In
mathematics, in the field of
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–Moore spectral sequence addresses the calculation of the
homology group
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 topolog ...
s of a
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
over a
fibration. The
spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces.
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to ...
and
John C. Moore's original paper addresses this for
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 H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
.
Motivation
Let
be a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and let
and
denote
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 H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
and
singular cohomology
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 ...
with coefficients in ''k'', respectively.
Consider the following pullback
of a continuous map ''p'':
:
A frequent question is how the homology of the fiber product,
, relates to the homology of ''B'', ''X'' and ''E''. For example, if ''B'' is a point, then the pullback is just the usual product
. In this case the
Künneth formula Künneth is a surname. Notable people with the surname include:
* Hermann Künneth (1892–1975), German mathematician
* Walter Künneth (1901–1997), German Protestant theologian
{{DEFAULTSORT:Kunneth
German-language surnames ...
says
:
However this relation is not true in more general situations. The Eilenberg−Moore spectral sequence is a device which allows the computation of the (co)homology of the fiber product in certain situations.
Statement
The Eilenberg−Moore spectral sequences generalizes the above isomorphism to the situation where ''p'' is a
fibration of topological spaces and the base ''B'' is
simply connected. Then there is a convergent spectral sequence with
:
This is a generalization insofar as the zeroeth
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to con ...
is just the tensor product and in the above special case the cohomology of the point ''B'' is just the coefficient field ''k'' (in degree 0).
Dually, we have the following homology spectral sequence:
:
Indications on the proof
The spectral sequence arises from the study of
differential graded objects (
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
es), not spaces. The following discusses the original homological construction of Eilenberg and Moore. The cohomology case is obtained in a similar manner.
Let
:
be the
singular chain
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 H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
functor with coefficients in
. By the
Eilenberg–Zilber theorem,
has a differential graded
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
structure over
with
structure maps
:
In down-to-earth terms, the map assigns to a singular chain ''s'': ''Δ
n'' → ''B'' the composition of ''s'' and the diagonal inclusion ''B'' ⊂ ''B'' × ''B''. Similarly, the maps
and
induce maps of differential graded coalgebras
,
.
In the language of
comodule In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.
Formal definition
Let ''K'' be a field, an ...
s, they endow
and
with differential graded comodule structures over
, with structure maps
:
and similarly for ''E'' instead of ''X''. It is now possible to construct the so-called
cobar resolution for
:
as a differential graded
comodule. The cobar resolution is a standard technique in differential homological algebra:
:
where the ''n''-th term
is given by
:
The maps
are given by
:
where
is the structure map for
as a left
comodule.
The cobar resolution is a
bicomplex, one degree coming from the grading of the chain complexes ''S''
∗(−), the other one is the simplicial degree ''n''. The
total complex of the bicomplex is denoted
.
The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map
:
that induces a
quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms
:H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bu ...
(i.e. inducing an isomorphism on homology groups)
where
is the
cotensor product and Cotor (cotorsion) is the
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
for the
cotensor product.
To calculate
:
,
view
:
as a
double complex.
For any bicomplex there are two
filtrations (see or the
spectral sequence of a filtered complex); in this case the Eilenberg−Moore spectral sequence results from filtering by increasing homological degree (by columns in the standard picture of a spectral sequence). This filtration yields
:
These results have been refined in various ways. For example, refined the convergence results to include spaces for which
:
acts
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
ly on
:
for all
and further generalized this to include arbitrary pullbacks.
The original construction does not lend itself to computations with other homology theories since there is no reason to expect that such a process would work for a homology theory not derived from chain complexes. However, it is possible to axiomatize the above procedure and give conditions under which the above spectral sequence holds for a general (co)homology theory, see Larry Smith's original work or the introduction in .
References
*
*
*
*
*
*
Further reading
* Allen Hatcher, Spectral Sequences in Algebraic Topology, Ch 3. Eilenberg–MacLane Space
{{DEFAULTSORT:Eilenberg-Moore spectral sequence
Spectral sequences
Homology theory