Chains, Homology, Cohomology, and Exact Sequences

Chains turn geometric pieces into an abelian-group complex

$$\cdots \longrightarrow C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \longrightarrow\cdots, \qquad \partial_n\partial_{n+1}=0.$$

An $n$-cycle lies in $\ker\partial_n$, an $n$-boundary lies in $\operatorname{im}\partial_{n+1}$, and

$$H_n = \frac{\ker\partial_n} {\operatorname{im}\partial_{n+1}}$$

retains the cycles not already explained as boundaries. Cochains reverse the arrows: cocycles modulo coboundaries form $H^n$. Continuous maps induce maps on these quotients, homotopic maps induce the same maps, and exact sequences identify which classes lift, extend, vanish, or leave a connecting obstruction. This is how local boundary data become topological invariants. Hatcher 2002, §§ 2.1 and 3.1, PDF develops these constructions and their invariance.

The quotient is essential. A cycle can be nonzero as a chain but zero in homology because it bounds a higher-dimensional chain. Dually, a cocycle can evaluate nontrivially on individual chains while its cohomology class is unchanged by adding a coboundary. The resulting pairing between cohomology and homology is the algebraic core of flux and charge pairings, but it does not by itself supply a field equation, a conserved physical charge, or a dynamical topological phase.

Chain-complex setting and coefficient choices

We use only kernels, images, quotient groups, and homomorphisms; each is recalled where it enters.

Unless stated otherwise,

• chains use integer coefficients;
• $G$ denotes an abelian coefficient group, written additively;
• $\partial_n:C_n\to C_{n-1}$ lowers degree, while $\delta^n:C^n\to C^{n+1}$ raises degree;
• $f_\#$ is a map of chains, $f_*$ the induced map on homology, and $f^*$ the induced map on cohomology; and
• orientations are chosen only for signed simplices and fundamental classes.

We begin with singular chains, which require no triangulation. Finite examples will use explicitly named simplicial or cellular chain complexes. For these finite simplicial or cellular examples, the comparison theorems identify the computed groups with singular homology. When homology uses an abelian coefficient group $G$, we mean

$$C_n(X;G) = C_n(X;\mathbb Z)\otimes_{\mathbb Z}G,$$

equivalently finite singular chains with coefficients in $G$. No spacetime metric or QFT sign convention enters this page.

Singular chains and the boundary cancellation

The standard $n$-simplex is

$$\Delta^n = \left\{ (t_0,\ldots,t_n)\in\mathbb R^{n+1} : t_i\geq0,\ \sum_{i=0}^n t_i=1 \right\}.$$

A singular $n$-simplex in $X$ is a continuous map $\sigma:\Delta^n\to X$. It need not be injective, and its image need not be an embedded submanifold. The singular chain group $C_n(X;\mathbb Z)$ is the free abelian group generated by all such maps, so an $n$-chain is a finite formal sum

$$c=\sum_\alpha m_\alpha\sigma_\alpha, \qquad m_\alpha\in\mathbb Z.$$

Let $\iota_i:\Delta^{n-1}\hookrightarrow\Delta^n$ include the face opposite vertex $i$. We fix the alternating-face convention

$$\partial_n\sigma = \sum_{i=0}^n (-1)^i\,\sigma\circ\iota_i$$

and extend it linearly. In oriented-simplex notation,

$$\partial[v_0v_1v_2] = [v_1v_2]-[v_0v_2]+[v_0v_1].$$

Applying $\partial$ again gives

$$\begin{aligned} \partial^2[v_0v_1v_2] &= ([v_2]-[v_1]) -([v_2]-[v_0]) +([v_1]-[v_0]) \\ &=0. \end{aligned}$$

In every dimension, each codimension-two face is deleted in two possible orders. The two terms have opposite signs and cancel, proving $\partial_{n-1}\partial_n=0$. This alternating boundary and its cancellation are worked out in Hatcher 2002, § 2.1, pp. 105–109, PDF, and in Frankel 2012, § 13.1a, pp. 333–337.

