nLab long exact sequence in generalized homology

Idea

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.

Definition

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

(1)$$\cdots \to X \overset{f}{\longrightarrow} Y \longrightarrow C_f \overset{\delta}{\longrightarrow} \Sigma X \overset{ \Sigma f }{ \longrightarrow } \Sigma Y \to \cdots$$

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:

$$\cdots \to \widetilde E_\bullet(X) \overset{f_\ast}{\longrightarrow} \widetilde E_\bullet(Y) \longrightarrow \widetilde E_\bullet(C_f) \overset{\delta_\ast}{\longrightarrow} \widetilde E_{\bullet-1}(X) \overset{ \Sigma f_\ast }{ \longrightarrow } \widetilde E_{\bullet-1}(Y) \to \cdots$$

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:

$$\cdots \leftarrow \widetilde E^\bullet(X) \overset{f^\ast}{\longleftarrow} \widetilde E^\bullet(Y) \longleftarrow \widetilde E^\bullet(C_f) \overset{\delta^\ast}{\longleftarrow} \widetilde E_{\bullet-1}(X) \overset{ \Sigma f^\ast }{ \longleftarrow} \widetilde E^{\bullet-1}(Y) \leftarrow \cdots$$

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.)

$$\cdots \to E \overset{\phi}{\longrightarrow} F \overset{}{\longrightarrow} G \overset{}{\longrightarrow} \Sigma E \to \cdots$$

of cohomology operations of coefficient theories (of spectra), in that

$$\cdots \to \widetilde E_\bullet(X) \overset{\phi_\ast}{\longrightarrow} \widetilde F_\bullet(X) \overset{}{\longrightarrow} \widetilde G_\bullet(X) \to \cdots$$

and

$$\cdots \to \widetilde E^\bullet(X) \overset{\phi_\ast}{\longrightarrow} \widetilde F^\bullet(X) \overset{}{\longrightarrow} \widetilde G^\bullet(X) \to \cdots$$

are long exact sequences of abelian groups.

Related concepts

• long exact sequence in homology

  • long exact sequence in chain homology

• long exact sequence of homotopy 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

  • the Serre spectral sequence.

  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.

References

Textbook accounts:

• Frank Adams, p. 197 in: Stable homotopy and generalised homology, Chicago Lectures in Mathematics, The University of Chicago Press 1974 (ucp:bo21302708, pdf)

For more references see those at Whitehead-generalized cohomology theory.