Contents

# Contents

## 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:

• 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

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

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

## References

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.