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Concept
Reduced homology
Summary
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres).
If P is a single-point space, then with the usual definitions the integral homology group
H0(P)
is isomorphic to (an infinite cyclic group), while for i ≥ 1 we have
Hi(P) = {0}.
More generally if X is a simplicial complex or finite CW complex, then the group H0(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.
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Singular homology
In algebraic topology, 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 (see also the related theory simplicial homology).
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with 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. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.