Algebraic Topology Appendix: From Homotopy to Cohomology

Standing notation

Throughout this appendix:

• \(I=[0,1]\).

• \(X,Y,Z\) denote topological spaces.

• A map between topological spaces is assumed continuous unless explicitly stated otherwise.

• A based space is a pair \((X,x_0)\) consisting of a topological space \(X\) and a chosen point \(x_0\in X\).

• \(S^n\) denotes the unit \(n\)-sphere \[S^n=\{x\in \mathbb{R}^{n+1}: \|x\|=1\}.\]

• \(D^n\) denotes the closed unit \(n\)-disk \[D^n=\{x\in \mathbb{R}^n: \|x\|\leq 1\}.\] Its boundary is \(\partial D^n=S^{n-1}\) for \(n\geq 1\).

• \(\mathbb{Z}\) is the additive group of integers, \(\mathbb{R}\) is the additive group of real numbers unless a ring structure is being used, and \(U(1)=\{z\in\mathbb{C}: |z|=1\}\) is the multiplicative circle group.

• \(\mathsf A\) denotes an abelian coefficient group. In singular cohomology notation we write \(H^n(X;\mathsf A)\), with a semicolon separating the space from the coefficients.

When a topological space is said to be a “nice space” below, the intended class is the class of spaces usually encountered in differential geometry and condensed matter applications: manifolds, finite CW complexes, simplicial complexes, and spaces homotopy equivalent to these. The definitions themselves do not require niceness, but some classification theorems do.

Homotopy of maps

Topology studies properties preserved by homeomorphism. Algebraic topology often studies weaker properties preserved by continuous deformation.

The parameter \(t\in I\) is deformation time. For fixed \(t\), the map \[H_t:X\to Y,\qquad H_t(x)=H(x,t)\] is an intermediate map between \(f_0\) and \(f_1\).

The phrase “relative to \(A\)” means that points of \(A\) are held fixed during the deformation.

A homeomorphism is a homotopy equivalence, but not conversely. For example, a solid disk \(D^2\) is not homeomorphic to a point, but it is homotopy equivalent to a point because it can be contracted continuously to its center.

Paths, loops, and the fundamental group

Homotopy becomes especially important when the domain is an interval or a sphere.

Thus the endpoints are not allowed to move during the deformation.

The operation is well-defined on homotopy classes. Strictly speaking, path concatenation is associative only up to reparametrization, but it becomes associative on homotopy classes. Hence \(\pi_1(X,x_0)\) is a group.

If \(X\) is path connected, then different basepoints give isomorphic fundamental groups. More precisely, if \(c\) is a path from \(x_0\) to \(x_1\), then \[\pi_1(X,x_0)\to \pi_1(X,x_1), \qquad [\gamma]\mapsto [\overline c*\gamma*c]\] is a group isomorphism. The isomorphism depends on the choice of \(c\) unless \(\pi_1(X,x_0)\) is abelian.

Computing the first examples of fundamental groups

The circle

Let \[p:\mathbb{R}\to S^1, \qquad p(\theta)=e^{i\theta}.\] This map wraps the real line around the circle. It is periodic: \[p(\theta+2\pi n)=p(\theta)\] for every \(n\in\mathbb{Z}\).

A loop \(\gamma:I\to S^1\) based at \(1\in S^1\) can be lifted to a path \(\widetilde\gamma:I\to\mathbb{R}\) satisfying \[p\circ\widetilde\gamma=\gamma, \qquad \widetilde\gamma(0)=0.\] Since \(\gamma(1)=1\), the endpoint \(\widetilde\gamma(1)\) must be an integer multiple of \(2\pi\): \[\widetilde\gamma(1)=2\pi n\] for a unique \(n\in\mathbb{Z}\).

Homotopic loops have the same winding number, and every integer occurs. Therefore \[\pi_1(S^1,1)\cong \mathbb{Z}.\] The isomorphism sends a loop class to its winding number.

The torus

The \(n\)-torus is \[T^n=(S^1)^n.\] Since loops in a product can wind independently around each \(S^1\) factor, \[\pi_1(T^n)\cong \mathbb{Z}^n.\] For the two-torus, \[\pi_1(T^2)\cong \mathbb{Z}^2.\] The two integer generators correspond to the two noncontractible cycles.

