nLab
connecting homomorphism

Idea

Generally, a connecting homomorphism is a morphism of the kind produced by the snake lemma.

Specifically, when the double complex that goes into the snake lemma is regarded as part of a short exact sequence $A_{•}\to B_{•}\to C_{•}$ of chain complexes, then the connecting homomorphisms induce morphisms ${\delta }_n:H_n(C)\to H_{n-1}(A)$ on the homology groups of these chain complexes which exhibit the corresponding long exact sequence in homology of the form

$$\cdots \to H_n(A)\to H_n(B)\to H_n(C)\stackrel{{\delta }_n}{\to }{H}_{n-1}(A)\to H_{n-1}(B)\to H_{n-1}(C)\to \cdots .$$\cdots \to H_n(A) \to H_n(B) \to H_n(C) \stackrel{\delta_n}{\to} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to \cdots \,.

This long exact sequence is the image under chain homology

$$H_0(-):{\mathrm{Ch}}_{•}(𝒜)\to 𝒜$$H_0(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

of the long homotopy fiber sequence of chain complexes induced by the short exact sequence. Hence the connecting homomorphism is the image under $H_{•}(-)$ of a mapping cone inclusion on chain complexes.

For long (co)homology exact sequences

In the case that $𝒜\simeq R$Mod for some ring $R$, the construction of the connecting homomorphism for homology long exact sequences is easily described in terms of elements and checking its properties is elementary, see In terms of elements below. By the embedding theorems the general case can be reduced to this case. But there is also a general abstract description without recourse to elements, which we discuss further below in General abstract construction .

In terms of elements

Let $R$ be a commutative ring and let $𝒜=R$Mod. Write ${\mathrm{Ch}}_{•}(𝒜)$ for the category of chain complexes in $𝒜$.

Let

$$0\to A_{•}\stackrel{i}{\to }{B}_{•}\stackrel{p}{\to }{C}_{•}\to 0$$0 \to A_\bullet \stackrel{i}{\to} B_\bullet \stackrel{p}{\to} C_\bullet \to 0

be a short exact sequence in ${\mathrm{Ch}}_{•}(𝒜)$.

Definition

For $n\in \mathbb{Z}$, define a group homomorphism

$${\delta }_n:H_n(C)\to H_{n-1}(A),$$\delta_n : H_n(C) \to H_{n-1}(A) \,,

called the $n$th connecting homomorphism of the short exact sequence, by sending

$${\delta }_n:[c]\mapsto [{\partial }^B\hat{c}{]}_A,$$\delta_n : [c] \mapsto [\partial^B \hat c]_A \,,

where

1. $c\in Z_n(C)$ is a cycle representing a given homology group;

2. $\hat{c}\in C_n(B)$ is any lift of that cycle to an element in $B_n$, which exists because $p$ is a surjection (but which no longer needs to be a cycle itself);

3. $[{\partial }^B\hat{c}{]}_A$ is the $A$-homology class of ${\partial }^B\hat{c}$ which is indeed in $A_{n-1}\hookrightarrow B_{n-1}$ by exactness (since $p({\partial }^B\hat{c})={\partial }^{C}p(\hat{c})={\partial }^{C}c=0$) and indeed in $Z_{n-1}(A)\hookrightarrow A_{n-1}$ since ${\partial }^A{\partial }^B\hat{c}={\partial }^B{\partial }^B\hat{c}=0$.

Proposition

Def. 1 is indeed well defined in that the given map is independent of the choice of lift $\hat{c}$ involved and in that the group structure is respected.

Proof

To see that the constructon is well-defined, let $\tilde{c}\in B_n$ be another lift. Then $p(\hat{c}-\tilde{c})=0$ and hence $\hat{c}-\tilde{c}\in A_n\hookrightarrow B_n$. This exhibits a homology-equivalence $[{\partial }^B\hat{c}{]}_A\simeq [{\partial }^B\tilde{c}{]}_A$ since ${\partial }^A(\hat{c}-\tilde{c})={\partial }^B\hat{c}-{\partial }^B\tilde{c}$.

To see that ${\delta }_n$ is a group homomorphism, let $[c]=[c_1]+[c_2]$ be a sum. Then $\hat{c}≔{\hat{c}}_1+{\hat{c}}_2$ is a lift and by linearity of $\partial$ we have $[{\partial }^B\hat{c}{]}_A=[{\partial }^B{\hat{c}}_1]+[{\partial }^B{\hat{c}}_2]$.

