A long exact sequence of cohomology groups (zig-zag and diagram-chasing)

## Exterior differentiation

*(This section is intended to introduce the background. Feel free to skip if you already know exterior differentiation.)*

There are several useful tools for vector calculus on $\mathbb{R}^3,$ namely gradient, curl, and divergence. It is possible to treat the gradient of a differentiable function $f$ on $\mathbb{R}^3$ at a point $x_0$ as the Fréchet derivative at $x_0$. But it does not work for curl and divergence at all. Fortunately there is another abstraction that works for all of them. It comes from differential forms.

Let $x_1,\cdots,x_n$ be the linear coordinates on $\mathbb{R}^n$ as usual. We define an *algebra* $\Omega^{\ast}$ over $\mathbb{R}$ generated by $dx_1,\cdots,dx_n$ with the following relations:

This is a vector space as well, and it’s easy to derive that it has a basis by

where $i<j<k$. The $C^{\infty}$ differential *forms* on $\mathbb{R}^n$ are defined to be the tensor product

As is can be shown, for $\omega \in \Omega^{\ast}(\mathbb{R}^n)$, we have a unique representation by

and in this case we also say $\omega$ is a $C^{\infty}$ $k$-form on $\mathbb{R}^n$ (for simplicity we also write $\omega=\sum f_Idx_I$). The algebra of all $k$-forms will be denoted by $\Omega^k(\mathbb{R}^n)$. And naturally we have $\Omega^{\ast}(\mathbb{R}^n)$ to be graded since

### The operator $d$

But if we have $\omega \in \Omega^0(\mathbb{R}^n)$, we see $\omega$ is merely a $C^{\infty}$ function. As taught in multivariable calculus course, for the differential of $\omega$ we have

and it turns out that $d\omega\in\Omega^{1}(\mathbb{R}^n)$. This inspires us to obtain a generalization onto the differential operator $d$:

and $d\omega$ is defined as follows. The case when $k=0$ is defined as usual (just the one above). For $k>0$ and $\omega=\sum f_I dx_I,$ $d\omega$ is defined ‘inductively’ by

This $d$ is the so-called *exterior differentiation*, which serves as the ultimate abstract extension of gradient, curl, divergence, etc. If we restrict ourself to $\mathbb{R}^3$, we see these vector calculus tools comes up in the nature of things.

**Functions**

**$1$-forms**

**$2$-forms**

The calculation is tedious but a nice exercise to understand the definition of $d$ and $\Omega^{\ast}$.

### Conservative field - on the kernel and image of $d$

By elementary computation we are also able to show that $d^2\omega=0$ for all $\omega \in \Omega^{\ast}(\mathbb{R}^n)$ (*Hint: $\frac{\partial^2 f}{\partial x_i \partial x_j}=\frac{\partial^2 f}{\partial x_j \partial x_i}$ but $dx_idx_j=-dx_idx_j$)*. Now we consider a vector field $\overrightarrow{v}=(v_1,v_2)$ of dimension $2$. If $C$ is an arbitrary simply closed smooth curve in $\mathbb{R}^2$, then we expect

to be $0$. If this happens (note the arbitrary of $C$), we say $\overrightarrow{v}$ to be a conservative field (path independent).

So when conservative? It happens when there is a function $f$ such that

This is equivalent to say that

If we use $C^{\ast}$ to denote the area enclosed by $C$, by Green’s theorem, we have

If you translate what you’ve learned in multivariable calculus course (path independence) into the language of differential form, you will see that the set of all conservative fields is precisely the *image* of $d_0:\Omega^0(\mathbb{R}^2) \to \Omega^1(\mathbb{R}^2)$. Also, they are in the *kernel* of the next $d_1:\Omega^1(\mathbb{R}^2) \to \Omega^2(\mathbb{R}^2)$. These $d$’s are naturally homomorphism, so it’s natural to discuss the *factor group*. But before that, we need some terminologies.

## de Rham complex and de Rham cohomology group

The complex $\Omega^{\ast}(\mathbb{R}^n)$ together with $d$ is called the *de Rham complex* on $\mathbb{R}^n$. Now consider the sequence

We say $\omega \in \Omega^k(\mathbb{R}^n)$ is *closed* if $d_k\omega=0$, or equivalently, $\omega \in \ker d_k$. Dually, we say $\omega$ is *exact* if there exists some $\mu \in \Omega^{k-1}(\mathbb{R}^n)$ such that $d\mu=\omega$, that is, $\omega \in \operatorname{im}d_{k-1}$. Of course all $d_k$’s can be written as $d$ but the index makes it easier to understand. Instead of doing integration or differentiation, which is ‘uninteresting’, we are going to discuss the abstract structure of it.

The $k$-th *de Rham cohomology* in $\mathbb{R}^n$ is defined to be the factor space

As an example, note that by the fundamental theorem of calculus, every $1$-form is exact, therefore $H_{DR}^1(\mathbb{R})=0$.

Since de Rham complex is a special case of *differential complex*, and other restrictions of de Rham complex plays no critical role thereafter, we are going discuss the algebraic structure of differential complex directly.

