nLab
long exact sequence in generalized homology

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Contents

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 XfYX \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)XfYC fδΣXΣfΣY \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 fcof(f)C_f \coloneqq cof(f) denotes the homotopy cofiber of ff and Σ()\Sigma(-) is reduced suspension. (If ff 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 fY/XC_f \,\simeq\, Y/X is represented simply by the quotient space of YY by XX under ff. )

Now for EE a Whitehead-generalized homology theory its evaluation on the sequence (1) yields a long exact sequence of abelian groups – the reduced EE-homology groups:

E˜ (X)f *E˜ (Y)E˜ (C f)δ *E˜ 1(X)Σf *E˜ 1(Y) \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 EE a Whitehead-generalized cohomology theory its evaluation on the sequence (1) yields a contravariant long exact sequence of abelian groups – the EE-cohomology groups:

E˜ (X)f *E˜ (Y)E˜ (C f)δ *E˜ 1(X)Σf *E˜ 1(Y) \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 *f^\ast denotes pullback in cohomology and δ *\delta^\ast is known as a connecting homomorphism.

That we have these long exact sequences in (co)homology is, depending on perspective:

The second perspective, via representing spectra, makes manifest that we also have long exact sequences over a fixed space XX, but now induced from a homotopy cofiber sequence (equivalently a homotopy fiber sequence by this Prop.)

EϕFGΣE \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

E˜ (X)ϕ *F˜ (X)G˜ (X) \cdots \to \widetilde E_\bullet(X) \overset{\phi_\ast}{\longrightarrow} \widetilde F_\bullet(X) \overset{}{\longrightarrow} \widetilde G_\bullet(X) \to \cdots

and

E˜ (X)ϕ *F˜ (X)G˜ (X) \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

References

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.