Higher spheres

For \(n\geq 2\), \[\pi_1(S^n)\cong 0,\] where \(0\) denotes the trivial group. Geometrically, any loop on \(S^n\) can be pulled away from at least one point and then contracted inside a copy of \(\mathbb{R}^n\) obtained by stereographic projection.

Rotation groups

The group \(SU(2)\) is diffeomorphic to \(S^3\), hence \[\pi_1(SU(2))\cong 0.\] The group \(SO(3)\) is obtained from \(SU(2)\) by identifying \(U\) and \(-U\): \[SU(2)\to SO(3)\] is a two-to-one covering map. As a result, \[\pi_1(SO(3))\cong \mathbb{Z}_2.\] This is the topology behind the distinction between integer-spin and half-integer-spin representations.

Covering spaces and universal covers

Covering spaces make the previous examples precise.

For connected and locally well-behaved spaces, the universal cover is unique up to isomorphism of covering spaces. The universal cover of \(S^1\) is \(\mathbb{R}\). The universal cover of \(SO(3)\) is \(SU(2)\).

Higher homotopy groups

The fundamental group uses loops, or maps from \(S^1\). Higher homotopy groups use maps from higher-dimensional spheres.

For \(n=1\), this recovers the fundamental group. For \(n\geq 2\), the group \(\pi_n(X,x_0)\) is abelian.

A useful model is to identify \(S^n\) with the quotient space \[I^n/\partial I^n,\] where the entire boundary of the cube is collapsed to one point. This makes the group operation visible: place two maps in adjacent subcubes and collapse the boundary.

Homotopy groups are powerful but hard to compute. Cohomology is usually easier to compute and is better adapted to differential forms, flux integrals, and Chern classes.

From homotopy to homology

The fundamental group detects noncontractible loops. Higher homotopy groups detect nontrivial maps from spheres. Homology takes a different approach: it studies cycles built from simple pieces and records which cycles are boundaries.

The guiding principle is: \[\text{homology detects holes by testing whether cycles bound.}\] For example:

• A nontrivial element of \(H_1(X;\mathbb{Z})\) is represented by a closed loop or collection of loops that is not the boundary of any surface in \(X\).

• A nontrivial element of \(H_2(X;\mathbb{Z})\) is represented by a closed surface that is not the boundary of any three-dimensional region in \(X\).

The most general definition is singular homology.

Singular chains and singular homology

The word “formal” means that the sum is not pointwise addition of maps. It is an algebraic record of oriented pieces with integer multiplicities.

For \(0\leq i\leq n\), define the \(i\)th face inclusion \[\iota_i:\Delta^{n-1}\to \Delta^n\] by inserting \(0\) in the \(i\)th coordinate: \[\iota_i(t_0,\ldots,t_{n-1})=(t_0,\ldots,t_{i-1},0,t_i,\ldots,t_{n-1}).\]

The alternating signs encode orientation. For example, if \(\sigma:\Delta^1\to X\) is a path, then \[\partial_1\sigma=\sigma(1)-\sigma(0),\] where the two endpoints are regarded as singular \(0\)-simplices.

The key identity is \[\partial_{n-1}\circ\partial_n=0.\] In words: the boundary of a boundary is zero.

The coefficient group can be changed. For an abelian group \(\mathsf A\), one defines \(C_n(X;\mathsf A)\) similarly, using coefficients in \(\mathsf A\). The resulting homology group is denoted \(H_n(X;\mathsf A)\).

Basic homology examples

Here \(f_*\) is induced by pushing singular simplices forward: \[f_\#(\sigma)=f\circ\sigma.\] The symbol \(f_\#\) denotes the chain-level map, while \(f_*\) denotes the induced map on homology classes.

Cochains and cohomology

Homology is covariant: a map \(f:X\to Y\) sends chains in \(X\) to chains in \(Y\). Cohomology reverses the direction: a map \(f:X\to Y\) pulls cochains on \(Y\) back to cochains on \(X\).