Proposition

Under chain homology $H_{•}(-)$ the morphisms in the short exact sequence together with the connecting homomorphisms yield the homology long exact sequence

$$\cdots \to H_n(A)\to H_n(B)\to H_n(C)\stackrel{{\delta }_n}{\to }{H}_{n-1}(A)\to H_{n-1}(B)\to H_{n-1}(C)\to \cdots .$$\cdots \to H_n(A) \to H_n(B) \to H_n(C) \stackrel{\delta_n}{\to} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to \cdots \,.

Proof

Consider first the exactness of $H_n(A)\stackrel{H_n(i)}{\to }{H}_n(B)\stackrel{H_n(p)}{\to }{H}_n(C)$.

It is clear that if $a\in Z_n(A)\hookrightarrow Z_n(B)$ then the image of $[a]\in H_n(B)$ is $[p(a)]=0\in H_n(C)$. Conversely, an element $[b]\in H_n(B)$ is in the kernel of $H_n(p)$ if there is $c\in C_{n+1}$ with ${\partial }^{C}c=p(b)$. Since $p$ is surjective let $\hat{c}\in B_{n+1}$ be any lift, then $[b]=[b-{\partial }^B\hat{c}]$ but $p(b-{\partial }^{B}c)=0$ hence by exactness $b-{\partial }^B\hat{c}\in Z_n(A)\hookrightarrow Z_n(B)$ and so $[b]$ is in the image of $H_n(A)\to H_n(B)$.

It remains to see that

1. the image of $H_n(B)\to H_n(C)$ is the kernel of ${\delta }_n$;

2. the kernel of $H_{n-1}(A)\to H_{n-1}(B)$ is the image of ${\delta }_n$.

This follows by inspection of the formula in def. 1. We spell out the first one:

If $[c]$ is in the image of $H_n(B)\to H_n(C)$ we have a lift $\hat{c}$ with ${\partial }^B\hat{c}=0$ and so ${\delta }_n[c]=[{\partial }^B\hat{c}{]}_A=0$. Conversely, if for a given lift $\hat{c}$ we have that $[{\partial }^B\hat{c}{]}_A=0$ this means there is $a\in A_n$ such that ${\partial }^{A}a≔{\partial }^{B}a={\partial }^B\hat{c}$. But then $\tilde{c}≔\hat{c}-a$ is another possible lift of $c$ for which ${\partial }^B\tilde{c}=0$ and so $[c]$ is in the image of $H_n(B)\to H_n(C)$.

Remark

Of course the situation for cochain cohomology is formally dual to this situation. For convenience we repeat the statement for dual chains:

Let $A^{•}\to B^{•}\to C^{•}$ be a short exact sequence of cochain complexes.

For $[c{]}_C\in H^n(C)$ the class of a closed element $c$, by surjectivity of $B\to C$ there is an element $\hat{c}\in B$ mapping to it. This need not be closed anymore, but of course $d_B\hat{c}$ is. By the fact that $B\to C$ is a chain map we have that the image of $d_B\hat{c}$ in $C$ vanishes. Therefore by the exactness of the sequence the element $d_B\hat{c}$ may be regarded as a closed element of $A$. The cohomology class $[d_B\hat{c}{]}_A$ of this is what the connecting homomorphism

$${\delta }^n:H^n(C)\to H^{n+1}(A)$$\delta^n : H^n(C) \to H^{n+1}(A)

assigns to $[c{]}_C$:

$$\delta :[c{]}_C\mapsto [d_B\hat{c}{]}_A.$$\delta : [c]_C \mapsto [d_B\hat c]_A \,.

This is indeed well defined, in that it is independent of the choice of $\hat{c}$: for $\hat{c}\prime$ another choice, we have that the difference $\hat{c}-\hat{c}\prime$ is in the kernel of $B\to C$ hence is in $A$. Then $d_B\hat{c}\prime =d_B\hat{c}+d_A(\hat{c}-\hat{c}\prime )$. Hence $[d_B\hat{c}{]}_A=[d_B\hat{c}\prime {]}_A$.

