nLab Serre long exact sequence

Idea

A Serre long exact sequence of a Serre fibration is a long exact sequences of ordinary cohomology/ordinary homology groups associated with a Serre fibration with sufficiently highly connected base and fibers. It arises as a special case of the information contained in the corresponding Serre spectral sequence.

Statement

Proposition

Let

$$\array{ F &\overset{i}{\longrightarrow}& X \\ && \downarrow^{\mathrlap{p}} \\ && B }$$

be a Serre fibration such that

1. the base space $B$ is $(n_1-1)$-connected for $n_1 \geq 2$;

2. the fiber $F$ is $(n_2-1)$-connected, for $n_1 \geq 1$;

then for every abelian group $A$ there is a long exact sequence of ordinary homology groups of the form

$$H_{n_1 + n_2 - 1}(F,A) \overset{i_\ast}{\longrightarrow} H_{n_1 + n_2 - 1}(X,A) \overset{p_\ast}{\longrightarrow} H_{n_1 + n_2 - 1}(B,A) \overset{\tau}{\longrightarrow} H_{n_1 + n_2 - 2}(F,A) \overset{i_\ast}{\longrightarrow} \cdots \overset{i_\ast}{\longrightarrow} H_1(X,A) \,.$$

Proof

Consider the homology Serre spectral sequence of the given fibration

$$E_2 = H_q(B, H_p(F,A)) \;\Rightarrow\; H_\bullet(X,A) \,.$$

By the connectedness assumptions and the Hurewicz theorem, then

$$H_{0 \lt \bullet \lt n_1}(B,-) = 0$$

and

$$H_{0 \lt \bullet \lt n_2}(F,-) = 0$$

and hence the only possibly non-vanishing groups on the second page of the spectral sequence (hence on every higher page) in total degree $k \leq n_1 + n_2 - 1$ are $E^r_{k,0}$ and $E^r_{0,k}$.

On the second page these groups are

$$E^2_{k,0} \simeq H_k(B,H_0(F,A)) \simeq H_k(B,A)$$

$$E^2_{0,k} \simeq H_0(B,H_k(F,A)) \simeq H_k(F,A)$$

by the fact that both $B$ and $F$ are connected by assumption.

Now, since the differentials on the $r$th page have bidegree $(-r,r+1)$, it follows that the only possibly non-vanishing differential on the $k$th page for $k \leq n_1 + n_2$ is

$$\partial_k \;\colon\; E^k_{k,0} \longrightarrow E^k_{0,k-1} \,.$$

This being so, it follows that the above non-vanishing groups on the $E^2$-page in total degree $k$ remain intact up to these pages, hence that

$$E^k_{k,0} \simeq H_k(B,A) \;\;\;\;\; \,, \;\;\;\;\; E^k_{0,k-1} \simeq H_{k-1}(F,A) \,.$$

Moreover, by convergence of the spectral sequence there are exact sequences of the form

$$0 \to E^\infty_{k,0} \longrightarrow E^k_{k,0} \overset{\partial_k}{\longrightarrow} E^k_{0,k-1} \longrightarrow E^\infty_{0,k-1} \to 0 \,,$$

hence, by the previous statement, of the form

$$0 \to E^\infty_{k,0} \longrightarrow H_k(B,A) \overset{\partial_k}{\longrightarrow} H_{k-1}(F,A) \longrightarrow E^\infty_{0,k-1} \to 0 \,,$$

Finally, by convergence and the filtering condition, the only non-vanishing filter contributions to $H_k(X,A)$ in degree $k \leq n_1 + n_2 - 1$ are in filtering degree 0 and $k$, and so projection to filtering degree $k$ gives a short exact sequence of the form

$$0 \to E^\infty_{0,k} \longrightarrow H_k(X,A) \longrightarrow E^\infty_{k,0} \to 0 \,.$$

Splicing together the exact sequences thus obtained yields the long exact sequence in question:

$$\array{ \cdots &\to& H_k(F,A) && \longrightarrow && H_k(X,A) && \longrightarrow && H_k(B,A) &\overset{}{\longrightarrow}& H_{k-1}(F,A) &\to& \cdots \\ && & \searrow && \nearrow & & \searrow && \nearrow \\ && && E^\infty_{0,k} && && E^\infty_{k,0} }$$

Examples and Applications

Proposition

For $X$ an n-connected topological space, then for $k \leq 2n$ there are isomorphisms

$$H_k(X) \simeq H_{k-1}(\Omega X)$$

between the ordinary homology of $X$ in degree $k$ and the ordinary homology of the loop space of $X$ in degree $k-1$.

Proof

The Serre long exact sequence from prop. applied to the based path space Serre fibration of $X$

$$\Omega X \longrightarrow Path_\ast(X) \longrightarrow X$$

is of the form

$$H_{2n}(\Omega X) \overset{i_\ast}{\longrightarrow} H_{2n}(Path_\ast(X)) \overset{p_\ast}{\longrightarrow} H_{2n}(X) \overset{\tau}{\longrightarrow} H_{2n-1}(\Omega X) \overset{i_\ast}{\longrightarrow} \cdots \overset{i_\ast}{\longrightarrow} H_1(X) \,.$$

Since $Path_\ast(X)$ is contractible, every third group in this sequence vanishes, and hence exactness gives the isomorphisms

$$\tau \colon H_k(X) \simeq H_{k-1}(\Omega X)$$

for $k \leq 2n$.

Related concepts

• Serre spectral sequence

• Thom-Gysin sequence

• long exact sequence in cohomology

• long exact sequence of homotopy groups

References

• Stanley Kochmann, prop. 3.2.1 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• John McCleary, example 5.D of_A User’s Guide to Spectral Sequences_, Cambridge University Press (2001)