Since \(\partial_n\circ\partial_{n+1}=0\), one has \[\delta^{n+1}\circ\delta^n=0.\]

Pairing with homology

There is a natural pairing between cochains and chains: \[\langle \varphi,c\rangle=\varphi(c), \qquad \varphi\in C^n(X;\mathsf A),\quad c\in C_n(X;\mathbb{Z}).\] If \(\varphi\) is a cocycle and \(c\) is a cycle, this pairing depends only on the cohomology class \([\varphi]\in H^n(X;\mathsf A)\) and the homology class \([c]\in H_n(X;\mathbb{Z})\). Indeed, if \(c=\partial b\) is a boundary and \(\varphi\) is a cocycle, then \[\langle\varphi,c\rangle =\varphi(\partial b) =(\delta\varphi)(b)=0.\] If \(\varphi=\delta\psi\) is a coboundary and \(c\) is a cycle, then \[\langle\varphi,c\rangle =(\delta\psi)(c) =\psi(\partial c)=0.\]

Therefore there is a well-defined pairing \[H^n(X;\mathsf A)\times H_n(X;\mathbb{Z})\to \mathsf A.\]

Pullback in cohomology

Let \(f:X\to Y\). The chain-level pushforward is \[f_\#:C_n(X;\mathbb{Z})\to C_n(Y;\mathbb{Z}), \qquad f_\#(\sigma)=f\circ\sigma.\] The cochain-level pullback is \[f^\#:C^n(Y;\mathsf A)\to C^n(X;\mathsf A), \qquad (f^\#\varphi)(c)=\varphi(f_\#c).\] This commutes with the coboundary operators, so it induces \[f^*:H^n(Y;\mathsf A)\to H^n(X;\mathsf A).\] The notation distinguishes the two levels:

• \(f^\#\) is the cochain-level pullback.

• \(f^*\) is the induced map on cohomology classes.

In many texts both are denoted \(f^*\), with context determining the meaning.

The homology–cohomology pairing: why only genuine topology survives

The definitions of cycles, boundaries, cocycles, and coboundaries are designed so that cohomology classes can be evaluated on homology classes. Let \(\mathsf A\) be an Abelian coefficient group. For simplicity, take singular cochains with coefficients in \(\mathsf A\) and singular chains with integer coefficients. An \(n\)-cochain \[\varphi\in C^n(X;\mathsf A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),\mathsf A)\] is a homomorphism from \(n\)-chains to \(\mathsf A\). Therefore it can be evaluated on an \(n\)-chain \(c\): \[\langle \varphi,c\rangle:=\varphi(c).\]

The important point is that this evaluation descends to a well-defined pairing \[\boxed{ H^n(X;\mathsf A)\times H_n(X;\mathbb{Z})\to\mathsf A, \qquad ([\varphi],[c])\mapsto \varphi(c). }\] Here is the proof. First suppose the cycle representative is changed by a boundary: \[c\mapsto c+\partial b.\] If \(\varphi\) is a cocycle, then \(\delta\varphi=0\), so \[\varphi(c+\partial b)=\varphi(c)+\varphi(\partial b) =\varphi(c)+(\delta\varphi)(b)=\varphi(c).\] Thus a cocycle cannot detect a boundary error in the chain representative.

Second suppose the cochain representative is changed by a coboundary: \[\varphi\mapsto \varphi+\delta\psi.\] If \(c\) is a cycle, then \(\partial c=0\), so \[(\varphi+\delta\psi)(c)=\varphi(c)+(\delta\psi)(c) =\varphi(c)+\psi(\partial c)=\varphi(c).\] Thus a coboundary cannot detect a cycle.

This is the algebraic version of the physical statement that a globally exact field integrates to zero on a closed cycle, and that a closed field integrates to zero on a cycle that is actually the boundary of a higher-dimensional region. Nonzero quantized flux appears only when a nontrivial field class is paired with a nontrivial cycle.

The cup product and the cohomology ring

Cohomology has a multiplication operation. This extra structure is one reason cohomology is so useful.

