In mathematics, **reduced homology** is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero. This change is required to make statements without some number of exceptional cases (Alexander duality being an example).

If *P* is a single-point space, then with the usual definitions the integral homology group

*H*_{0}(*P*)

is isomorphic to (an infinite cyclic group), while for *i* ≥ 1 we have

*H*_{i}(*P*) = {0}.

More generally if *X* is a simplicial complex or finite CW complex, then the group *H*_{0}(*X*) is the free abelian group with the connected components of *X* as generators. The reduced homology should replace this group, of rank *r* say, by one of rank *r*− 1. Otherwise the homology groups should remain unchanged. An *ad hoc* way to do this is to think of a 0-th homology class not as a formal sum of connected components, but as such a formal sum where the coefficients add up to zero.

In the usual definition of homology of a space *X*, we consider the chain complex

and define the homology groups by .

To define reduced homology, we start with the *augmented* chain complex

where . Now we define the *reduced* homology groups by

- for positive
*n*and .

One can show that ; evidently for all positive *n*.

Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the tensor product, or *reduced* cohomology groups from the cochain complex made by using a Hom functor, can be applied.

An **exact sequence** is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

**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 and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to 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 functions on the group of chains in homology theory.

In mathematics, **group cohomology** is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group *G* in an associated *G*-module*M* to elucidate the properties of the group. By treating the *G*-module as a kind of topological space with elements of representing *n*-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group *G* and *G*-module *M* themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called *group homology*. The techniques of group cohomology can also be extended to the case that instead of a *G*-module, *G* acts on a nonabelian *G*-group; in effect, a generalization of a module to non-Abelian coefficients.

In algebraic topology, the **Betti numbers** are used to distinguish topological spaces based on the connectivity of *n*-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces, the sequence of Betti numbers is 0 from some point onward, and they are all finite.

In mathematics, particularly algebraic topology and homology theory, the **Mayer–Vietoris sequence** is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

In algebraic topology, a branch of mathematics, **singular homology** refers to the study of a certain set of algebraic invariants of a topological space *X*, the so-called **homology groups** Intuitively, singular homology counts, for each dimension *n*, the *n*-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In algebraic topology, a branch of mathematics, the **(singular) homology** of a topological space **relative to** a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

In mathematics, **cellular homology** in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.

In algebraic topology the **cap product** is a method of adjoining a chain of degree *p* with a cochain of degree *q*, such that *q* ≤ *p*, to form a composite chain of degree *p* − *q*. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.

In mathematics, **Hochschild homology ** is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).

In mathematics, **Maass forms** or **Maass wave forms** are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.

In mathematics, and more specifically in homological algebra, a **resolution** is an exact sequence of modules, which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a **finite resolution** is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object or the rightmost object is the zero-object.

In mathematics, a **Δ-set***S*, often called a **semi-simplicial set**, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.

In mathematics, **cohomology with compact support** refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.

In mathematics, specifically in surgery theory, the **surgery obstructions** define a map from the normal invariants to the L-groups which is in the first instance a set-theoretic map with the following property when :

In mathematics, the **Bockstein spectral sequence** is a spectral sequence relating the homology with mod *p* coefficients and the homology reduced mod *p*. It is named after Meyer Bockstein.

In algebraic topology and graph theory, **graph homology** describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1-complex, the only non-trivial homology groups are the 0-th group and the 1-th group.

In algebraic topology, **homological connectivity** is a property describing a topological space based on its homology groups. This property is related, but more general, than the properties of graph connectivity and topological connectivity. There are many definitions of homological connectivity of a topological space *X*.

- Hatcher, A., (2002)
*Algebraic Topology*Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.

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