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 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 denotes the homotopy cofiber of and is reduced suspension. (If 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 is represented simply by the quotient space of by under . )
Now for a Whitehead-generalized homology theory its evaluation on the sequence (1) yields a long exact sequence of abelian groups – the reduced -homology groups:
Dually, for a Whitehead-generalized cohomology theory its evaluation on the sequence (1) yields a contravariant long exact sequence of abelian groups – the -cohomology groups:
Here denotes pullback in cohomology and 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
,
where
denotes now the spectrum representing the (co)homology theory,
denotes the homotopy groups of spectra,
applied here to smash product of spectra and to mapping spectra , respectively.
The second perspective, via representing spectra, makes manifest that we also have long exact sequences over a fixed space , 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 11:06:05. See the history of this page for a list of all contributions to it.