group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
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.
Let
be a Serre fibration such that
the base space $B$ is $(n_1-1)$-connected for $n_1 \geq 2$;
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
Consider the homology Serre spectral sequence of the given fibration
By the connectedness assumptions and the Hurewicz theorem, then
and
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
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
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
Moreover, by convergence of the spectral sequence there are exact sequences of the form
hence, by the previous statement, of the form
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
Splicing together the exact sequences thus obtained yields the long exact sequence in question:
For $X$ ab n-connected topological space, then for $k \leq 2n$ there are isomorphisms
between the ordinary homology of $X$ in degree $k$ and the ordinary homology of the loop space of $X$ in degree $k-1$.
The Serre long exact sequence from prop. applied to the based path space Serre fibration of $X$
is of the form
Since $Path_\ast(X)$ is contractible, every third group in this sequence vanishes, and hence exactness gives the isomorphisms
for $k \leq 2n$.
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)
Last revised on January 17, 2021 at 11:04:06. See the history of this page for a list of all contributions to it.