In
mathematics, specifically in
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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the Eilenberg–Steenrod axioms are properties that
homology theories
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 topol ...
of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s have in common. The quintessential example of a homology theory satisfying the axioms is
singular homology, developed by
Samuel Eilenberg and
Norman Steenrod.
One can define a homology theory as a
sequence of
functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the
Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.
[http://www.math.uiuc.edu/K-theory/0245/survey.pdf ]
If one omits the dimension axiom (described below), then the remaining axioms define what is called an
extraordinary homology theory
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 ...
. Extraordinary cohomology theories first arose in
K-theory and
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same d ...
.
Formal definition
The Eilenberg–Steenrod axioms apply to a sequence of functors
from the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
of
pairs
Concentration, also known as Memory, Shinkei-suijaku (Japanese meaning "nervous breakdown"), Matching Pairs, Match Match, Match Up, Pelmanism, Pexeso or simply Pairs, is a card game in which all of the cards are laid face down on a surface and tw ...
of topological spaces to the category of abelian
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
s, together with a
natural transformation called the boundary map (here
is a shorthand for
. The axioms are:
# Homotopy: Homotopic maps induce the same map in homology. That is, if
is
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
to
, then their induced
homomorphisms are the same.
#
Excision: If
is a pair and ''U'' is a subset of ''A'' such that the closure of ''U'' is contained in the interior of ''A'', then the inclusion map
induces an
isomorphism in homology.
# Dimension: Let ''P'' be the one-point space; then
for all
.
# Additivity: If
, the disjoint union of a family of topological spaces
, then
# Exactness: Each pair ''(X, A)'' induces a
long exact sequence in homology, via the inclusions
and
:
::
If ''P'' is the one point space, then
is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.
Consequences
Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.
The homology of some relatively simple spaces, such as
n-spheres, can be calculated directly from the axioms. From this it can be easily shown that the (''n'' − 1)-sphere is not a
retract of the ''n''-disk. This is used in a proof of the
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simple ...
.
Dimension axiom
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an
extraordinary homology theory
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 ...
(dually,
extraordinary cohomology theory). Important examples of these were found in the 1950s, such as
topological K-theory and
cobordism theory, which are extraordinary ''co''homology theories, and come with homology theories dual to them.
See also
*
Zig-zag lemma
Notes
References
*
*
*
{{DEFAULTSORT:Eilenberg-Steenrod axioms
Homology theory
Mathematical axioms