Homology measures the failure of exactness

Define the groups of cycles and boundaries by

$$Z_n(X;\mathbb Z)=\ker\partial_n, \qquad B_n(X;\mathbb Z)=\operatorname{im}\partial_{n+1}.$$

Because $\partial_n\partial_{n+1}=0$, every boundary is a cycle:

$$B_n(X;\mathbb Z)\subseteq Z_n(X;\mathbb Z).$$

The $n$th homology group is therefore the quotient

$$H_n(X;\mathbb Z) = \frac{Z_n(X;\mathbb Z)} {B_n(X;\mathbb Z)}.$$

Two cycles represent the same class precisely when

$$z'\sim z \quad\Longleftrightarrow\quad z'-z=\partial b$$

for some $(n+1)$-chain $b$. Thus $H_n=0$ means that every $n$-cycle bounds; it does not mean that there are no cycles. In the language introduced below, the chain complex is exact at $C_n$ exactly when $H_n=0$.

At degree zero, singular homology has a useful concrete description:

$$H_0(X;\mathbb Z) \cong \bigoplus_{\pi_0^{\mathrm{path}}(X)}\mathbb Z.$$

It is the free abelian group on the path components. Higher homology detects some higher-dimensional failures to fill, but the collection of homology groups is not a complete classifier of spaces.

A finite simplicial circle calculation

Let $K$ be the boundary of a triangle, with vertices $v_0,v_1,v_2$ and cyclically oriented edges

$$e_{01}=[v_0v_1], \qquad e_{12}=[v_1v_2], \qquad e_{20}=[v_2v_0].$$

We now use the simplicial chain groups $C_n^\Delta(K;\mathbb Z)$. There are no $2$-simplices. For the simplicial chain $u=ae_{01}+be_{12}+ce_{20}$,

$$\partial u = (-a+c)v_0+(a-b)v_1+(b-c)v_2.$$

Hence $\partial u=0$ exactly when $a=b=c$, so

$$Z_1^\Delta(K;\mathbb Z) = \mathbb Z\,z, \qquad z=e_{01}+e_{12}+e_{20}.$$

Since $C_2^\Delta(K)=0$, also $B_1^\Delta(K)=0$, and therefore

$$H_1^\Delta(K;\mathbb Z)\cong\mathbb Z.$$

Nakahara 2003, §§ 3.2–3.4, pp. 98–119 gives a complementary simplicial construction and explicit boundary calculations.

The class $[z]$ is a generator. By comparison, the same result holds for singular homology $H_1(K;\mathbb Z)$.

Now add the oriented face $t=[v_0v_1v_2]$. Because $-[v_0v_2]=[v_2v_0]=e_{20}$,

$$\partial t=e_{12}+e_{20}+e_{01}=z.$$

The same triangular loop is now a boundary, so its class vanishes. The boundary complex is a circle, while the filled complex is a disk. Homology records the presence or absence of the filling, not merely the visible 1-chain.

Induced maps make homology topological

A continuous map $f:X\to Y$ sends a singular simplex to its composite with $f$:

$$f_\#(\sigma)=f\circ\sigma.$$

Faces commute with composition, so

$$\partial f_\#=f_\#\partial.$$

A homomorphism satisfying this identity is a chain map. It sends cycles to cycles and boundaries to boundaries, and therefore induces

$$f_*:H_n(X;\mathbb Z)\longrightarrow H_n(Y;\mathbb Z), \qquad f_*[z]=[f_\#z].$$

The construction is functorial: $(g\circ f)_*=g_*\circ f_*$ and $(\operatorname{id}_X)_*=\operatorname{id}_{H_n(X)}$.

