely two such differing by sign. Thus can also be formed by identifying antipodal points of the unit -sphere, , in \R^. One can further restrict to the upper hemisphere of and merely identify antipodal points on the bounding equator. This shows that is also equivalent to the closed -dimensional disk, , with antipodal points on the boundary, \partial D^n=S^, identified. Low-dimensional examples * is called the real projective line, which is topologically equivalent to a circle. Thinking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects c in \mathcal and d in \mathcal, a bijection between the respective morphism sets :\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. Assuming that ''G'' is abelian in the case that n > 1, Eilenberg–MacLane spaces of type K(G,n) always exist, and are all weak homotopy equivalent. Thus, one may consider K(G,n) as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a K(G,n)" or as "a model of K(G,n)". Moreover, it is comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ext Functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The group cohomology, cohomology of groups, Lie algebra cohomology, Lie algebras, and Hochschild cohomology, associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies group extension, extensions of one module (mathematics), module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring (mathematics), ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let R be a ring and let R\text be the category (mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are function (mathematics), functions on the group of chain (algebraic topology), chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and abstract algebra, algebra. The terminology tends to hide the fact that cohomology, a Covariance and contravariance of functors, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
