Everything about Homology Mathematics totally explained
In
mathematics (especially
algebraic topology and
abstract algebra),
homology (in
Greek homos = identical) is a certain general procedure to associate a
sequence of
abelian groups or
modules with a given mathematical object such as a
topological space or a
group. See
homology theory for more background, or
singular homology for a concrete version for topological spaces, or
group cohomology for a concrete version for groups.
For a topological space, the homology groups are generally much easier to compute than the
homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.
Construction of homology groups
The procedure works as follows: Given an object such as a topological space
, one first defines a
chain complex
that encodes information about
.
A chain complex is a sequence of abelian groups or modules
connected by
homomorphisms The latter are called
connecting homomorphisms and are provided by the
snake lemma.
Further Information
Get more info on 'Homology Mathematics'.
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