If $f_0,f_1:X\to Y$ are homotopic, the prism construction produces homomorphisms $P_n:C_n(X)\to C_{n+1}(Y)$ satisfying

$$f_{1\#}-f_{0\#} = \partial P+P\partial.$$

For a cycle $z$, this reduces to

$$f_{1\#}z-f_{0\#}z=\partial(Pz),$$

so $f_{0*}[z]=f_{1*}[z]$. Homotopic maps induce the same homology map, and homotopy-equivalent spaces have isomorphic homology. The converse fails: isomorphic homology groups do not imply homotopy equivalence, much less homeomorphism. Hatcher 2002, § 2.1, pp. 110–113, PDF gives the induced-map construction and the prism proof of homotopy invariance.

This also supplies the algebra promised by the preceding page. If $M$ and $N$ are connected, closed, oriented $n$-manifolds, their chosen orientations determine fundamental classes

$$[M]\in H_n(M;\mathbb Z), \qquad [N]\in H_n(N;\mathbb Z).$$

For a continuous map $f:M\to N$, the degree is the unique integer satisfying

$$f_*[M]=\deg(f)\,[N].$$

Changing either chosen orientation changes the corresponding sign. See Homotopy, Degree, Winding, and Covering Spaces for the geometric and integral formulas for degree; here the formula is the induced-map definition.

Cochains reverse the arrows

Let $G$ be an abelian group. An $n$-cochain is a homomorphism from integer $n$-chains to $G$:

$$C^n(X;G) = \operatorname{Hom}_{\mathbb Z} \bigl(C_n(X;\mathbb Z),G\bigr).$$

We write its value on a chain as $\langle\alpha,c\rangle=\alpha(c)$. The coboundary is the dual of the boundary, with no additional sign:

$$\begin{aligned} \delta^n:C^n(X;G)&\longrightarrow C^{n+1}(X;G),\\ \langle\delta^n\alpha,c\rangle &= \langle\alpha,\partial_{n+1}c\rangle. \end{aligned}$$

Since $\partial^2=0$, also $\delta^{n+1}\delta^n=0$. Define

$$Z^n(X;G)=\ker\delta^n, \qquad B^n(X;G)=\operatorname{im}\delta^{n-1},$$

and

$$H^n(X;G) = \frac{Z^n(X;G)} {B^n(X;G)}.$$

Elements of $Z^n$ are cocycles and elements of $B^n$ are coboundaries. A continuous map $f:X\to Y$ now reverses direction:

$$f^*:C^n(Y;G)\longrightarrow C^n(X;G), \qquad f^*\alpha=\alpha\circ f_\#.$$

It commutes with $\delta$ and induces $f^*:H^n(Y;G)\to H^n(X;G)$. Homology is covariant; cohomology is contravariant. Hatcher 2002, § 3.1, pp. 189–190 and 197–201, PDF develops the cochain complex and its functoriality.

Evaluation descends to classes

If $\alpha$ is a cocycle and $z$ is a cycle, define

$$\langle[\alpha],[z]\rangle = \langle\alpha,z\rangle\in G.$$

This does not depend on either representative. Replacing $\alpha$ by $\alpha+\delta\lambda$ changes the value by

$$\langle\delta\lambda,z\rangle = \langle\lambda,\partial z\rangle =0,$$

while replacing $z$ by $z+\partial b$ changes it by

$$\langle\alpha,\partial b\rangle = \langle\delta\alpha,b\rangle =0.$$

Thus there is a natural evaluation, or Kronecker, pairing

$$H^n(X;G)\times H_n(X;\mathbb Z) \longrightarrow G.$$

This pairing need not be perfect. In particular, integral cohomology is not generally just the homomorphism dual of integral homology.