Let \(\varphi\in C^p(X;R)\) and \(\psi\in C^q(X;R)\), where \(R\) is a commutative ring such as \(\mathbb{Z}\), \(\mathbb{R}\), or \(\mathbb{Z}_2\). There is a cochain \[\varphi\smile\psi\in C^{p+q}(X;R)\] called the cup product. One standard definition is the Alexander-Whitney formula. If \(\sigma:\Delta^{p+q}\to X\) is a singular \((p+q)\)-simplex with ordered vertices \(v_0,\ldots,v_{p+q}\), then \[(\varphi\smile\psi)(\sigma)= \varphi(\sigma|[v_0,\ldots,v_p])\,\psi(\sigma|[v_p,\ldots,v_{p+q}]),\] where \(\sigma|[v_0,\ldots,v_p]\) and \(\sigma|[v_p,\ldots,v_{p+q}]\) denote the restrictions of \(\sigma\) to the corresponding front and back faces.

This product descends to cohomology: \[H^p(X;R)\times H^q(X;R)\to H^{p+q}(X;R).\] The direct sum \[H^*(X;R)=\bigoplus_{n\geq 0}H^n(X;R)\] with this product is called the cohomology ring.

On cohomology classes, the cup product is graded-commutative: \[[\alpha]\smile[\beta]=(-1)^{pq}[\beta]\smile[\alpha]\] for \([\alpha]\in H^p(X;R)\) and \([\beta]\in H^q(X;R)\).

Orientations and fundamental classes

For integration and Chern numbers, one often pairs a top-degree cohomology class with a fundamental homology class.

The fundamental class is the rigorous homology object behind notation such as \[\int_M \omega.\] If \([\alpha]\in H^n(M;\mathbb{Z})\), then \[\langle [\alpha],[M]\rangle\in\mathbb{Z}\] is the integer obtained by evaluating the top-degree cohomology class on the whole manifold.

If \(\Sigma\) is a connected, compact, oriented surface, then \[[\Sigma]\in H_2(\Sigma;\mathbb{Z})\] is its fundamental class. A first Chern number is an evaluation of the form \[\langle c_1(L),[\Sigma]\rangle.\]

De Rham cohomology

Singular cohomology is defined for topological spaces. On smooth manifolds there is another cohomology theory built from differential forms.

Let \(M\) be a smooth manifold. Recall the notation \[\Omega^k(M)=\Gamma(\Lambda^kT^*M).\] This means that \(\Omega^k(M)\) is the real vector space of smooth differential \(k\)-forms on \(M\).

The exterior derivative is a linear map \[d_k:\Omega^k(M)\to\Omega^{k+1}(M)\] satisfying \[d_{k+1}\circ d_k=0.\] When the degree is clear, one writes \(d\) instead of \(d_k\).

Pullback of forms and de Rham cohomology

If \(f:M\to N\) is smooth, then pullback gives \[f^*:\Omega^k(N)\to\Omega^k(M).\] The exterior derivative commutes with pullback: \[f^*(d\omega)=d(f^*\omega).\] Therefore \(f^*\) sends closed forms to closed forms and exact forms to exact forms. It induces a well-defined map \[f^*:H^k_{\mathrm{dR}}(N)\to H^k_{\mathrm{dR}}(M).\] This is the smooth-manifold version of the contravariance of singular cohomology.

Integration pairing

Let \(c=\sum_{a=1}^N m_a\sigma_a\) be a smooth singular \(k\)-chain in \(M\), where each \[\sigma_a:\Delta^k\to M\] is smooth. For \(\omega\in\Omega^k(M)\), define \[\int_c\omega=\sum_{a=1}^N m_a\int_{\Delta^k}\sigma_a^*\omega.\] Stokes’ theorem says \[\int_c d\eta=\int_{\partial c}\eta.\] Consequently:

• if \(\omega\) is closed and \(c\) and \(c'\) differ by a boundary, then \[\int_c\omega=\int_{c'}\omega;\]

• if \(\omega\) is exact and \(c\) is a cycle, then \[\int_c\omega=0.\]

Thus integration gives a well-defined pairing \[H^k_{\mathrm{dR}}(M)\times H_k(M;\mathbb{Z})\to\mathbb{R}, \qquad ([\omega]_{\mathrm{dR}},[c])\mapsto \int_c\omega.\]

The theorem says that closed differential forms modulo exact differential forms compute the same real cohomology as singular cochains with real coefficients.