General abstract

Theorem

Let $0\to A_{•}\stackrel{f}{\to }{B}_{•}\stackrel{g}{\to }{C}_{•}\to 0$ be a short exact sequence of chain complexes in some abelian category $𝒜$. Then for all $n\in \mathbb{Z}$ there are natural connecting homomorphisms $\partial :H_n(C)\to H_{n-1}(A)$ such that we have a long exact sequence of the form

$$\cdots \stackrel{g}{\to }{H}_{n+1}(C)\stackrel{\partial }{\to }{H}_n(A)\stackrel{f}{\to }{H}_n(B)\stackrel{g}{\to }{H}_n(C)\stackrel{\partial }{\to }{H}_{n-1}(A)\stackrel{f}{\to }\cdots$$\cdots \stackrel{g}{\to} H_{n+1}(C) \stackrel{\partial}{\to} H_n(A) \stackrel{f}{\to} H_n(B) \stackrel{g}{\to} H_n(C) \stackrel{\partial}{\to} H_{n-1}(A) \stackrel{f}{\to} \cdots

in chain homology.

Proof

Applying the snake lemma to the commuting diagram

$$\begin{array}{ccccccc}& & 0& & 0& & 0\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& \to & Z_{n}A& \to & Z_{n}B& \to & Z_{n}C\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& \to & A_n& \to & B_n& \to & C_n& \to & 0\\ & & {\downarrow }^d& & {\downarrow }^d& & {\downarrow }^d\\ 0& \to & A_{n-1}& \to & B_{n-1}& \to & C_{n-1}& \to & 0\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& \to & \frac{A_{n-1}}{\mathrm{im}(d)(A_n)}& \to & \frac{B_{n-1}}{\mathrm{im}(d)(B_n)}& \to & \frac{C_{n-1}}{\mathrm{im}(d)(C_n)}& \to & 0\\ & & \downarrow & & \downarrow & & \downarrow \\ & & 0& & 0& & 0\end{array}$$\array{ && 0 && 0 && 0 \\ && \downarrow && \downarrow && \downarrow \\ 0 &\to& Z_n A &\to& Z_n B &\to & Z_n C \\ && \downarrow && \downarrow && \downarrow \\ 0 &\to& A_n &\to& B_n &\to & C_n &\to & 0 \\ && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} \\ 0 &\to& A_{n-1} &\to& B_{n-1} &\to & C_{n-1} &\to& 0 \\ && \downarrow && \downarrow && \downarrow \\ 0 &\to& \frac{A_{n-1}}{im(d)(A_n)} &\to& \frac{B_{n-1}}{im(d)(B_n)} &\to & \frac{C_{n-1}}{im(d)(C_n)} &\to & 0 \\ && \downarrow && \downarrow && \downarrow \\ && 0 && 0 && 0 }

shows that the rows in the commuting diagram

$$\begin{array}{ccccccccc}& & \frac{A_n}{\mathrm{im}(d)(A_{n+1})}& \to & \frac{B_n}{\mathrm{im}(d)(B_{n+1})}& \to & \frac{C_n}{\mathrm{im}(d)(C_{n+1})}& \to & 0\\ & & {\downarrow }^d& & {\downarrow }^d& & {\downarrow }^d\\ 0& \to & Z_{n-1}A& \stackrel{f}{\to }& Z_{n-1}B& \stackrel{g}{\to }& Z_{n-1}C\end{array}$$\array{ && \frac{A_{n}}{im(d)(A_{n+1})} &\to& \frac{B_{n}}{im(d)(B_{n+1})} &\to & \frac{C_{n}}{im(d)(C_{n+1})} &\to & 0 \\ && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} \\ 0 &\to& Z_{n-1} A &\stackrel{f}{\to}& Z_{n-1} B &\stackrel{g}{\to}& Z_{n-1} C }

are exact sequences. Therefore applying the snake lemma to this, once more, yields the desired long exact sequence.

Examples

Example

The connecting homomorphism of the long exact sequence in homology induced from short exact sequences of the form

$$A/A_{n\mathrm{tor}}\stackrel{(-)\cdot n}{\to }A\to A/(nA)$$A/A_{n tor} \stackrel{(-)\cdot n}{\to} A \to A/(n A)

is called a Bockstein homomorphism.

Properties

Relation to homotopy fiber sequences

The connecting homomorphism in a long exact sequence in homology induced from a short exact sequence $A_{•}\stackrel{f}{\to }{B}_{•}\to C_{•}$ is equivalently the image under the homology group functor of the homotopy cofiber sequence induced by $f$. This is discussed in detail at mapping cone in the section homology exact sequences.

References

For instance section 1.3 of

• Charles Weibel, An Introduction to Homological Algebra