Coefficients and torsion matter

For singular chains, the universal coefficient theorem gives a natural short exact sequence

$$\begin{aligned} 0\longrightarrow{}& \operatorname{Ext}_{\mathbb Z}^{1} \bigl(H_{n-1}(X;\mathbb Z),G\bigr)\\ \longrightarrow{}& H^n(X;G)\\ \longrightarrow{}& \operatorname{Hom}_{\mathbb Z} \bigl(H_n(X;\mathbb Z),G\bigr) \longrightarrow0. \end{aligned}$$

The right-hand map is evaluation. The sequence splits as a sequence of groups, but the splitting is not natural in general. The $\operatorname{Ext}_{\mathbb Z}^{1}$ symbol denotes the extension term in the universal coefficient theorem; it measures the possible failure of dualization by $\operatorname{Hom}_{\mathbb Z}(-,G)$ to preserve exactness. It records information that naïve evaluation on $H_n$ cannot see. If chains and cochains are instead taken over a field $\mathbb F$, linear algebra gives

$$H^n(X;\mathbb F) \cong \operatorname{Hom}_{\mathbb F} \bigl(H_n(X;\mathbb F),\mathbb F\bigr).$$

The change of coefficient category is part of that statement.

For a concrete warning, the cellular chain complex of the real projective plane over $\mathbb Z$ is

$$0\longrightarrow \mathbb Z \xrightarrow{\times2} \mathbb Z \xrightarrow{0} \mathbb Z \longrightarrow0.$$

Consequently,

$$H_1(\mathbb{RP}^2;\mathbb Z)\cong\mathbb Z_2, \qquad H_2(\mathbb{RP}^2;\mathbb Z)=0,$$

whereas the dual cochain complex gives

$$H^1(\mathbb{RP}^2;\mathbb Z)=0, \qquad H^2(\mathbb{RP}^2;\mathbb Z)\cong\mathbb Z_2.$$

The degree-two class comes from $\operatorname{Ext}_{\mathbb Z}^{1}(\mathbb Z_2,\mathbb Z)$; it cannot be detected by evaluating on $H_2$, which is zero. With $\mathbb Z_2$ coefficients, multiplication by $2$ becomes zero, and both $H_1(\mathbb{RP}^2;\mathbb Z_2)$ and $H_2(\mathbb{RP}^2;\mathbb Z_2)$ are $\mathbb Z_2$. See Hatcher 2002, §§ 2.2 and 3.1, especially pp. 140–144 and 190–196, PDF for the cellular calculation and the universal coefficient theorem.

Exact sequences expose lifting obstructions

A sequence

$$A\xrightarrow{i}B\xrightarrow{q}C$$

is exact at $B$ when

$$\operatorname{im}i=\ker q.$$

A short exact sequence

$$0\longrightarrow A \xrightarrow{i}B \xrightarrow{q}C \longrightarrow0$$

therefore says that $i$ is injective, $q$ is surjective, and $C$ is canonically isomorphic to the quotient $B/i(A)$. Exact does not mean split. For $N>1$,

$$0\longrightarrow\mathbb Z \xrightarrow{\times N} \mathbb Z \xrightarrow{\bmod N} \mathbb Z_N \longrightarrow0$$

is exact but not split: a homomorphism $\mathbb Z_N\to\mathbb Z$ must send a finite-order element to zero, so it cannot be a right inverse to reduction modulo $N$.

Now take a degreewise short exact sequence of chain complexes,

$$0\longrightarrow A_\bullet \xrightarrow{i}B_\bullet \xrightarrow{q}C_\bullet \longrightarrow0,$$

where $i$ and $q$ commute with the differentials $d$. It induces the long exact sequence

$$\begin{aligned} \cdots\longrightarrow{}& H_n(A) \xrightarrow{i_*} H_n(B) \xrightarrow{q_*} H_n(C)\\ \xrightarrow{\Delta_{\mathrm{conn}}}{}& H_{n-1}(A) \xrightarrow{i_*} H_{n-1}(B) \longrightarrow\cdots . \end{aligned}$$

The connecting map is the important new arrow. Given $[c]\in H_n(C)$:

1. choose a cycle $c\in C_n$ representing the class;
2. lift it to $b\in B_n$, so $q(b)=c$;
3. since $q(db)=d\,q(b)=dc=0$, exactness gives $db=i(a)$ for some $a\in A_{n-1}$; and
4. $i(da)=d\,i(a)=d^2b=0$, so injectivity of $i$ gives $da=0$.