## The long exact sequence of cohomology groups

We are going to show that, there exists a long exact sequence of cohomology groups after a short exact sequence is defined. For the convenience let’s recall here some basic definitions

### Exact sequence

A sequence of vector spaces (or groups)

is said to be *exact* if the image of $f_{k-1}$ is the kernel of $f_k$ for all $k$. Sometimes we need to discuss a extremely short one by

As one can see, $f$ is injective and $g$ is surjective.

### Differential complex

A direct sum of vector spaces $C=\oplus_{k \in \mathbb{Z}}C^k$ is called a *differential complex* if there are homomorphisms by

such that $d_{k-1}d_k=0$. Sometimes we write $d$ instead of $d_{k}$ since this *differential operator* of $C$ is universal. Therefore we may also say that $d^2=0$. The cohomology of $C$ is the direct sum of vector spaces $H(C)=\oplus_{k \in \mathbb{Z}}H^k(C) $ where

A map $f: A \to B$ where $A$ and $B$ are differential complexes, is called a *chain map* if we have $fd_A=d_Bf$.

### The sequence

Now consider a short exact sequence of differential complexes

where both $f$ and $g$ are chain maps (this is important). Then there exists a long exact sequence by

Here, $f^{\ast}$ and $g^{\ast}$ are the naturally induced maps. For $c \in C^q$, $d^{\ast}[c]$ is defined to be the cohomology class $[a]$ where $a \in A^{q+1}$, and that $f(a)=db$, and that $g(b)=c$. The sequence can be described using the two-layer commutative diagram below.

The long exact sequence is actually the purple one (you see why people may call this zig-zag lemma). This sequence is ‘based on’ the blue diagram, which can be considered naturally as an expansion of the short exact sequence. The method that will be used in the following proof is called diagram-chasing, whose importance has already been described by Professor James Munkres: *master* this. We will be *abusing* the properties of almost every homomorphism and group appeared in this commutative diagram to trace the elements.

#### Proof

First, we give a precise definition of $d^{\ast}$. For a closed $c \in C^q$, by the surjectivity of $g$ (note this sequence is exact), there exists some $b \in B^q$ such that $g(b)=c$. But $g(db)=d(g(b))=dc=0$, we see for $db \in B^{q+1}$ we have $db \in \ker g$. By the exactness of the sequence, we see $db \in \operatorname{im}{f}$, that is, there exists some $a \in A^{q+1}$ such that $f(a)=db$. Further, $a$ is closed since

and we already know that $f$ has trivial kernel (which contains $da$).

$d^{\ast}$ is therefore defined by

where $[\cdot]$ means “the homology class of”.

But it is expected that $d^{\ast}$ is a well-defined homomorphism. Let $c_q$ and $c_q’$ be two closed forms in $C^q$. To show $d^{\ast}$ is well-defined, we suppose $[c_q]=[c_q’]$ (i.e. they are homologous). Choose $b_q$ and $b_q’$ so that $g(b_q)=c_q$ and $g(b_q’)=c_q’$. Accordingly, we also pick $a_{q+1}$ and $a_{q+1}’$ such that $f(a_{q+1})=db_q$ and $f(a_{q+1}’)=db_q’$. By definition of $d^{\ast}$, we need to show that $[a_{q+1}]=[a_{q+1}’]$.

Recall the properties of factor group. $[c_q]=[c_q’]$ if and only if $c_q-c_q’ \in \operatorname{im}d$. Therefore we can pick some $c_{q-1} \in C^{q-1}$ such that $c_q-c_q’=dc_{q-1}$. Again, by the surjectivity of $g$, there is some $b_{q-1}$ such that $g(b_{q-1})=c_{q-1}$.

Note that

Therefore $b_q-b_q’-db_{q-1} \in \operatorname{im} f$. We are able to pick some $a_q \in A^{q}$ such that $f(a_q)=b_q-b_q’-db_{q-1}$. But now we have

Since $f$ is injective, we have $da_q=a_{q+1}-a_{q+1}’$, which implies that $a_{q+1}-a_{q+1}’ \in \operatorname{im}d$. Hence $[a_{q+1}]=[a_{q+1}’]$.

To show that $d^{\ast}$ is a homomorphism, note that $g(b_q+b_q’)=c_q+c_q’$ and $f(a_{q+1}+a_{q+1}’)=d(b_q+b_q’)$. Thus we have

The latter equals $[a_{q+1}]+[a_{q+1}’]$ since the canonical map is a homomorphism. Therefore we have

Therefore the long sequence exists. It remains to prove exactness. Firstly we need to prove exactness at $H^q(B)$. Pick $[b] \in H^q(B)$. If there is some $a \in A^q$ such that $f(a)=b$, then $g(f(a))=0$. Therefore $g^{\ast}[b]=g^{\ast}[f(a)]=[g(f(a))]=[0]$; hence $\operatorname{im}f \subset \ker g$.