De Rham examples

Integral cohomology, periods, and quantization

De Rham cohomology uses real coefficients. Chern classes live naturally in integral cohomology.

There is a homomorphism of coefficient groups \[\mathbb{Z}\to\mathbb{R}, \qquad n\mapsto n.\] It induces a natural map \[H^k(X;\mathbb{Z})\to H^k(X;\mathbb{R}).\] A real cohomology class is called integral if it lies in the image of this map.

On a smooth manifold, the de Rham theorem identifies \(H^k(M;\mathbb{R})\) with \(H^k_{\mathrm{dR}}(M)\). Under this identification, a de Rham class \([\omega]_{\mathrm{dR}}\) is integral if and only if all its periods over integral \(k\)-cycles are integers: \[\int_c\omega\in\mathbb{Z}\] for every \([c]\in H_k(M;\mathbb{Z})\).

This criterion is the mathematical source of flux quantization. In physics conventions for a \(U(1)\) connection, the curvature two-form \(F\) is real and the normalized class \[\left[\frac{F}{2\pi}\right]_{\mathrm{dR}}\] is integral when it comes from a genuine complex line bundle.

Thus for every closed oriented surface \(\Sigma\subset M\), \[\frac{1}{2\pi}\int_\Sigma F\in\mathbb{Z}.\] This integer is the first Chern number of the line bundle restricted to \(\Sigma\).

Cech cohomology and transition functions

Singular cohomology is defined using simplices. Fibre bundles are often described using open covers and transition functions. Cech cohomology is the cohomology theory naturally adapted to open covers.

Let \[\mathcal U=\{U_i\}_{i\in I}\] be an open cover of \(X\). For indices \(i_0,\ldots,i_p\), write \[U_{i_0\cdots i_p}=U_{i_0}\cap\cdots\cap U_{i_p}.\]

The Cech coboundary is \[(\delta a)_{i_0\cdots i_{p+1}} = \sum_{m=0}^{p+1}(-1)^m a_{i_0\cdots \widehat{i_m}\cdots i_{p+1}},\] where the hat means that the index is omitted.

A Cech cocycle satisfies \(\delta a=0\). A Cech coboundary is a cochain of the form \(a=\delta b\). The quotient of cocycles by coboundaries gives Cech cohomology for the cover.

For good covers of nice spaces, Cech cohomology agrees with singular cohomology. A good cover is an open cover for which every nonempty finite intersection \(U_{i_0\cdots i_p}\) is contractible.

Multiplicative notation for \(U(1)\)

For \(U(1)\)-valued transition functions, multiplicative notation is more natural. A Cech \(1\)-cochain is a collection of functions \[g_{ij}:U_{ij}\to U(1).\] The cocycle condition is \[g_{ij}g_{jk}=g_{ik}\] on triple overlaps \(U_{ijk}\). Equivalently, \[g_{ij}g_{jk}g_{ki}=1.\] This is exactly the transition-function consistency condition for a complex line bundle.

Complex line bundles and the first Chern class

A complex line bundle is a rank-\(1\) complex vector bundle. Thus each fibre is a one-dimensional complex vector space.

Let \(L\to X\) be a complex line bundle and let \(\mathcal U=\{U_i\}\) be a trivializing open cover. Choose a nonzero local frame \(e^{(i)}\) over \(U_i\). On an overlap \(U_{ij}\), the frames differ by a function \[g_{ij}:U_{ij}\to \mathbb{C}^*=\mathbb{C}\setminus\{0\}\] such that \[e^{(j)}=e^{(i)}g_{ij}.\] If the bundle has a Hermitian metric and the frames are chosen to have unit length, then \[g_{ij}:U_{ij}\to U(1).\] On triple overlaps, \[e^{(k)}=e^{(j)}g_{jk}=e^{(i)}g_{ij}g_{jk},\] while also \[e^{(k)}=e^{(i)}g_{ik}.\] Therefore \[g_{ij}g_{jk}=g_{ik}.\]

If we change local frames by unitary functions \[e'^{(i)}=e^{(i)}h_i, \qquad h_i:U_i\to U(1),\] then the transition functions change to \[g'_{ij}=h_i^{-1}g_{ij}h_j.\] For \(U(1)\) this is commutative, but the displayed order is the order dictated by the frame convention.

