In
mathematics, the Serre
spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
Life and career
He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
in the
Leray spectral sequence) is an important tool 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 ...
. It expresses, in the language of
homological algebra, the singular (co)homology of the total space ''X'' of a (Serre)
fibration in terms of the (co)homology of the
base space ''B'' and the fiber ''F''. The result is due to
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
in his doctoral dissertation.
Cohomology spectral sequence
Let
be a
Serre fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
of topological spaces, and let ''F'' be the (path-connected)
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
. The Serre cohomology spectral sequence is the following:
:
Here, at least under standard simplifying conditions, the coefficient group in the
-term is the ''q''-th
integral cohomology group of ''F'', and the outer group is the
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 ...
of ''B'' with coefficients in that group.
Strictly speaking, what is meant is cohomology with respect to the
local coefficient system on ''B'' given by the cohomology of the various fibers. Assuming for example, that ''B'' is
simply connected, this collapses to the usual cohomology. For a
path connected base, all the different fibers are
homotopy equivalent
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 deforma ...
. In particular, their cohomology is isomorphic, so the choice of "the" fiber does not give any ambiguity.
The
abutment
An abutment is the substructure at the ends of a bridge span or dam supporting its superstructure. Single-span bridges have abutments at each end which provide vertical and lateral support for the span, as well as acting as retaining walls ...
means integral cohomology of the total space ''X''.
This spectral sequence can be derived from an
exact couple In mathematics, an exact couple, due to , is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple.
For the definition ...
built out of the
long exact sequences of the cohomology of the pair
, where
is the restriction of the fibration over the ''p''-skeleton of ''B''. More precisely, using
this notation,
:
''f'' is defined by restricting each piece on
to
, ''g'' is defined using the coboundary map in the
long exact sequence of the pair, and ''h'' is defined by restricting
to
There is a multiplicative structure
:
coinciding on the ''E''
2-term with (−1)
''qs'' times the cup product, and with respect to which the differentials
are
(graded) derivations inducing the product on the
-page from the one on the
-page.
Homology spectral sequence
Similarly to the cohomology spectral sequence, there is one for homology:
:
where the notations are dual to the ones above.
Example computations
Hopf fibration
Recall that the Hopf fibration is given by
. The
-page of the Leray–Serre Spectral sequence reads
:
The differential
goes
down and
right. Thus the only differential which is not necessarily is , because the rest have domain or codomain ''0'' (since they are on the ''E''
2-page). In particular, this sequence degenerates at ''E''
2 = ''E''
∞. The ''E''
3-page reads
:
The spectral sequence abuts to
i.e.
Evaluating at the interesting parts, we have
and
Knowing the cohomology of
both are zero, so the differential
is an isomorphism.
Sphere bundle on a complex projective variety
Given a complex ''n''-dimensional projective variety there is a canonical family of line bundles
for
coming from the embedding
. This is given by the global sections
which send
: