nLab
Serre long exact sequence

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Cohomology

cohomology

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Algebraic topology

Contents

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

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

be a Serre fibration such that

  1. the base space BB is (n 11)(n_1-1)-connected for n 12n_1 \geq 2;

  2. the fiber FF is (n 21)(n_2-1)-connected, for n 11n_1 \geq 1;

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

H n 1+n 21(F,A)i *H n 1+n 21(X,A)p *H n 1+n 21(B,A)τH n 1+n 22(F,A)i *i *H 1(X,A). 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))H (X,A). 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<<n 1(B,)=0 H_{0 \lt \bullet \lt n_1}(B,-) = 0

and

H 0<<n 2(F,)=0 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 kn 1+n 21k \leq n_1 + n_2 - 1 are E k,0 rE^r_{k,0} and E 0,k rE^r_{0,k}.

On the second page these groups are

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

by the fact that both BB and FF are connected by assumption.

Now, since the differentials on the rrth page have bidegree (r,r+1)(-r,r+1), it follows that the only possibly non-vanishing differential on the kkth page for kn 1+n 2k \leq n_1 + n_2 is

k:E k,0 kE 0,k1 k. \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 2E^2-page in total degree kk remain intact up to these pages, hence that

E k,0 kH k(B,A),E 0,k1 kH k1(F,A). 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

0E k,0 E k,0 k kE 0,k1 kE 0,k1 0, 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

0E k,0 H k(B,A) kH k1(F,A)E 0,k1 0, 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)H_k(X,A) in degree kn 1+n 21k \leq n_1 + n_2 - 1 are in filtering degree 0 and kk, and so projection to filtering degree kk gives a short exact sequence of the form

0E 0,k H k(X,A)E k,0 0. 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:

H k(F,A) H k(X,A) H k(B,A) H k1(F,A) E 0,k E k,0 \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 XX ab n-connected topological space, then for k2nk \leq 2n there are isomorphisms

H k(X)H k1(ΩX) H_k(X) \simeq H_{k-1}(\Omega X)

between the ordinary homology of XX in degree kk and the ordinary homology of the loop space of XX in degree k1k-1.

Proof

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

ΩXPath *(X)X \Omega X \longrightarrow Path_\ast(X) \longrightarrow X

is of the form

H 2n(ΩX)i *H 2n(Path *(X))p *H 2n(X)τH 2n1(ΩX)i *i *H 1(X). 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 *(X)Path_\ast(X) is contractible, every third group in this sequence vanishes, and hence exactness gives the isomorphisms

τ:H k(X)H k1(ΩX) \tau \colon H_k(X) \simeq H_{k-1}(\Omega X)

for k2nk \leq 2n.

References

Last revised on May 10, 2016 at 10:31:00. See the history of this page for a list of all contributions to it.