Define

$$\Delta_{\mathrm{conn}}[c]=[a].$$

The result is independent of the choices. If $b'=b+i(a')$, then $db'=i(a+da')$, which changes $a$ by a boundary. If the cycle representative changes to $c+d\gamma$, lift $\gamma$ to $\eta\in B_{n+1}$ and use $b+d\eta$; its differential is still $db$.

This construction also verifies exactness at $H_n(C)$. If $[c]=q_*[b]$ for a cycle $b$, then $db=0$ and $\Delta_{\mathrm{conn}}[c]=0$. Conversely, if $\Delta_{\mathrm{conn}}[c]=0$, write $a=da'$ and replace the chosen lift by $b-i(a')$. The replacement is a cycle mapping to $c$, so $[c]\in\operatorname{im}q_*$. Therefore

$$\ker\Delta_{\mathrm{conn}} = \operatorname{im}q_*.$$

The connecting class is precisely the obstruction to lifting a cycle class in $C_\bullet$ to a cycle class in $B_\bullet$. Hatcher 2002, § 2.1, pp. 113–117, PDF gives this lift–differentiate–identify construction and proves the full long sequence exact.

Relative homology remembers where a boundary may lie

For a subspace $A\subseteq X$, define the relative chain complex

$$C_n(X,A;\mathbb Z) = \frac{C_n(X;\mathbb Z)} {C_n(A;\mathbb Z)}.$$

A relative cycle is represented by a chain $c$ in $X$ whose boundary lies in $A$. Relative homology is not the homology of the complement $X\setminus A$; it declares chains contained in $A$ to be zero.

The short exact sequence

$$0\longrightarrow C_\bullet(A) \longrightarrow C_\bullet(X) \longrightarrow C_\bullet(X,A) \longrightarrow0$$

produces

$$\begin{aligned} \cdots\longrightarrow{}& H_n(A) \longrightarrow H_n(X) \longrightarrow H_n(X,A)\\ \xrightarrow{\Delta_{\mathrm{conn}}}{}& H_{n-1}(A) \longrightarrow H_{n-1}(X) \longrightarrow\cdots . \end{aligned}$$

Here $\Delta_{\mathrm{conn}}[c]=[\partial c]$: the boundary of a relative cycle is an ordinary cycle in $A$.

For the oriented pair $(D^2,S^1)$, the relevant part is

$$0=H_2(D^2) \longrightarrow H_2(D^2,S^1) \xrightarrow{\ \Delta_{\mathrm{conn}}\ } H_1(S^1) \longrightarrow H_1(D^2)=0.$$

Thus the connecting map is an isomorphism,

$$H_2(D^2,S^1;\mathbb Z) \cong H_1(S^1;\mathbb Z) \cong\mathbb Z.$$

With the boundary orientation on $S^1$,

$$\Delta_{\mathrm{conn}}[D^2,S^1]=[S^1].$$

The circle is nontrivial inside the boundary subspace, yet it is the boundary of the relative disk. Reversing the disk orientation reverses both displayed classes.

Relative cochains can be defined by

$$C^n(X,A;G) = \operatorname{Hom}_{\mathbb Z} \bigl(C_n(X,A;\mathbb Z),G\bigr).$$

The cohomology sequence reverses the restriction arrows, while its connecting map raises degree:

$$\begin{aligned} \cdots\longrightarrow{}& H^{n-1}(A;G) \xrightarrow{\Delta_{\mathrm{conn}}} H^n(X,A;G)\\ \longrightarrow{}& H^n(X;G) \longrightarrow H^n(A;G) \xrightarrow{\Delta_{\mathrm{conn}}} H^{n+1}(X,A;G) \longrightarrow\cdots . \end{aligned}$$

A class in $H^n(A;G)$ extends from $A$ to a class on $X$ exactly when its connecting class in $H^{n+1}(X,A;G)$ vanishes. This is exactness at $H^n(A;G)$: the kernel of $\Delta_{\mathrm{conn}}$ is the image of the restriction map $H^n(X;G)\to H^n(A;G)$.

