Reduced homology
Encyclopedia
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
is an infinite cyclic group, while for i ≥ 1 we have
More generally if X is a simplicial complex
or finite CW complex
, then the group H0(X) is the free abelian group
on generators the connected component
s of X. 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.
A more fundamental way to do the same thing is to go back to the chain complex
defining homology, and tweak the C0 term in it. Namely, define the augmentation ε from C0 to the integers, which expresses the sum of coefficients. Replace C0 by the kernel of ε. Then calculate homology groups as usual, with the modified chain complex. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, reduced homology is a minor modification made to homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although 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
Alexander duality
In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin...
being an example).
If P is a single-point space, then with the usual definitions the integral homology group
- H0(P)
is an infinite cyclic group, while for i ≥ 1 we have
- Hi(P) = {0}.
More generally if X is a simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
or finite CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
, then the group H0(X) is the free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
on generators the connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
s of X. 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.
A more fundamental way to do the same thing is to go back to the chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
defining homology, and tweak the C0 term in it. Namely, define the augmentation ε from C0 to the integers, which expresses the sum of coefficients. Replace C0 by the kernel of ε. Then calculate homology groups as usual, with the modified chain complex. Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
, or reduced cohomology groups from the cochain complex made by using a Hom functor
Hom functor
In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
, can be applied.