algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
The homology groups/cohomology groups of a (reduced) Whitehead-generalized homology theory/cohomology theory evaluated on an long homotopy cofiber sequence of pointed spaces form a long exact sequence of abelian groups.
Let $X \overset{f}{\longrightarrow} Y$ be a morphism of pointed homotopy types (typically presented as a continuous function of pointed topological spaces or a morphism of pointed simplicial sets) and write
for its induced long homotopy cofiber sequence, where $C_f \coloneqq cof(f)$ denotes the homotopy cofiber of $f$ and $\Sigma(-)$ is reduced suspension. (If $f$ is presented by an inclusion/cofibration of cell complexes in the classical model structure on topological spaces or the classical model structure on simplicial sets then $C_f \,\simeq\, Y/X$ is represented simply by the quotient space of $Y$ by $X$ under $f$. )
Now for $E$ a Whitehead-generalized homology theory its evaluation on the sequence (1) yields a long exact sequence of abelian groups – the reduced $E$-homology groups:
Dually, for $E$ a Whitehead-generalized cohomology theory its evaluation on the sequence (1) yields a contravariant long exact sequence of abelian groups – the $E$-cohomology groups:
Here $f^\ast$ denotes pullback in cohomology and $\delta^\ast$ is known as a connecting homomorphism.
That we have these long exact sequences in (co)homology is, depending on perspective:
an axiom in Whitehead-generalized homology theory/cohomology theory;
a consequence of the long exact sequence of homotopy groups under the identifications
$\widetilde E_\bullet(X) \;\simeq\; \pi_\bullet \big( \Sigma^\infty X \wedge E \big)$
$\widetilde E^\bullet(X) \;\simeq\; \pi_{-\bullet} Maps\big( \Sigma^\infty X, E \big)$,
where
$E$ denotes now the spectrum representing the (co)homology theory,
$\pi_\bullet(-)$ denotes the homotopy groups of spectra,
applied here to smash product of spectra $(-)\wedge E$ and to mapping spectra $Maps(-,E)$, respectively.
The second perspective, via representing spectra, makes manifest that we also have long exact sequences over a fixed space $X$, but now induced from a homotopy cofiber sequence (equivalently a homotopy fiber sequence by this Prop.)
of cohomology operations of coefficient theories (of spectra), in that
and
are long exact sequences of abelian groups.
On a homotopy fiber sequence of spaces (instead of a cofiber sequence) there is, in general, no exact sequence of cohomolgy groups; but instead there is
However, in special cases this reduces again to a long exact sequence in cohomology:
the Serre long exact sequence for sufficiently highly connected base and fiber spaces;
the Thom-Gysin sequence for spherical fibrations.
Textbook accounts:
For more references see those at Whitehead-generalized cohomology theory.
Last revised on January 17, 2021 at 06:06:05. See the history of this page for a list of all contributions to it.