The classification theorem is:

This theorem explains why \(H^2\) appears in the study of line bundles. A line bundle is not merely described by local vector spaces; its global twisting is measured by an integral degree-two cohomology class.

How transition functions produce an integer on \(S^2\)

Cover \(S^2\) by two open sets: \[U_N=S^2\setminus\{\text{south pole}\}, \qquad U_S=S^2\setminus\{\text{north pole}\}.\] Their overlap deformation retracts to the equator \(S^1\). A complex line bundle is described by one transition function \[g_{NS}:U_N\cap U_S\to U(1).\] Restricting to the equator gives a map \[g_{NS}|_{S^1}:S^1\to U(1).\] Such maps are classified up to homotopy by their winding number: \[[S^1,U(1)]\cong \pi_1(U(1))\cong \mathbb{Z}.\] If \[g_{NS}(e^{i\phi})=e^{in\phi},\] then the winding number is \(n\). The corresponding line bundle has \[\langle c_1(L),[S^2]\rangle=n.\] Since \[H^2(S^2;\mathbb{Z})\cong\mathbb{Z},\] this integer fully classifies complex line bundles over \(S^2\).

Connections, curvature, and the de Rham representative of \(c_1\)

Let \(L\to M\) be a Hermitian complex line bundle over a smooth manifold. Choose a unit local frame \(e^{(i)}\) over \(U_i\). A section can be written locally as \[s=e^{(i)}\psi_i,\] where \(\psi_i:U_i\to\mathbb{C}\).

Use the physics convention \[D_i=d-iA_i,\] where \(A_i\in\Omega^1(U_i)\) is a real one-form. The covariant derivative is \[D_i\psi_i=d\psi_i-iA_i\psi_i.\] On overlaps, suppose \[e^{(j)}=e^{(i)}g_{ij}, \qquad \psi_i=g_{ij}\psi_j, \qquad g_{ij}:U_{ij}\to U(1).\] The requirement that covariant derivatives transform in the same way as sections is \[D_i\psi_i=g_{ij}D_j\psi_j.\] Substituting \(\psi_i=g_{ij}\psi_j\) gives \[d(g_{ij}\psi_j)-iA_i g_{ij}\psi_j = g_{ij}(d\psi_j-iA_j\psi_j).\] After cancellation of the \(g_{ij}d\psi_j\) terms, this becomes \[dg_{ij}-iA_i g_{ij}=-i g_{ij}A_j.\] Multiplying by \(g_{ij}^{-1}\) gives \[g_{ij}^{-1}dg_{ij}-iA_i=-iA_j.\] Therefore \[A_j=A_i+i g_{ij}^{-1}dg_{ij}.\] If \(g_{ij}=e^{i\chi_{ij}}\), then \[i g_{ij}^{-1}dg_{ij}=-d\chi_{ij},\] so \[A_j=A_i-d\chi_{ij}.\]

The local curvature is \[F_i=dA_i.\] On overlaps, \[F_j=dA_j=dA_i+d(i g_{ij}^{-1}dg_{ij})=dA_i=F_i,\] because \(g_{ij}^{-1}dg_{ij}=i\,d\chi_{ij}\) locally and \(d^2\chi_{ij}=0\). Hence the \(F_i\) glue to a globally defined closed two-form \[F\in\Omega^2(M), \qquad dF=0.\]

The first Chern class satisfies \[\left[\frac{F}{2\pi}\right]_{\mathrm{dR}} = \text{image of }c_1(L)\text{ under }H^2(M;\mathbb{Z})\to H^2(M;\mathbb{R})\cong H^2_{\mathrm{dR}}(M).\] Equivalently, for every closed oriented surface \(\Sigma\subset M\), \[\langle c_1(L),[\Sigma]\rangle = \frac{1}{2\pi}\int_\Sigma F \in\mathbb{Z}.\]