These relative homology and cohomology sequences are developed in Hatcher 2002, § 2.1, pp. 115–118, and § 3.1, pp. 199–201, PDF.

A finite cochain paired with a cycle on a spatial slice

Let $\Sigma$ be a finite cellulation of a spatial slice, fix $N>1$, and take $q\geq1$. Consider a $\mathbb Z_N$-valued cellular $q$-cochain

$$\begin{gathered} a\in C^q(\Sigma;\mathbb Z_N), \qquad \delta a=0,\\ a\sim a+\delta\lambda, \qquad \lambda\in C^{q-1}(\Sigma;\mathbb Z_N). \end{gathered}$$

The cocycle condition and equivalence by exact cochains define $[a]\in H^q(\Sigma;\mathbb Z_N)$. In the bounded QFT interpretation used here, this class labels topological symmetry-defect data on a spatial slice. Gaiotto, Kapustin, Seiberg, and Willett 2015, § 3, especially equation (3.4), pp. 12–13, PDF pair such a cohomology class with a charged support in $H_q(\Sigma;\widehat{\mathbb Z}_N)$.

There is a related but distinct spacetime statement: a flat background for a $q$-form symmetry is represented one degree higher, by a $(q+1)$-cocycle modulo exact cocycles in $H^{q+1}(M_d;\mathbb Z_N)$. This is Gaiotto, Kapustin, Seiberg, and Willett 2015, § 6, equation (6.1), PDF. We keep the spatial class $[a]$ and the spacetime background class separate.

The character group is

$$\widehat{\mathbb Z}_N = \operatorname{Hom}(\mathbb Z_N,U(1)).$$

Choose the standard identification $Q\in\mathbb Z_N\leftrightarrow\chi_Q\in\widehat{\mathbb Z}_N$ with

$$\chi_Q(t) = \exp\!\left(\frac{2\pi iQ}{N}\,t\right).$$

Here $Q$ and $t$ are represented by integers; changing either representative by a multiple of $N$ leaves the character unchanged. If $z$ is an integral cellular $q$-cycle, reducing its coefficients modulo $N$ and decorating it with $\chi_Q$ gives a class $[\beta_Q]\in H_q(\Sigma;\widehat{\mathbb Z}_N)$. Define

$$W_Q([a],[z]) = \chi_Q\!\left(\langle a,z\rangle\right) = \exp\!\left[ \frac{2\pi iQ}{N} \langle a,z\rangle \right].$$

Both quotient relations are visible in one line:

$$\begin{aligned} \langle a+\delta\lambda,z\rangle -\langle a,z\rangle &= \langle\lambda,\partial z\rangle=0,\\ \langle a,z+\partial c\rangle -\langle a,z\rangle &= \langle\delta a,c\rangle=0. \end{aligned}$$

Thus the phase depends only on the cohomology class of $a$, the homology class of the character-decorated support $\beta_Q$, and the chosen standard character pairing. If $z$ is open, then $\partial z\neq0$ and a gauge change contributes $\chi_Q(\langle\lambda,\partial z\rangle)$. That failure does not mean the calculation is inconsistent; it says that endpoint or boundary data are needed before the open support defines an invariant observable.

For a check with two independent cycles, take $\Sigma=T^2$ with its standard CW decomposition. It has one vertex, two oriented $1$-cells $x,y$, and one $2$-cell attached by the commutator. Its cellular boundary maps vanish, so

$$H_1(T^2;\mathbb Z)\cong\mathbb Z^2, \qquad H^1(T^2;\mathbb Z_N)\cong\mathbb Z_N^2.$$

Writing

$$[z]=m[x]+n[y], \qquad [a]=(r,s),$$

gives

$$W_Q = \exp\!\left[ \frac{2\pi iQ}{N}(rm+sn) \right].$$

This is the algebraic skeleton of a topological flux–charge pairing. The developed field theory, including BF actions, coefficient quantization, equations of motion, and linking observables, belongs to BF Couplings and Discrete Topological Data.