Conversely, suppose now $g^{\ast}[b]=[0]$, we shall show that there exists some $[a] \in H^q(A)$ such that $f^{\ast}[a]=[b]$. Note $g^{\ast}[b]=\operatorname{im}d$ where $d$ is the differential operator of $C$ (why?). Therefore there exists some $c_{q-1} \in C^{q-1}$ such that $g(b)=dc_{q-1}$. Pick some $b_{q-1}$ such that $g(b_{q-1})=c_{q-1}$. Then we have

Therefore $f(a)=b-db_{q-1}$ for some $a \in A^q$. Note $a$ is closed since

and $f$ is injective. $db=0$ since we have

Furthermore,

Therefore $\ker g^{\ast} \subset \operatorname{im} f$ as desired.

Now we prove exactness at $H^q(C)$. (Notation:) pick $[c_q] \in H^q(C)$, there exists some $b_q$ such that $g(b_q)=c_q$; choose $a_{q+1}$ such that $f(a_{q+1})=db_q$. Then $d^{\ast}[c_q]=[a_{q+1}]$ by definition.

If $[c_q] \in \operatorname{im}g^{\ast}$, we see $[c_q]=[g(b_q)]=g^{\ast}[b_q]$. But $b_q$ is closed since $[b_q] \in H^q(B)$, we see $f(a_{q+1})=db_q=0$, therefore $d^{\ast}[c_q]=[a_{q+1}]=[0]$ since $f$ is injective. Therefore $\operatorname{im}g^{\ast} \subset \ker d^{\ast}$.

Conversely, suppose $d^{\ast}[c^q]=[0]$. By definition of $H^{q+1}(A)$, there is some $a_q \in A$ such that $da_q = a_{q+1}$ (can you see why?). We claim that $b_q-f(a_q)$ is closed and we have $[c_q]=g^{\ast}[b_q-f(a_q)]$.

By direct computation,

Meanwhile

Therefore $\ker d^{\ast} \subset \operatorname{im}g^{\ast}$. Note that $g(f(a_q))=0$ by exactness.

Finally, we prove exactness at $H^{q+1}(A)$. Pick $\alpha \in H^{q+1}(A)$. If $\alpha \in \operatorname{im}d^{\ast}$, then $\alpha=[a_{q+1}]$ where $f(a_{q+1})=db_q$ by definition. Then

Therefore $\alpha \in \ker f^{\ast}$. Conversely, if we have $f^{\ast}(\alpha)=[0]$, pick the representative element of $\alpha$, namely we write $\alpha=[a]$; then $[f(a)]=[0]$. But this implies that $f(a) \in \operatorname{im}d$ where $d$ denotes the differential operator of $B$. There exists some $b_{q+1} \in B^{q+1}$ and $b_q \in B^q$ such that $db_{q}=b_{q+1}$. Suppose now $c_q=g(b_q)$. $c_q$ is closed since $dc_q=g(db_q)=g(b_{q+1})=g(f(a))=0$. By definition, $\alpha=d^{\ast}[c_q]$. Therefore $\ker f^{\ast} \subset \operatorname{im}d^{\ast}$.

### Remarks

As you may see, almost every property of the diagram has been used. The exactness at $B^q$ ensures that $g(f(a))=0$. The definition of $H^q(A)$ ensures that we can simplify the meaning of $[0]$. We even use the injectivity of $f$ and the surjectivity of $g$.

This proof is also a demonstration of diagram-chasing technique. As you have seen, we keep running through the diagram to ensure that there is “someone waiting” at the destination.

This long exact group is useful. Here is an example.

## Application: Mayer-Vietoris Sequence

By differential forms on a open set $U \subset \mathbb{R}^n$, we mean

And the de Rham cohomology of $U$ comes up in the nature of things.

We are able to compute the cohomology of the union of two open sets. Suppose $M=U \cup V$ is a manifold with $U$ and $V$ open, and $U \amalg V$ is the disjoint union of $U$ and $V$ (the coproduct in the category of sets). $\partial_0$ and $\partial_1$ are inclusions of $U \cap V$ in $U$ and $V$ respectively. We have a natural sequence of inclusions

Since $\Omega^{*}$ can also be treated as a contravariant functor from the category of Euclidean spaces with smooth maps to the category of commutative differential graded algebras and their homomorphisms, we have

By taking the difference of the last two maps, we have

The sequence above is a short exact sequence. Therefore we may use the zig-zag lemma to find a long exact sequence (which is also called the Mayer-Vietoris sequence) by

### An example

This sequence allows one to compute the cohomology of two union of two open sets. For example, for $H^{*}_{DR}(\mathbb{R}^2-P-Q)$, where $P(x_p,y_p)$ and $Q(x_q,y_q)$ are two distinct points in $\mathbb{R}^2$, we may write

and

Therefore we may write $M=\mathbb{R}^2$, $U=\mathbb{R}^2-P$ and $V=\mathbb{R}^2-Q$. For $U$ and $V$, we have another decomposition by

where

But

is a four-time (homeomorphic) copy of $\mathbb{R}^2$. So things become clear after we compute $H^{\ast}_{DR}(\mathbb{R}^2)$.

## References / Further reading

- Raoul Bott, Loring W. Tu,
*Differential Forms in Algebraic Topology* - Munkres J. R.,
*Elements of Algebraic Topology* - Micheal Spivak,
*Calculus on Manifolds* - Serge Lang,
*Algebra*