Dirac monopole as the model computation

Let \(L\to S^2\) be the line bundle whose transition function on the equator is \[g_{NS}=e^{in\phi}.\] Choose local connection one-forms \[A_N=\frac{n}{2}(1-\cos\theta)d\phi, \qquad A_S=-\frac{n}{2}(1+\cos\theta)d\phi.\] On the overlap, \[A_S=A_N-n\,d\phi.\] This agrees with the transition rule because \(g_{NS}=e^{in\phi}\) gives \(\chi_{NS}=n\phi\) and hence \[A_S=A_N-d\chi_{NS}.\] The curvature is \[F=dA_N=dA_S=\frac{n}{2}\sin\theta\,d\theta\wedge d\phi.\] Therefore \[\frac{1}{2\pi}\int_{S^2}F = \frac{1}{2\pi}\int_0^{2\pi}\int_0^\pi \frac{n}{2}\sin\theta\,d\theta\,d\phi =n.\] The same integer appears in two ways: \[\begin{align*} \text{transition function winding:}&& g_{NS}:S^1\to U(1),&& \operatorname{wind}(g_{NS})&=n,\\ \text{curvature integral:}&& \frac{1}{2\pi}\int_{S^2}F&& &=n. \end{align*}\] These are not two unrelated facts. They are two representatives of the same first Chern class.

Why cohomology appears in Chern classes

The first Chern class is a cohomology class because it must be independent of the local choices used to describe the bundle.

A connection is described locally by one-forms \(A_i\). These one-forms are not globally defined in general. On overlaps they differ by gauge transformations. The curvature \(F\) is globally defined, but it depends on the chosen connection. However, changing the connection changes \(F\) by an exact form at the level of the normalized curvature class. Thus the de Rham cohomology class \[\left[\frac{F}{2\pi}\right]_{\mathrm{dR}}\] depends only on the bundle, not on the particular connection.

Cohomology is exactly the language for this kind of object:

A differential form representative may change, but its class modulo exact forms remains fixed.

The integral cohomology class \(c_1(L)\in H^2(M;\mathbb{Z})\) is stronger than the real de Rham class because it remembers quantization. The de Rham class records the real fluxes. The integral class says those normalized fluxes are integers on closed surfaces.

Homotopy, homology, and cohomology: comparison

The following table separates the main ideas.

For fibre bundles and Berry phases, the most important chain of ideas is \[\text{transition functions} \longrightarrow \text{Cech cocycles} \longrightarrow H^2(M;\mathbb{Z}) \longrightarrow c_1(L) \longrightarrow \left[\frac{F}{2\pi}\right]_{\mathrm{dR}} \longrightarrow \frac{1}{2\pi}\int_\Sigma F\in\mathbb{Z}.\]

Minimal checklist for the main notes

After reading this appendix, the following statements should have precise meanings:

1. A loop is a map \(S^1\to X\), equivalently a path \(I\to X\) whose endpoints agree.

2. \(\pi_1(X,x_0)\) is the group of loops based at \(x_0\) modulo homotopy relative to endpoints.

3. \(\pi_1(S^1)\cong\mathbb{Z}\) because loops around a circle have an integer winding number.

4. \(\pi_1(SO(3))\cong\mathbb{Z}_2\) because \(SU(2)\to SO(3)\) is a two-to-one universal covering map.

5. \(H_n(X;\mathbb{Z})\) is the group of \(n\)-cycles modulo \(n\)-boundaries.

6. \(H^n(X;\mathsf A)\) is the group of \(n\)-cocycles modulo \(n\)-coboundaries.

7. \(H^k_{\mathrm{dR}}(M)\) is the group of closed \(k\)-forms modulo exact \(k\)-forms.

8. De Rham cohomology agrees with singular cohomology with real coefficients.

9. Integral cohomology is needed to state flux quantization.

10. The first Chern class \(c_1(L)\in H^2(M;\mathbb{Z})\) is represented in de Rham cohomology by \(F/(2\pi)\).

Exercises for the appendix