algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
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The Serre spectral sequence or Leray-Serre spectral sequence is a spectral sequence for computation of ordinary cohomology (ordinary homology) of topological spaces in a Serre-fiber sequence of topological spaces.
(ordinary cohomology Serre spectral sequence)
Given a homotopy fiber sequence
over a connected topological space $X$, such that the canonical group action of the fundamental group $\pi_1(X)$ on the ordinary cohomology of the fiber $F$ is trivial (for instance if $X$ is a simply connected topological space), then there exists a cohomology Serre spectral sequence of the form:
(e.g. Hatcher, Thm. 1.14)
Hence for $n \in \mathbb{N}$ we have a filtration of abelian groups
where
hence that – iteratively as $p$ decreases – $F^n_p$ is an extension of $E^{p,n-p}_\infty$ by $F^n_{p+1}$.
The generalization of this from ordinary cohomology to generalized (Eilenberg-Steenrod) cohomology is the Atiyah-Hirzebruch spectral sequence, see there for details.
There are two kinds of relative Serre spectral sequences.
For $F \to E \to X$ as above and $A \hookrightarrow X$ a subspace, the induced restriction of the fibration
induces a spectral sequence in relative cohomology of the base space of the form
(e.g. Davis 91, theorem 9.33)
Conversely, for
a sub-fibration over the same base, then this induces a spectral sequence for relative cohomology of the the total space in terms of ordinary cohomology with coefficients in the relative cohomology of the fibers:
(e.g. Kochman 96, theorem 2.6.3, Davis 91, theorem 9.34)
There is also a generalization to equivariant cohomology: for cohomology with coefficients in a Mackey functor withRO(G)-grading for representation spheres $S^V$, then for $E \to X$ an $F$-fibration of topological G-spaces and for $A$ any $G$-Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):
where on the left in the $E_2$-page we have ordinary cohomology with coefficients in the genuine equivariant cohomology groups of the fiber.
For details on the plain Serre spectral sequence see at Atiyah-Hirzebruch spectral sequence and take $E = H R$ to be ordinary cohomology.
(integral cohomology of homotopy quotient of 4-sphere by finite subgroup of SU(2))
Let
$G \xhookrightarrow{\;i\;} Sp(1) \simeq SU(2) \simeq Spin(3)$ be a finite subgroup of SU(2);
$\mathbb{H} \,\in\, G Actions(VectorSpaces_{\mathbb{R}})$ the linear representation of $G$ on the quaternions, regarded as a 4d real vector space, which is the restricted representation of the defining representation of Sp(1);
$S^{\mathbb{H}} \in G Actions(TopologicalSpaces)$ the corresponding representation sphere.
Then the integral cohomology in degree 4 of the homotopy quotient is the direct sum
of the integers with the cyclic group of order that of $G$.
By the Borel construction we have a homotopy fiber sequence of the form
over the classifying space of $G$.
Here the integral cohomology of the 4-sphere fiber is (e.g. by the nature of the Eilenberg-MacLane space $K(\mathbb{Z},4)$)
We claim that the group action of $\pi_1(B G) \simeq G$ (by this Prop.) on the integral cohomology of the fiber is trivial. This follows by observing that:
we have an isomorphism of topological G-spaces between the representation sphere of $\mathbb{H}$ and the unit sphere in $\mathbb{R} \oplus \mathbb{H}$ (by this Prop.):
the group action of Sp(1) on $\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4$ is through the defining action of SO(4), hence the action on $\mathbb{R} \oplus \mathbb{H}$ is through SO(5),
because quaternions are a normed division algebra, so that left-multiplication by unit-norm quaternions $q \in$ Sp(1) $= S(\mathbb{H})$ preserves the norm (e.g HSS 18, Rem. A.8);
the generator of $H^4(S^4,\mathbb{Z})$ may be identified with the volume form (under the Hopf degree theorem and the de Rham theorem) which is manifestly preserved by the action of the special orthogonal group $SO(5)$.
Therefore, the integral-cohomological Serre spectral sequence (Prop. ) applies to the Borel fiber sequence (4).
Now, noticing that the integral cohomology of a classifying space of a discrete group is its group cohomology
we have for the given finite subgroup of SU(2) (by this Prop) that:
where $G^{ab} \coloneqq G / [G,G]$ denotes the abelianization of $G$.
Using the cohomology groups (5) and (6) in the fomula (1) for the second page $E_2^{\bullet, \bullet}$ of the cohomology Serre spectral sequence (Prop. ) shows that this is of the following form:
Since the codomains of the differentials on all the following pages are translated diagonally (downwards and rightwards, by the general formula) from the codomains seen above, one sees that for every differential on every page, the domain or the codomain is the zero group.
This means that all differentials are the zero morphism, hence that the spectral sequence collapses already on this second page:
Therefore the convergence statement (2) says that the degree-4 cohomology group in question is a group extension of
by
in that we have a short exact sequence of the form
But since the Ext-group of the integers is trivial (this Expl.) this extension must be the direct sum
This is the claim (3) to be proven.
The original article is
Textbook accounts:
Alan Hatcher, Spectral sequences in algebraic topology I: The Serre spectral sequence (pdf, pdf)
Stanley Kochman, section 2.2. of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Davis, Lecture notes in algebraic topology, 1991
Lecture notes etc. includes
Discussion in homotopy type theory includes
and implementation in Lean is in
Floris van Doorn, Egbert Rijke, Ulrik Buchholtz, Favonia, Steve Awodey, Jeremy Avigad, Mike Shulman, Jonas Frey, Spectral (github.com/cmu-phil/Spectral)
Floris van Doorn, §5 in: On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory (2018) [arXiv:1808.10690]
In equivariant cohomology, for Bredon cohomology:
and for genuine equivariant cohomology, i.e. for RO(G)-graded cohomology with coefficients in a Mackey functor:
See also
Last revised on January 1, 2023 at 19:54:57. See the history of this page for a list of all contributions to it.