Serre spectral sequence

For a fibration $F\to E\to B$ with suitable connectivity, the cohomological Serre spectral sequence haswith differential $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$.

For a Serre fibration, this first-quadrant spectral sequence has differential of bidegree $(r,1-r)$ and a fiber-cohomology local coefficient system on the base. A simply connected base makes that system constant. Its edge map is restriction to the fiber; a surviving fiber class is the restriction of a total-space class. For commutative ring coefficients it is multiplicative and its differentials are graded derivations.

There is no fiber bundle $F\to S^m\to S^q$ when $m>q\ge 3$ and $q$ is odd. A local trivialization makes $F$ a retract of an open subset of $S^m$, so $H^j(F;\mathbb{Q})=0$ for $j>m$; the long exact sequence of homotopy groups of a fibration makes $F$ connected. In the multiplicative rational Serre spectral sequence, only base columns zero and $q$ occur. Since $H^q(S^m)=0$, a fibre class $a\in H^{q-1}(F;\mathbb{Q})$ must satisfy $d_{q}a=u$, the base generator. Its degree is even. If $a^s=0$ is its first vanishing power, the derivation law gives $0=d_q(a^s)=su{a}^{s-1}\ne 0$ on the $E_q$ page, a contradiction. Bounded fibre cohomology ensures such a vanishing power exists. In particular $S^4$ cannot be the total space of a bundle over $S^3$.

A Serre fibration is a map having the homotopy lifting property with respect to every disc, equivalently every CW complex.

For a based Serre fibration with fibre $F$ over the basepoint, lift a based cube in $B$ from its constant bottom face, keeping the side faces fixed. Its final face lies in $F$, giving surjectivity of the relative map. Homotopies in $B$ lift; two lifts of the same cube are relatively homotopic by lifting with both side liftings prescribed. This gives injectivity. For $n\ge 2$ this is a group isomorphism; at $n=1$ it is initially a bijection of pointed sets, unless the relative set is endowed with the group law transported through the fibration.

For a Serre fibration over $S^r$, cover the base by a contractible punctured sphere and a contractible ball. Excision relative to the preimage of the punctured sphere identifies the relative pair with the product of $F$ and $(D^r,S^{r-1})$ up to homotopy. The base pair has a single free homology group in degree $r$, so the Künneth theorem gives the displayed shift without a torsion term. The relative long exact sequence then gives $\cdots \to H_i(F)\to H_i(E)\to H_{i-r}(F)\to H_{i-1}(F)\to \cdots$. The connecting map retains the twisting of the original Serre fibration.

For a Serre fibration with this fiber over a simply connected CW base, the fiber identity class has only one possible nonzero outgoing differential in the cohomological Serre spectral sequence, landing in $H^{n+1}(B;G)$. If that group vanishes, the class extends to the CW total space. Its representing map, paired with projection to $B$, is an isomorphism on fiber and base homotopy groups and hence a weak homotopy equivalence to the product.

For a map $f:X\to B$, let $E_f$ consist of points of $X$ and paths starting at their images, with projection $p(x,\gamma )=\gamma (1)$. Appending segments of a base homotopy proves the homotopy lifting property, so $p$ is a Serre fibration. The constant-path inclusion $s:X\to E_f$ satisfies $ps=f$ and is a homotopy equivalence: forget the path for an inverse, and shorten paths to prove the homotopy. The homotopy need not preserve the endpoint in $B$, so the statement concerns an equivalence of total spaces over a commuting triangle, not a fibrewise inverse. The fibre over $b_0$ is the homotopy fibre of $f$.

A map $p:E\to B$ has the homotopy lifting property with respect to $X$ when every homotopy $X\times I\to B$ whose initial map lifts to $E$ has a lift extending that initial lift.

For a based fibration $F\to E\to B$, lifting representatives defines connecting maps and an exact sequence

For a fibration over $S^n$, the only possible differential in the homological Serre spectral sequence is $d_n$. Identifying its two nonzero columns with the homology of the fiber gives the Wang homomorphism $\Delta$.

For a fibration $F\to E\to S^n$, the Wang homomorphism belongs to the exact sequence

A transgression is a spectral-sequence differential from a fiber class on the vertical edge to a base class on the horizontal edge.

For a fibration $F\to E\to B$, classes $x\in H^r(F)$ and $y\in H^{r+1}(B)$ form a transgressive pair when their images in relative cohomology satisfy $\delta x={\pi }^{\ast }y$. In the Serre spectral sequence this relation is represented by $d_{r+1}x=y$.

Steenrod operations preserve transgressive pairs. With mod-two coefficients, if $(x,y)$ is transgressive, then so is $({\mathrm{Sq}}^{j}x,{\mathrm{Sq}}^{j}y)$ whenever the square is defined by the instability range. The corresponding statement holds for odd-primary reduced powers and the Bockstein, with the appropriate page shift.