What the invariants do and do not establish

Topological conclusions and the separate physical questions

Algebraic-topological conclusion	Additional physical question
A cycle represents a nonzero homology class, so it does not bound in the chosen chain complex.	Does the theory contain an operator, defect, flux, or state supported by that class?
A cocycle evaluates consistently on homology classes.	Which coefficient group and character pairing are selected by the physical theory?
A connecting class obstructs a mathematical lift or extension.	Is that obstruction a physical anomaly, a boundary charge, or merely irrelevant to the model?
A finite-cochain phase is invariant under changes of representatives.	Does an action realize it, and what dynamics, spectrum, or superselection structure follows?
A torsion class survives in integral or finite-coefficient topology.	Can the chosen continuum variables detect it, or is additional discrete data required?

Topology supplies the left column under stated coefficient, boundary, and equivalence choices. The action, gauge quotient, boundary conditions, state space, and observable algebra decide the right. In particular, a nonzero homology or cohomology class does not by itself prove charge conservation, quantization, stability, or physical existence.

Common pitfalls

Replacing inclusion by equality. The identity $\partial^2=0$ proves only $B_n\subseteq Z_n$. Equality is the additional statement $H_n=0$.

Treating every cycle as a submanifold. A singular cycle is a finite formal sum of parameterized simplices. It can self-overlap, have multiplicities, or fail to be embedded.

Forgetting the variance. A map $f:X\to Y$ gives $f_*:H_n(X)\to H_n(Y)$ but $f^*:H^n(Y)\to H^n(X)$. The cochain arrow reverses because a cochain is precomposed with $f_\#$.

Using homology as a complete classifier. Homotopy-equivalent spaces have isomorphic homology, but the converse is false. Even all homology groups can miss distinctions detected by other invariants.

Dropping the coefficient group. Integral, real, and finite coefficients can retain different information. Over $\mathbb Z$, the $\operatorname{Ext}$ term prevents cohomology from being merely the dual of homology.

Calling relative homology the homology of a complement. The complex $C_\bullet(X,A)$ is a quotient by chains in $A$. It does not remove $A$ from the space.

Reading “exact” as “split.” Exactness specifies images and kernels of the given maps. A splitting is extra structure and, when one exists, need not be canonical.

Promoting a kinematic pairing to a field theory. The phase $W_Q([a],[z])$ is well defined on classes. It does not derive a BF action, linking law, conserved current, or spectrum.

Exercises

Each optional check is followed by a solution.

1. Retrieve the quotient construction

State $Z_n$, $B_n$, and $H_n$. Why is the quotient defined, and what does $H_n=0$ say?

Solution

They are

$$Z_n=\ker\partial_n, \qquad B_n=\operatorname{im}\partial_{n+1}, \qquad H_n=Z_n/B_n.$$

The identity $\partial_n\partial_{n+1}=0$ implies $B_n\subseteq Z_n$, so the quotient is defined. The statement $H_n=0$ means every cycle is a boundary. It does not say that the cycle group is zero.

2. Test the hypothesis “exact means split”

Show that

$$0\longrightarrow\mathbb Z \xrightarrow{\times2} \mathbb Z \xrightarrow{\bmod2} \mathbb Z_2 \longrightarrow0$$

is exact but cannot split.

Solution

Multiplication by $2$ is injective. Its image is $2\mathbb Z$, which is the kernel of reduction modulo $2$, and reduction modulo $2$ is surjective. Hence the sequence is exact.

A splitting would require a homomorphism $s:\mathbb Z_2\to\mathbb Z$ with $(\bmod2)\circ s$ equal to the identity. But $2[1]=0$ in $\mathbb Z_2$, so $2s([1])=s(0)=0$ in the torsion-free group $\mathbb Z$. Thus $s([1])=0$, whose reduction is not $[1]$. No splitting exists.

3. Calculate the relative disk groups

Use the long exact sequence of $(D^2,S^1)$ to compute $H_2(D^2,S^1;\mathbb Z)$ and $H_1(D^2,S^1;\mathbb Z)$. Identify the connecting image of the oriented relative disk.

Solution

Contractibility gives $H_2(D^2)=H_1(D^2)=0$, while $H_1(S^1)\cong\mathbb Z$ and the map $H_0(S^1)\to H_0(D^2)$ is an isomorphism. Exactness in

$$0\to H_2(D^2,S^1) \xrightarrow{\Delta_{\mathrm{conn}}} H_1(S^1)\to0$$

therefore gives

$$H_2(D^2,S^1;\mathbb Z)\cong\mathbb Z.$$

The next part of the sequence and the isomorphism on $H_0$ give $H_1(D^2,S^1;\mathbb Z)=0$. With the induced boundary orientation,

$$\Delta_{\mathrm{conn}}[D^2,S^1]=[S^1].$$

4. Transfer the pairing to the torus

For $[a]=(r,s)\in H^1(T^2;\mathbb Z_N)$, $[z]=m[x]+n[y]\in H_1(T^2;\mathbb Z)$, and $Q\in\mathbb Z_N$ labelling $\chi_Q\in\widehat{\mathbb Z}_N$, compute $W_Q$. Check invariance under changing $a$ by a coboundary and $z$ by a boundary. What changes for an open chain?

Solution

The evaluation is $rm+sn$ modulo $N$, so

$$W_Q = \exp\!\left[ \frac{2\pi iQ}{N}(rm+sn) \right].$$

For a cycle $z$,

$$\langle\delta\lambda,z\rangle = \langle\lambda,\partial z\rangle=0,$$

and for a cocycle $a$,

$$\langle a,\partial c\rangle = \langle\delta a,c\rangle=0.$$

The phase is therefore unchanged by both representative choices. If $\partial z\neq0$, a change $a\mapsto a+\delta\lambda$ instead multiplies the phase by $\chi_Q(\langle\lambda,\partial z\rangle)$. Endpoint or boundary data are then required to construct an invariant quantity.

Synthesis and continuations

The relation $\partial^2=0$ places boundaries inside cycles, and homology is the quotient that measures the remaining failure to fill. Dualizing produces cocycles modulo coboundaries. Induced maps and the prism identity make these quotients homotopy invariants; the fundamental-class formula $f_*[M]=\deg(f)[N]$ recovers degree algebraically. Exact sequences then make the invariants computable by converting a failed lift into a connecting class. Relative groups record boundaries permitted to lie in a chosen subspace, and evaluation pairs cohomology classes with homology cycles.

For smooth real representatives, periods, Poincaré duality, and intersection pairings, continue to de Rham Cohomology, Periods, Poincaré Duality, and Intersection. Real differential forms will not retain all torsion information displayed here. For the developed physical realization, continue to BF Couplings and Discrete Topological Data. For chain homotopy and quasi-isomorphisms as algebraic subjects, continue to Chain Homotopy, Quasi-Isomorphisms, and Derived Vocabulary.

References

• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, §§ 13.1a and 13.2c, pp. 333–346, and Appendix B, pp. 628–631. Frankel gives a geometric route through chains, homology, and chain and cochain complexes.
• Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett, “Generalized Global Symmetries”, Journal of High Energy Physics 02 (2015) 172, §§ 3 and 6, especially pp. 12–13 and equation (6.1), p. 33. These passages support the finite cohomology– homology pairing on a spatial slice and the one-degree-higher flat spacetime cocycle modulo exact cocycles used in the bounded QFT application.
• Allen Hatcher, Algebraic Topology — Open PDF, Cambridge University Press, 2002, §§ 2.1–2.2 and 3.1, especially pp. 105–118, 128–144, and 189–201. These sections supply homology, induced maps, relative and exact sequences, cochains, and the universal coefficient theorem.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Institute of Physics Publishing, 2003, Chapter 3, especially §§ 3.2–3.4, pp. 98–119. This gives a complementary simplicial development with explicit boundary and homology calculations.